Limits And Derivatives Questions

We provide limits and derivatives practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on limits and derivatives skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.

List of limits and derivatives Questions

Question No Questions Class
1 ( lim _{x rightarrow 0}left[left(1-e^{x}right) frac{sin x}{|x|}right] ) is (where []
represents the greatest integer function)
A . -1
B. 1
( c cdot 0 )
D. Does not exist
11
2 ( lim _{n rightarrow infty} n^{2}{sqrt{left.left(1-cos frac{1}{n}right) sqrt{left(1-cos frac{1}{n}right)}right)} )
( A )
( B )
( c )
( D )
11
3 If ( lim _{x rightarrow 0}(cos x+a sin b x)^{frac{1}{x}}=e^{2} ) then
the possible values of ( ^{prime} a^{prime} &^{prime} b^{prime} a r e: )
This question has multiple correct options
A ( . a=1, b=2 )
В. ( a=2, b=1 )
c. ( a=3, b=2 / 3 )
D. ( a=2 / 3, b=3 )
11
4 ( lim _{x rightarrow 2} frac{x^{7}-128}{x^{5}-32}= )
A ( cdot frac{5}{28} )
в. ( frac{7}{5} )
c. ( frac{28}{5} )
D. 5 ( overline{7} )
11
5 Evaluate ( : l t_{x rightarrow 0} x operatorname{cosec} x ) 11
6 Evaluate :
[
lim _{y rightarrow 0} frac{(x+y)}{y} sec (x+y)-x sec x
]
11
7 ( lim _{x rightarrow infty}left(frac{3 x^{2}+2 x+1}{x^{2}+x+2}right)^{frac{6 x+1}{3 x+2}} ) is equal to
( A cdot 3 )
B. 9
( c )
D. none of these
11
8 Evaluate:
( lim _{x rightarrow 0} frac{sqrt{1+2 x^{2}}-sqrt{1-2 x^{2}}}{x^{2}} )
11
9 ( lim _{x rightarrow 0} frac{{sin (alpha+beta) x+sin (alpha-beta) x+sin 2}{cos ^{2} beta x-cos ^{2} alpha x} ) 11
10 Find the left and right hand limits of ( f(x)=left{begin{array}{ll}frac{3 x^{2}+2}{3 x-2} & x1end{array} text { at } x=1right. )
( A )
(a) 5 and ( frac{1}{7} )
B. ( (b)-frac{1}{2} ) and -1
C ( cdot(c)-frac{7}{2} ) and 2
D cdot (d) ( -frac{9}{2} ) and 3
11
11 Find the value of
( lim _{x rightarrow 3} frac{sqrt{16 x^{2}+112}-16}{3 x^{2}-15 x+18} )
11
12 solve the limit
( lim _{x rightarrow 3} frac{2}{x-3} )
( A cdot 2 )
B. 3
( c cdot 4 )
D. Does not exist
11
13 The value of ( lim _{x rightarrow 0} frac{sin left(pi cos ^{2} xright)}{x^{2}} ) is
A. ( -pi )
в. ( frac{pi}{2} )
c. ( pi )
D. ( frac{3 pi}{2} )
11
14 Solve
( lim _{x rightarrow 0}left(frac{1^{x}+2^{x}+3^{x}+ldots+n^{x}}{n}right)^{1 / 2}= )
11
15 Solve ( lim _{x rightarrow 0} frac{sqrt{2+x}-sqrt{2}}{x} ) 11
16 The value of ( lim _{n rightarrow infty} frac{sqrt[4]{n^{5}+2}-sqrt[3]{n^{2}+1}}{sqrt[5]{n^{4}+2}-sqrt[2]{n^{3}+1}} )
is
( mathbf{A} cdot mathbf{1} )
B.
c. -1
D. ( infty )
11
17 If the function ( f(x) ) satisfies ( lim _{x rightarrow 1} frac{f(x)-2}{x^{2}-1}=pi, ) then ( lim _{x rightarrow 1} f(x)= )
A . 2
B. 3
c. 1
D.
11
18 ( lim _{x rightarrow 0}left{(1+x)^{frac{2}{x}}right} ) (where {} denotes the
fractional part of ( x ) ) is equal to:
A ( cdot e^{2}-7 )
B ( cdot e^{2} )
c. ( e^{7}-7 )
D. limit does not exist
11
19 Evaluate the Given limit: ( lim _{x rightarrow 3}(x+3) ) 11
20 The value of ( lim _{x rightarrow infty}left(x sin left(frac{3}{x}right)right) ) 11
21 ( lim _{x rightarrow infty} int_{0}^{x / 2} frac{t^{2}}{x^{2}left(1+t^{2}right)} d t ) is equal to
A ( cdot frac{1}{4} )
B.
c. 1
D. None of these
11
22 Evaluate: ( lim _{boldsymbol{pi}} frac{boldsymbol{c o t}^{3} boldsymbol{x}-boldsymbol{t a n} boldsymbol{x}}{cos (boldsymbol{x}+boldsymbol{pi} / mathbf{4})} )
( boldsymbol{x} rightarrow boldsymbol{boldsymbol { 4 }} )
( A )
B. ( 8 sqrt{2} )
( c cdot 4 )
D. ( 4 sqrt{2} )
11
23 Solve ( : lim _{x rightarrow 1} frac{x^{431}+3 . x^{221}-2 . x^{39}-2}{x-1}= ) 11
24 ( lim _{x rightarrow 1}left(frac{1}{x-1}-frac{2}{x^{2}-1}right)= )
A . 1 3
B. ( frac{-1}{2} )
( c cdot frac{1}{2} )
D. ( frac{-1}{3} )
11
25 The value of the expression ( lim _{x rightarrow 0} frac{x sin (sin x)}{1-cos x} ) is
( A )
B. 2
( c cdot 4 )
D.
11
26 Evaluate:
( lim _{x rightarrow infty} frac{2+cos ^{2} x}{x+2007} )
11
27 Evaluate:
( lim _{x rightarrow frac{pi}{3}} frac{sin left(frac{pi}{3}-xright)}{2 cos x-1} )
11
28 If ( alpha ) and ( beta ) are the roots of the equation
( 375 x^{2}-25 x-2=0, ) then the value of
( lim _{n rightarrow infty}left(sum_{r=1}^{n} alpha^{r}+sum_{r=1}^{n} beta^{r}right) ) the value of is?
A ( cdot frac{29}{248} )
в. ( frac{17}{348} )
c. ( frac{29}{358} )
D. ( frac{11}{34} )
11
29 Evaluate: ( lim _{x rightarrow 2} frac{sin left(x^{2}-5 x+6right)}{x^{2}-7 x+10} ) 11
30 ( lim _{x rightarrow 1} frac{sqrt{1-cos 2(x-1)}}{x-1} )
A. exists and it equals ( sqrt{2} )
B. exists and it equals ( -sqrt{2} )
c. does not exist because ( x-1 rightarrow 0 )
D. does not exist because left hand limit is not equal to right hand linitt
11
31 ( lim _{x rightarrow infty} sin x ) is equal to:
( mathbf{A} cdot 0 )
( B . infty )
C. Exists is finite and non-zone
D. Does not exist
11
32 ( lim _{x rightarrow frac{pi}{2}} frac{tan 2 x}{x-frac{pi}{2}} ) 11
33 Let ( boldsymbol{a}=boldsymbol{m} boldsymbol{i n}left{boldsymbol{x}^{2}+boldsymbol{2} boldsymbol{x}+boldsymbol{3}, boldsymbol{x} epsilon boldsymbol{R}right} ) and
( boldsymbol{b}=lim _{boldsymbol{theta} rightarrow mathbf{0}} frac{mathbf{1}-cos boldsymbol{theta}}{boldsymbol{theta}^{2}} )
The value of ( sum_{r=0}^{n} a^{r} . b^{n-r} ) is?
( ^{text {A } cdot frac{2^{n+1}-1}{3.2^{n}}} )
B. ( frac{2^{n+1}+1}{3.2^{n}} )
c. ( frac{4^{n+1}-1}{3.2^{n}} )
D. none of these
11
34 Evaluate:
( lim _{x rightarrow 2} frac{sqrt{3-x}-1}{2-x} )
11
35 ( lim _{n rightarrow infty} sum_{n=1}^{20} cos ^{2 n}(x-10) ) is equal to
( A )
B.
( c cdot 19 )
D. 20
11
36 Solve:
( lim _{boldsymbol{y} rightarrow mathbf{7}} frac{boldsymbol{y}^{2}-boldsymbol{4} boldsymbol{y}-boldsymbol{2} mathbf{1}}{3 boldsymbol{y}^{2}-17 boldsymbol{y}-mathbf{2 8}} )
11
37 State when a function ( f(x) ) is said to be
increasing on an interval ( [a, b] . ) Test whether the function ( f(x)=x^{2}-6 x+ )
3 is increasing on the interval [4,6]
11
38 ( lim _{n rightarrow infty} frac{sqrt{x^{2}+1}-sqrt[3]{x^{3}+1}}{sqrt[4]{x^{4}+1}-sqrt[5]{x^{4}+1}} ) equals
( mathbf{A} cdot mathbf{1} )
B.
( c .-1 )
D. none of these
11
39 Solve:
( lim _{u rightarrow 1} frac{sqrt{u^{2}+8}}{sqrt{u^{2}+3}}-frac{sqrt{10-u^{2}}}{sqrt{5-u^{2}}} )
11
40 ( operatorname{Let} f(x)=frac{sin {x}}{x^{2}+a x+b} . ) If ( fleft(5^{+}right) & )
( fleft(3^{+}right) ) exists finitely and are not zero, then the value of
( (a+b) ) is (where ( {cdot} ) represents
fractional part function).
( A cdot 7 )
B . 10
c. 11
D. 20
11
41 Evaluate the following limit ( lim _{x rightarrow 0} frac{(1+x)^{6}-1}{x} ) 11
42 Evaluate :
( lim _{n rightarrow infty} frac{n !}{(n+1) !-n !} )
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. 3
11
43 ( lim _{x rightarrow 0} frac{x tan 2 x-2 x tan x}{(1-cos 2 x)^{2}}= )
( A cdot 2 )
B.
( c .-2 )
D. ( -frac{1}{2} )
11
44 Evaluate :
( lim _{n rightarrow infty} frac{[1 . x]+[2 . x]+[3 . x]+ldots ldots+[n . x]}{n^{2}}, ) where
denotes the greatest integer function.
11
45 Which one of the following statements
is true?
( mathbf{A} cdot operatorname{If} lim _{x rightarrow c} f(x) cdot g(x) ) and ( lim _{x rightarrow c} f(x) ) exist, then ( lim _{x rightarrow c} g(x) ) exists.
( mathbf{B} cdot operatorname{If} lim _{x rightarrow c} f(x) cdot g(x) ) exists, then ( lim _{x rightarrow c} f(x) ) and ( lim _{x rightarrow c} g(x) ) exist.
C. ( operatorname{If} lim _{x rightarrow c} f(x)+g(x) ) and ( lim _{x rightarrow c} f(x) ) exist, then ( lim _{x rightarrow c} g(x) ) also
exists.
( mathbf{D} cdot operatorname{If} lim _{x rightarrow c} f(x)+g(x) ) exists, then ( lim _{x rightarrow c} f(x) ) and ( lim _{x rightarrow c} g(x) ) also
exist.
11
46 Evaluate:-
( lim _{t rightarrow-3} frac{6+4 t}{t^{2}+1} )
11
47 Evaluate :
( lim _{x rightarrow infty} frac{3 x^{2}+4 x+5}{4 x^{2}+7} )
11
48 Show that ( lim _{x rightarrow 0} frac{e^{1 / x}-1}{e^{1 / x}+1} ) does not exist. 11
49 Prove that
( boldsymbol{L}=lim _{boldsymbol{n} rightarrow infty}left(1+frac{4}{n}right)^{3 n}=mathbf{1 2} )
11
50 25.
02
lim
(2h+2+h) –
given that f(2)=6 and f (1)=4
ho fch-h²+1) – f (1)
(a) does not exist
(c) is equal to 3/2
(b) is equal to – 3/2
(d) is equal to 3 (2003)
i dhe
11
11
51 ( lim _{x rightarrow 0} frac{sqrt{1+x}-1}{x} ) equals to
A .
B.
( c cdot 0 )
D. none of these
11
52 26. If f(x – y) = f(x) g(y)-f(y).g(x) and
g(x – y) = g(x) g(y)-f(x).fly) for all x, y eR.
Ifright hand derivative at x=0 exists for f(x). Find derivative
of g(x) at x=0
(2005 – 4 Marks)
11
53 Find ( b lim _{x rightarrow 2 a} frac{sqrt{(x-2 a)}+sqrt{x}-sqrt{2 a}}{sqrt{left(x^{2}-4 a^{2}right)}} ) is
( frac{1}{b sqrt{a}} )
11
54 ( lim _{x rightarrow 1}{1-x+[x+1]+[1-x]}, ) where
( x ) ] denotes greatest integer function, is
( A cdot 0 )
B.
( c .-1 )
( D .2 )
11
55 ( lim _{boldsymbol{x} rightarrow mathbf{0}} frac{(mathbf{1}+boldsymbol{x})^{mathbf{1} / mathbf{6}}-(mathbf{1}-boldsymbol{x})^{mathbf{1} / mathbf{6}}}{boldsymbol{x}} ) 11
56 3. Let f(x) = 4 and f'(x)=4. Then lim
xf (2)-2f (x)
1S
X-2
x2
given by
(a) 2
[2002]
(6) 2
(0) -4
C) -4
() 3
11
57 ( lim _{x rightarrow infty} frac{sqrt{x^{2}+1}-sqrt[3]{x^{3}+1}}{sqrt[4]{x^{4}+1}-sqrt[5]{x^{4}+1}} ) is
equals to
A .
B.
c. -1
D. None of these
11
58 Evaluate the following limits. ( lim _{x rightarrow 0}(cos x+sin x)^{1 / x} ) 11
59 Evaluate the following question. ( lim _{x rightarrow a} frac{(x)^{3 / 2}-(a)^{3 / 2}}{x-a} ) 11
60 23. Let
:
R
R
be such that f(1) = 3 and f ‘(1) = 6. Then
lim
x ol
(2002S)
f(1+x))
f(1)
equals
(b) e12
(2) 1
(c) e
(d) e3
11
61 ( lim _{x rightarrow 3} frac{sqrt{3 x}-3}{sqrt{2 x-4}-sqrt{2}} ) is equal to.
A ( cdot frac{1}{sqrt{2}} )
B. ( sqrt{3} )
( c cdot frac{1}{2 sqrt{2}} )
D. ( frac{sqrt{3}}{2} )
11
62 ( lim _{x rightarrow frac{pi}{2}}left(lim _{x rightarrow infty} cos frac{x}{2} cos frac{x}{2^{2}} cos frac{x}{2^{3}} ldots ldots cos frac{x}{2^{n}}right) )
equals to
11
63 ( lim _{x rightarrow a} frac{x^{7}-a^{7}}{x-a} ) 11
64 Find ( K ) where
( lim _{x rightarrow k}left[frac{x^{4}-1}{x-1}right]=lim _{x rightarrow k}left(frac{x^{3}-k^{3}}{x^{2}-k^{2}}right) )
11
65 V2 (1-сos 2x)
(1991 – 2 Marks)
The value of lim
x>0
(a) 1
(c) 0
(b)-1
(d) none of these
11
66 Evaluate: ( lim _{n rightarrow infty} frac{1+3+5+ldots . n t e r m s}{2+4+6+ldots n t e r m s} )
( A cdot 2 )
в.
( c cdot 3 )
D.
11
67 ( lim _{x rightarrow 2}left[frac{1}{x-2}-frac{2(2 x-3)}{x^{3}-3 x^{2}+2 x}right] ) 11
68 If function ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x}^{3}-boldsymbol{a}^{3}}{boldsymbol{x}-boldsymbol{a}}, ) is
continuous at ( x=a ) then the value of
( f(a) ) is
( mathbf{A} cdot 2 a )
В ( .2 a^{2} )
( c cdot 3 a )
D. ( 3 a^{2} )
11
69 ( lim _{x rightarrow 0} frac{cos a x-cos b x}{x^{2}}= ) 11
70 ( lim _{x rightarrow frac{pi}{2}} frac{left[1-tan left(frac{x}{2}right)right][1-sin x]}{1+tan left(frac{x}{2}right)[pi-2 x]^{3}} ) is
A ( cdot infty )
B.
c. 0
D. ( frac{1}{32} )
11
71 If ( lim _{x rightarrow a}left{frac{f(x)}{g(x)}right} ) exists, then
( mathbf{A} cdot operatorname{both} lim _{x rightarrow a} f(x) ) and ( lim _{x rightarrow a} g(x) ) must exist
B. ( lim _{x rightarrow a} f(x) ) need not exist but ( lim _{x rightarrow a} g(x) ) exists
C. neither ( lim _{x rightarrow a} f(x) ) nor ( lim _{x rightarrow a} g(x) ) may exist
D. ( lim _{x rightarrow a} f(x) ) exists but ( lim _{x rightarrow a} g(x) ) need not exist
11
72 Find the following limit:
( lim _{x rightarrow 4} frac{sqrt{1+2 x}-3}{sqrt{x}-2} )
11
73 The value of ( lim _{x rightarrow 0}left[frac{tan x}{x}+frac{sin x}{x}right] ) is 11
74 Evaluate:
( lim _{x rightarrow infty}left(frac{2 x-3}{sqrt{x^{2}-1}}right) )
11
75 Find the value of limit
( lim _{x rightarrow frac{pi}{6}} frac{2 sin ^{2} x+sin x-1}{2 sin ^{2} x-3 sin x+1}= )
( mathbf{A} cdot mathbf{0} )
B. 3
( c .-3 )
D.
11
76 If ( f(x)=left{begin{array}{cl}x-1, & x geq 1 \ 2 x^{2}-2, & xmathbf{0} \ -boldsymbol{x}^{2}+mathbf{1}, & boldsymbol{x} leq mathbf{0}end{array}, text { and } boldsymbol{h}(boldsymbol{x})=|boldsymbol{x}| ), then right.
( lim _{x rightarrow 0} f(g(h(x))) ) is
A.
B.
( c cdot 2 )
( D )
11
77 ( lim _{x rightarrow 0} frac{sqrt{1-sqrt{cos x}}}{x}= )
A ( cdot frac{1}{2} )
B. ( -frac{1}{2} )
c. Does not exist
D. None of these
11
78 Evaluate: ( lim _{x rightarrow frac{pi}{2}}left(2 x tan x-frac{pi}{cos x}right) ) 11
79 fig a polynomial satisfying ( boldsymbol{g}(boldsymbol{x}) boldsymbol{g}(boldsymbol{y})=boldsymbol{g}(boldsymbol{x})+boldsymbol{g}(boldsymbol{y})+boldsymbol{g}(boldsymbol{x} boldsymbol{y})-boldsymbol{2} )
for all real ( x ) and ( y ) and ( g(2)=5 ) then
( lim _{x rightarrow 3} g(x) ) is
( A cdot 9 )
B. 25
c. 10
D. none of these
11
80 ( lim _{x rightarrow infty} sin x ) equals
( mathbf{A} cdot 1 )
B. 0
( c cdot infty )
D. does not exist
11
81 Evaluate the following limits. ( lim _{x rightarrow 0} frac{xleft(e^{x}-1right)}{1-cos x} ) 11
82 ( lim _{x rightarrow infty}(sqrt{x+sqrt{x}}-sqrt{x}) ) is equal to
( mathbf{A} cdot mathbf{1} )
B.
( c cdot frac{1}{2} )
D. none of these
11
83 Find the limits of the following expression ( frac{(x-3)(2-5 x)(3 x+1)}{(2 x-1)^{3}} )
(1) when ( x=infty,(2) ) when ( x=0 )
11
84 ( lim _{x rightarrow 0} frac{sin left(pi cos ^{2} xright)}{x^{2}}= ) 11
85 Evaluate ( lim _{x rightarrow 0} frac{a^{sin x}-1}{sin x} ) 11
86 The value of ( lim _{x rightarrow 0} frac{(1-cos 2 x) sin 5 x}{x^{2} sin 3 x} )
equal to
A ( cdot frac{10}{3} )
в. ( frac{3}{10} )
( c cdot frac{6}{5} )
D.
11
87 Solve the following:
( lim _{x rightarrow 0} frac{2 sin ^{2} 3 x}{3 x^{2}} )
( mathbf{A} cdot mathbf{6} )
B. 9
c. 18
D. 3
11
88 Evaluate the following limits. ( lim _{x rightarrow 0} frac{sqrt{1+3 x}-sqrt{1-3 x}}{x} ) 11
89 ( lim _{x rightarrow 0} frac{sin left(pi cos ^{2} piright)}{x^{2}} ) is equal to
( ^{A} cdot frac{pi}{2} )
B.
( c .-pi )
( D )
11
90 ( f(x)=left|begin{array}{ccc}sin x & cos x & tan x \ x^{3} & x^{2} & x \ 2 x & 1 & 1end{array}right|, ) then
( lim _{x rightarrow 0} frac{f(x)}{x^{2}} ) is
( A )
B. 3
( c cdot 1 )
D. zero
11
91 Evaluate ( : lim _{x rightarrow a} frac{x^{m}-a^{m}}{x^{n}-a^{n}}= ) 11
92 Solve:
( lim _{h rightarrow 0} frac{1}{h}left[frac{1}{cos (x+h)}-frac{1}{cos x}right] )
11
93 ( boldsymbol{R}=lim _{boldsymbol{x} rightarrow mathbf{0}^{+}} boldsymbol{f}(boldsymbol{x}) ) is equal to
A ( cdot frac{pi}{2} )
в. ( frac{pi}{2 sqrt{2}} )
c. ( frac{pi}{sqrt{2}} )
D. ( sqrt{2} pi )
11
94 What is ( lim _{h rightarrow 0} frac{sqrt{2 x+3 h}-sqrt{2 x}}{2 h} ) equal to?
( ^{mathbf{A}} cdot frac{1}{2 sqrt{2 x}} )
в. ( frac{3}{sqrt{2 x}} )
c. ( frac{3}{2 sqrt{2 x}} )
D. ( frac{3}{4 sqrt{2 x}} )
11
95 Evaluate:
( lim _{x rightarrow a} frac{sqrt{x}-sqrt{a}}{x-a} )
11
96 ( lim _{x rightarrow a^{+}} frac{{x} sin (x-a)}{(x-a)^{2}}= ) where ( {x} )
denotes fractional part of ( x ) and ( a epsilon N )
A . 0
B. 1
( c cdot a )
D. 5
11
97 ( lim _{x rightarrow infty}left[frac{2+2 x+sin 2 x}{(2 x+sin 2 x) e^{sin x}}right] ) is equal to
A.
в.
( c cdot-1 )
D. Does not exist
11
98 ( lim _{x rightarrow 0} frac{x cos x-log (1+x)}{x^{2}} ) is equal to
A ( cdot frac{1}{2} )
B.
c. 1
D. None of these
11
99 Solve:
( lim _{x rightarrow} frac{sin (2+x)-sin (2-x)}{x} )
11
100 ( operatorname{Let} f(x)=frac{x^{2}-9 x+20}{x-[x]} ) where ( [x] ) is
the greatest integer not greater than ( x ) then
This question has multiple correct options
A ( cdot lim _{x rightarrow 5^{-}} f(x)=0 )
B . ( lim _{x rightarrow 5^{+}} f(x)=1 )
c. ( lim _{x rightarrow 5} f(x) ) does not exists
D. none of these
11
101 Evaluate ( lim _{x rightarrow 0} log frac{sin x}{x} ) 11
102 The integer ( n ) for which ( lim _{x rightarrow 0} frac{(cos x-1)left(cos x-e^{x}right)}{x^{n}} ) is finite non
zero number is
A . 1
B. 2
( c .3 )
( D )
11
103 ( lim _{x rightarrow 3} 2 x^{2}-3 x-5= )
( mathbf{A} cdot mathbf{4} )
B. 3
( c .-4 )
D. -3
11
104 Evaluate the following limits. ( lim _{x rightarrow 0} frac{sqrt{a^{2}+x^{2}}-a}{x^{2}} )
A ( cdot frac{1}{sqrt{a}} )
в. ( frac{1}{sqrt{2 a}} )
( c cdot frac{1}{a} )
D. ( frac{1}{2 a} )
11
105 ( lim _{x rightarrow 2}left[frac{1}{x-2}-frac{2}{x(x-1)(x-2)}right]= )
( A )
2
B. ( frac{2}{3} )
( c cdot alpha )
D.
11
106 ( lim _{x rightarrow frac{pi}{2}} frac{left(1-tan frac{x}{2}right)(1-sin x)}{left[1+tan frac{x}{2}right][pi-2 x]^{3}} ) 11
107 (cos x – 1)(cosx-e) is a finite
22.
The integer n for which lim
-0
(2002)
non-zero number is
(a) 1 (6) 2
(c) 3
.
(d) 4
11
108 Consider the function ( f ) defined by ( f(x) )
( =x-x(x), ) where ( x ) is a positive
veriable,and (x) denotes the integral part of ( x ) and show that it is
discontinuous for intergral values of x,and continuous for all others. Is the function periodic? If periodic,what is its
period? Draw its graph.
11
109 Evaluate the following limits. ( lim _{x rightarrow 0} frac{(1+x)^{6}-1}{x} ) 11
110 Evaluate the following limits. ( lim _{x rightarrow 1} frac{sqrt{5 x-4}-sqrt{x}}{x^{3}-1} )
A ( cdot frac{2}{5} )
в. ( frac{1}{3} )
( c cdot frac{2}{3} )
D. None of these
11
111 Evaluate:
( lim _{x rightarrow 0} tan ^{-1} frac{a}{x^{2}}, ) where ( a in mathbb{R} )
11
112 1P +2P
3P +….. + n P
4.
lim
n->00
nP+1
[2002]
1
1
1
p+1
1-P
Pp-1
P +2
11
113 JEE M 2019-9 Jan (MI
(a) exists and equals a da
(b)
exists and equals 2
(c) exists and equals 2/2
(d) does not exist
11
114 Evaluate the following limits. ( lim _{x rightarrow 1} frac{sqrt{x+8}}{sqrt{x}} ) 11
115 Evaluate the following limits. ( lim _{x rightarrow 0} frac{(a+x)^{2}-a^{2}}{x} ) 11
116 Evaluate ( : lim _{x rightarrow a} frac{x-a}{|x-a|} )
( A )
B.
( c .-c )
D. Does not exist
11
117 Evaluate: ( lim _{x rightarrow 2} frac{x-2}{sqrt{x+2}-2} ) 11
118 Evaluate ( lim _{x rightarrow I}left(tan frac{pi x}{4}right)^{tan frac{pi x}{2}} ) 11
119 Arrange the following limits in the ascending order:
(1) ( lim _{x rightarrow infty}left(frac{1+x}{2+x}right)^{x+2} )
(2) ( lim _{x rightarrow 0}(1+2 x)^{3 / x} )
(3) ( lim _{boldsymbol{theta} rightarrow mathbf{0}} frac{sin boldsymbol{theta}}{mathbf{2} boldsymbol{theta}} )
(4) ( lim _{x rightarrow 0} frac{log _{e}(1+x)}{x} )
A. 1,2,3,4
B. 1,3,4,2
c. 1,4,3,2
D. 3,4,1,2
11
120 2
lim
oo 11-n?
n
2 + …. +
1-₂2
1,2
1-n²s
is equal to
(1984 – 2 Marks)
(d) none of these
11
121 Find the value of ( k ) so that the function ( f )
is continuous at the indicated point. ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{cc}boldsymbol{k} boldsymbol{x}^{2} & , boldsymbol{x} leq mathbf{2} \ boldsymbol{3} & , boldsymbol{x}>mathbf{2}end{array}right} ) at ( boldsymbol{x}=mathbf{2} )
11
122 Solve:
( int_{0}^{1} frac{boldsymbol{d} boldsymbol{x}}{sqrt{boldsymbol{x}+mathbf{1}}+sqrt{boldsymbol{x}}} boldsymbol{d} boldsymbol{x}= )
A ( cdot frac{4}{3}(sqrt{2}+1) )
B. ( frac{4}{3}(sqrt{2}-1) )
c. ( frac{3}{4}(sqrt{2}-1) )
D. ( frac{3}{4}(sqrt{2}-2) )
11
123 ( lim _{x rightarrow 0} frac{sin m x}{sin n x}[m / n] ) 11
124 ( lim _{n rightarrow infty} frac{2^{3}-1^{3}}{2^{3}+1^{3}} cdot frac{3^{3}-1^{3}}{3^{3}+1^{3}} cdots frac{n^{3}-1}{n^{3}+1} )
equals
A ( cdot frac{1}{3} )
B. ( frac{2}{3} )
c.
D.
11
125 ( lim _{x rightarrow o}left(frac{1}{x^{2}}-cot xright) ) 11
126 ( lim _{x rightarrow 0} frac{(1-cos 2 x)(3+cos x)}{x tan 4 x} ) is equal to:
A. ( -frac{1}{4} )
B.
c.
D.
11
127 The value of ( lim _{n rightarrow infty} frac{3^{n}+2^{n}}{3^{n}-2^{n}} ) is
A . -1
B.
( c cdot 0 )
D.
11
128 | 15.
(1999 – 2 Marks)
Tin x tan 2x– 2 tan x
*10 (1 – cos 2x)
(a) 2 (6) 2
(c) 1/2
(d) -1/2
11
129 ( lim _{x rightarrow 0} frac{x}{sqrt{x+4}-2} ) is equal to
( mathbf{A} cdot mathbf{4} )
B. ( sqrt{2} )
c. ( 2 sqrt{2} )
D.
11
130 ( A B C ) is an isosceles triangle inscribed in a circle of radius ( r . ) If ( A B=A C ) and ( h )
is the altitude from ( A ) to ( B C, ) then evaluate ( lim _{h rightarrow 0} frac{Delta}{P^{3}}, ) where ( Delta ) is area of the triangle and ( P ) its perimeter.
A ( cdot frac{1}{128 r} )
в. ( frac{1}{64 r} )
c. ( frac{1}{32 r} )
D. ( frac{1}{256 r} )
11
131 ( lim _{x rightarrow 1} frac{1}{sqrt{|x|-{-x}}}(text { where }{x} ) denotes
the fractional part of ( x ) ) is equal to
A. does not exists
B. 1
( c cdot alpha )
D. ( frac{1}{2} )
11
132 If ( f(x) ) is the integral of ( frac{2 sin x-sin 2 x}{x^{3}}, x neq 0 . ) Find
( lim _{x rightarrow 0} f^{prime}(x), ) where ( f^{prime}(x)=frac{d f(x)}{d x} )
A ( cdot frac{1}{2} )
B.
( c cdot frac{1}{3} )
( D )
11
133 Solve ( lim _{x rightarrow 0} frac{sin x}{sqrt{x^{2}}} )
( mathbf{A} cdot 1 )
B . -1
( mathbf{c} cdot 0 )
D. doesn’t exist
11
134 The left-hand derivative of f(x) = [x] sin(nt x) at x =k, kan
integer, is
(20015)
(a) (-1) (k-1)
(b) (-1)k-1 (k-1)
(c) (-1){kt
(d) (-1)&- lkn
TO
11
135 ( lim _{x rightarrow infty} x^{2}left(1-cos frac{2}{x}right) ) 11
136 ( lim _{x rightarrow infty} frac{2 sqrt{x}+3 sqrt[3]{x}+4 sqrt[4]{x}}{sqrt{(2 x-3)}+sqrt[3]{(2 x-3)}+sqrt[4]{(2}} )
is equal to
A . 1
B. ( alpha )
c. ( sqrt{2} )
D. None of these
11
137 ( lim _{x rightarrow 0} frac{int_{0}^{t^{2}} cos t^{2} d t}{x sin x} ) is equal to:
( A cdot-1 )
B. +1
( c cdot 2 )
( D ldots-2 )
11
138 ( lim _{x rightarrow 3}left(x^{2}-9right)left[frac{1}{x+3}+frac{1}{x-3}right] ) 11
139 ( lim _{x rightarrow 0} frac{sqrt{4+sin 3 x}-2}{log (1+tan 2 x)}=frac{3}{a} cdot ) Find 11
140 ( lim _{x rightarrow infty}left{left(e^{x}+piright)^{frac{1}{x}}right} ) ( where {} denotes the
fractional part of ( x ) ) is equal to:
( mathbf{A} cdot pi-e )
B . ( pi-3 )
( mathbf{c} cdot e-2 )
D. ( 3-e )
11
141 The value of
( lim _{x rightarrow 0} frac{1}{x}left(tan ^{-1}left(frac{x+1}{2 x+1}right)-frac{pi}{4}right) ) is equal
to.
A .
B.
( c cdot frac{-1}{2} )
( D )
11
142 ( lim _{boldsymbol{y} rightarrow 0} frac{sqrt{1+sqrt{1+boldsymbol{y}^{4}}}-sqrt{mathbf{2}}}{boldsymbol{y}^{4}} )
A ( cdot ) Exists and equals ( frac{1}{4 sqrt{2}} )
B. Does not exist
C – Exist and equals ( frac{1}{2 sqrt{2}} )
D. Exists and equals ( frac{1}{2 sqrt{2}(sqrt{2}+1)} )
11
143 Solve ( lim _{x rightarrow a} frac{sqrt{1+a x}-sqrt{1-a x}}{x} ) 11
144 ( lim _{x rightarrow 0} frac{sqrt{1-cos x}}{x} ) is equal to
A. ( -frac{1}{sqrt{2}} )
B. ( frac{1}{sqrt{2}} )
c. 0
D. Does not exist
11
145 ( lim _{h rightarrow 0}left{frac{1}{h^{3} sqrt{8+h}}-frac{1}{2 h}right}= )
A ( cdot frac{-1}{12} )
B. ( frac{-4}{3} )
( c cdot frac{-16}{3} )
D. ( frac{-1}{48} )
11
146 1.
SuolellVETU
Find the derivative of sin (x2 + 1) with respect to x from first
principle.
(1978)
Tidalad
11
147 The value of ( lim _{x rightarrow a} frac{sqrt{x-b}-sqrt{a-b}}{x^{2}-a^{2}}(a> )
( b) ) is
A ( cdot frac{1}{4 a} )
B. ( frac{1}{a sqrt{a-b}} )
c. ( frac{1}{2 a sqrt{a-b}} )
D. ( frac{1}{4 a sqrt{a-b}} )
11
148 ( lim _{x rightarrow 2} frac{sum 32 x}{x^{3}-p} ) 11
149 ( lim _{x rightarrow 0} sin ^{-1}(sin x) ) is equal to
A ( cdot frac{pi}{2} )
B. 1
c. zero
D. None of the above
11
150 ( lim _{x rightarrow 0}left(frac{sqrt{a+2 x}-sqrt{3 a}}{sqrt{3 a+x}}right) ) 11
151 ( operatorname{Let} f(x)=frac{1-tan x}{4 x-pi}, x neq frac{pi}{4}, ) then
( lim _{x rightarrow frac{pi}{4}} f(x)= )
( mathbf{A} cdot mathbf{1} )
в. ( frac{1}{2} )
( c cdot-frac{1}{2} )
D. –
11
152 Find the absolute maximum and the
absolute minimum values of the
following function in the given intervals. ( boldsymbol{f}(boldsymbol{x})=(boldsymbol{x}-mathbf{2}) sqrt{boldsymbol{x}-mathbf{1}} ) in ( [mathbf{1}, boldsymbol{9}] )
11
153 Evaluate: ( lim _{x rightarrow 0} frac{sqrt[3]{1+sin x}-sqrt[3]{1-sin x}}{x} )
A . 0
B.
( c cdot frac{2}{3} )
D. ( frac{3}{2} )
11
154 ( lim _{h rightarrow 0} frac{sqrt{x+h}-sqrt{x}}{h}, x neq 0 ) 11
155 ( lim _{x rightarrow 0} frac{(x+2)^{10}-2^{10}}{(x+2)^{5}-2^{5}} ) is
A . 16
B. 32
( c cdot 64 )
D. 128
11
156 Evaluate ( : lim _{x rightarrow 0}left(frac{1-cos x}{x^{2}}right) ) 11
157 Let ( alpha ) and ( beta ) be the distinct roots of
( a x^{2}+b x+c=0, ) then
( lim _{x rightarrow alpha} frac{1-cos left(a x^{2}+b x+cright)}{(x-alpha)^{2}} ) is equal to
( mathbf{A} cdot mathbf{0} )
B ( cdot frac{1}{2} a^{2}(alpha-beta)^{2} )
c. ( frac{1}{2}(alpha-beta)^{2} )
D. ( -frac{1}{2} a^{2}(alpha-beta)^{2} )
11
158 If ( lim _{x rightarrow 2} frac{x^{n}-2^{n}}{x-2}=80 ) and ( n in N, ) find
( mathbf{n} )
11
159 This question has four choices ( (A),(B) )
(C) and (D) out of which ONE or MORE
are correct.
Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}+sqrt{boldsymbol{x}^{2}+boldsymbol{2} boldsymbol{x}} ) and ( boldsymbol{g}(boldsymbol{x})= )
( sqrt{x^{2}+2 x}-x, ) then
This question has multiple correct options
A ( cdot lim _{x rightarrow infty} g(x)=1 )
B. ( lim _{x rightarrow-infty} f(x)=1 )
c. ( lim _{x rightarrow-infty} f(x)=-1 )
D. ( lim _{x rightarrow infty} g(x)=-1 )
11
160 Solve: ( lim _{x rightarrow 2} frac{left(1-3^{x}-4^{x}+12^{x}right)}{sqrt{(2 cos x+7)}-} ) 11
161 For ( boldsymbol{m}, boldsymbol{n} in boldsymbol{N}, ) if ( lim _{x rightarrow 0} frac{boldsymbol{x}^{n}-(sin boldsymbol{x})^{m}}{boldsymbol{x}^{m}}=boldsymbol{L} )
where ( L=frac{n}{6}, ) then ( m-n= )
A .2
B. –
( c cdot 0 )
D. –
11
162 Evaluate: ( lim _{x rightarrow 0}left[tan left(frac{pi}{4}+xright)right]^{frac{1}{sin 2 x}} ) 11
163 ( lim _{x rightarrow 0} frac{sin ^{2} x}{sqrt{2}-sqrt{1+cos x}} ) equals:
A ( .2 sqrt{2} )
B. ( 4 sqrt{2} )
( c cdot sqrt{2} )
( D )
11
164 Let ( a, b, c ) are non zero constant number
then the value of
( lim _{r rightarrow infty} frac{cos frac{a}{r}-cos frac{b}{r} cos frac{c}{r}}{sin frac{b}{r} sin frac{c}{r}} ) is?
( ^{mathbf{A}} cdot frac{a^{2}+b^{2}-c^{2}}{2 b c} )
B. ( frac{c^{2}+a^{2}-b^{2}}{2 b c} )
c. ( frac{b^{2}+c^{2}-a^{2}}{2 b c} )
D. Independent of ( a, b ) and ( c )
11
165 ( lim _{x rightarrow 3} frac{sqrt{4-x}-sqrt{x-2}}{15-5 x} ) 11
166 Evaluate ( boldsymbol{x} stackrel{lim }{boldsymbol{a}} frac{boldsymbol{x}^{boldsymbol{7}}-boldsymbol{a} boldsymbol{7}}{boldsymbol{x}-boldsymbol{a}} ) 11
167 Evaluate: ( lim _{x rightarrow 1} frac{1-x^{frac{-1}{3}}}{1-x^{frac{-2}{3}}} ) 11
168 The value of ( lim _{x rightarrow infty} sqrt{x}(sqrt{x+c}-sqrt{x}) ) is
A ( cdot frac{c}{2} )
B. ( frac{c}{3} )
( c cdot frac{c}{4} )
D. none of these
11
169 ( lim _{x rightarrow 0^{+}} frac{sqrt{1+x}-sqrt{1-x}}{sqrt{1+x^{2}}-sqrt{1-x^{2}}} ) equals to
A. 1
B. ( frac{1}{2} )
( c cdot infty )
( D )
11
170 ( lim _{x rightarrow a} frac{x^{m}-a^{m}}{x^{n}-a^{n}} ) is equal to.
A ( . m n a^{m-n} )
в. ( frac{m}{n} a^{m-n} )
c. ( frac{n}{m} a^{m-n} )
D. ( m n a^{m+n} )
11
171 ( f(x)=left{begin{array}{l}x-5 ; x leq 1 \ 4 x^{2}-9 ; 12end{array}right. )
Then ( boldsymbol{f}left(boldsymbol{2}^{+}right)-boldsymbol{f}left(boldsymbol{2}^{-}right)= )
( A )
B. 2
( c )
( D )
11
172 Evaluate: ( lim _{x rightarrow frac{pi}{2}} frac{left(1-tan frac{x}{2}right)(1-sin x)}{left(1+tan frac{x}{2}right)(pi-2 x)^{3}} )
A ( cdot frac{1}{16} )
( B cdot frac{1}{8} )
( c cdot 0 )
D. ( frac{1}{32} )
11
173 Solve:
( lim _{x rightarrow 0} 9 )
11
174 ( f(x)=left{begin{array}{cc}-1, & x<-1 \ x^{3}, & -1 leq x leq 1 \ 1-x, & 1<x2end{array}, ) then right.
A. ( lim _{x rightarrow-1} f(x)=1 )
B ( cdot lim _{x rightarrow 1} f(x)=1 )
c. ( lim _{x rightarrow 2} f(x)=-1 )
D. ( lim _{x rightarrow 2^{-}} f(x)=0 )
11
175 36. Let a(a) and B(a) be the roots of the equation
(11+a -1)x2 +(71+a-1)x +(81+a – 1) =0 where
a>-1. Then lim a(a) and lim B(a) are (2012)
@) and 1 and 1
(0) – and 2 (2) 2 and 3
a->0+
a>0+
11
176 Solve ( lim _{x rightarrow infty} frac{sqrt{x^{2}+1}-sqrt[3]{x^{2}+1}}{sqrt[4]{x^{4}+1}-sqrt[5]{x^{4}+1}} ) 11
177 ( lim _{x rightarrow 0^{-}} frac{3 sin left(2 x^{2}right)}{x^{2}}=A )
then the value of ( boldsymbol{A} ) is
A . 2
B. 4
( c cdot 6 )
D. 8
11
178 If ( lim _{x rightarrow 0}left(x^{-3} sin 3 x+a x^{-2}+bright) ) exists
and is equal to ( 0, ) then
( ^{mathbf{A}} cdot_{a}=-3 ) and ( b=frac{9}{2} )
в. ( _{a=3} ) and ( b=frac{9}{2} )
c. ( _{a=-3 text { and } b=-frac{9}{2}} )
D. ( _{a=3} ) and ( b=-frac{9}{2} )
11
179 Prove that:
( lim _{x rightarrow 0} frac{a x+x cos x}{b sin x} )
11
180 ( lim _{x rightarrow x_{1}} frac{x}{x-x_{1}} int_{x_{1}}^{x} f(t) d t ) is equal to
В ( cdot x_{1} fleft(x_{1}right) )
( mathbf{c} cdot fleft(x_{1}right) )
D. Does not exist
11
181 ( lim _{x rightarrow infty} xleft(a^{frac{1}{x}}-b^{frac{1}{x}}right)= )
A .
в. ( log _{e} a / b )
c. ( log _{e}(a b) )
D.
11
182 ( lim _{x rightarrow 0} frac{sin |x|}{x} ) is equal to
( mathbf{A} cdot 1 )
B. 0
C. Positive infinity
D. Does not exist
11
183 ( underset{n rightarrow infty}{L t}left[frac{n^{1 / 2}}{(n+3)^{3 / 2}}+frac{n^{1 / 2}}{(n+6)^{3 / 2}}+frac{n^{1 / 2}}{(n+9)}right. )
is equal to :
( A cdot frac{1}{2} )
B.
( c cdot frac{1}{3} )
D. None of these
11
184 Find :
( lim _{x rightarrow 0^{+}} x^{x} )
11
185 ( operatorname{Let} f(x)=left{begin{array}{cc}x^{2} & x<1 \ x & 1<x4end{array}right. )
This question has multiple correct options
A ( cdot lim _{x rightarrow 1^{-}} f(x)=1 )
B. ( lim _{x rightarrow 1^{+}} f(x)=1 )
( mathbf{c} cdot lim _{x rightarrow 4^{-}} f(x)=4 )
D. ( lim _{x rightarrow+4^{+}} f(x)=4 )
11
186 Let ( boldsymbol{f}:(1,2) rightarrow mathbb{R} ) satisfies the
inequality ( frac{cos (2 x-4)-33}{2}<f(x)<frac{x^{2}|4 x-8|}{x-2} mid forall x in )
( (1,2) . ) then find ( lim _{x rightarrow 2^{-}} f(x) )
11
187 ( lim _{x rightarrow infty}(sqrt{x^{2}+8 x+3}-sqrt{x^{2}+4 x+3}) )
equals
A .
B . ( infty )
( c cdot 2 )
D.
11
188 The fraction ( frac{sqrt{mathbf{3 x – a}}-sqrt{x+a}}{x-a} )
becomes ( frac{mathbf{0}}{mathbf{0}} ) when ( boldsymbol{x}=boldsymbol{a} )
11
189 Evaluate ( lim _{x rightarrow a} frac{x^{7}-a^{7}}{x-a} ) 11
190 Solve the following limit ( lim _{h rightarrow 0} frac{sqrt{h}}{sqrt{16+sqrt{h}-4}} ) 11
191 Find the value of ( lim _{x rightarrow 0} frac{2 x^{2}+3 x+4}{2} )
A .2
B.
c. ( 3 sqrt{5} )
D. ( 2 sqrt{5} )
11
192 Evaluate ( lim _{x rightarrow 8} frac{sqrt{1+sqrt{1+x}-2}}{x-8} )
A ( cdot frac{3}{2} )
B. ( frac{1}{4} )
c. ( frac{1}{24} )
D. None of these
11
193 Evaluate: ( lim _{x rightarrow 2} frac{x^{2}+5}{x^{2}-3} ) 11
194 ( lim _{x rightarrow 0} frac{1}{x^{2}} frac{1}{sin ^{2} x} )
A ( cdot infty )
B. ( -frac{1}{3} )
( c cdot frac{1}{3} )
D. does not exist
11
195 ( lim _{n rightarrow infty} frac{n^{p} sin ^{2}(n !)}{n+1}, 0<p<1, ) is equal to
( mathbf{A} cdot mathbf{0} )
B. ( infty )
c. 1
D. none of these
11
196 ( lim _{x rightarrow 0} frac{(1-cos 2 x)^{2}}{2 x tan x-x tan 2 x} ) is :-
A . -2
B. ( -frac{1}{2} )
( c cdot frac{1}{2} )
D. 2
11
197 The value of ( lim _{x rightarrow 2} frac{sqrt{1+sqrt{2+x}}-sqrt{3}}{x-2} ) is
( A cdot frac{1}{8 sqrt{3}} )
в. ( frac{1}{4 sqrt{3}} )
c. 0
D. None of these
11
198 ( f(x)=frac{3 x^{2}+a x+a+1}{x^{2}+x-2}, ) then which
of the following can be correct? This question has multiple correct options
A ( cdot lim _{x rightarrow 1} f(x) ) exists ( Rightarrow a=-2 )
B. ( lim _{x rightarrow-2} f(x) ) exists ( Rightarrow a=13 )
C if the limit exists, lim ( f(x)=frac{4}{3} )
D. If the limit exists, ( lim _{x rightarrow-2} f(x)=-frac{1}{3} )
11
199 ( eleft(lim _{x rightarrow beta}(tan x cot beta-1)right) )
is equal to
A ( cdot frac{1}{sin beta cos beta} )
( mathbf{B} cdot sin beta cos beta )
c. ( frac{-1}{sin beta cos beta} )
D ( cdot e^{sec ^{2} beta cot beta} )
11
200 Evaluate: ( lim _{x rightarrow 0} frac{sin 3 x^{2}}{cos left(2 x^{2}-xright)} )
( mathbf{A} cdot mathbf{0} )
B. –
( c cdot 4 )
( D )
11
201 ( lim _{x rightarrow 0} frac{(1+x)^{5}-1}{(1+x)^{3}-1}= )
A . 0
B.
( c cdot frac{5}{3} )
D. 3
11
202 ( lim _{x rightarrow infty} frac{cos x+sin ^{2} x}{x+1} ) 11
203 Evaluate the following limit :
( lim _{x rightarrow 0} frac{sin ^{2} 3 x}{x^{2}} )
( A )
B. 3
( c cdot 9 )
D.
11
204 If ( |boldsymbol{x}|<1, ) then ( lim _{n rightarrow infty}{(1+x)(1+ )
( left.left.boldsymbol{x}^{2}right)left(1+boldsymbol{x}^{4}right) ldots . .left(1+boldsymbol{x}^{2 n}right)right} ) is equal to
A ( cdot frac{1}{x-1} )
в. ( frac{1}{1-x} )
c. ( 1-x )
D. ( x-1 )
11
205 The value of ( lim _{x rightarrow 0} frac{(1+x)^{1 / 4}-(1-x)^{1 / 4}}{x} )
is
A ( cdot frac{1}{2} )
B.
( c cdot-1 )
D. ( -frac{1}{2} )
11
206 ( lim _{boldsymbol{x} rightarrow mathbf{5}}left(frac{sqrt{1-cos (2 boldsymbol{x}-mathbf{1 0})}}{sin (boldsymbol{x}-mathbf{5})}right) )
( A cdot-sqrt{2} )
B. ( sqrt{2} )
c. does not exist
D. none of these
11
207 ( lim _{x rightarrow infty} frac{x^{3}+x^{2}+1}{2 x^{2}+3 x+4}= )
A.
B.
( c cdot alpha )
D.
11
208 Compute ( : lim _{x rightarrow infty} frac{(sqrt{x^{2}}+x-x)}{x} ) 11
209 ( lim _{x rightarrow sqrt{10}} frac{sqrt{7+2 x}-(sqrt{5}+sqrt{2})}{x^{2}-10} ) is equal to
A ( cdot frac{1}{sqrt{40}(sqrt{5}+sqrt{2})} )
B. ( frac{-1}{sqrt{10}[sqrt{7+2 sqrt{10}}+sqrt{5}+2]} )
c. 1
D. ( frac{1}{sqrt{10}(sqrt{5}+sqrt{2})} )
11
210 Find the limit :-
( lim _{x rightarrow infty}left{frac{x^{2}+2 x+3}{2 x^{2}+x+5}right}^{frac{3 x-2}{3 x+2}} )
11
211 Evaluate the following limits. ( lim _{x rightarrow sqrt{3}} frac{x^{4}-9}{x^{2}+4 sqrt{3} x-15} ) 11
212 Evaluate:
( lim _{n rightarrow infty}left[left{1+left(frac{1}{n}right)^{4}right}left{1+left(frac{2}{n}right)^{4}right}^{1 / 2}right} )
11
213 ( lim _{x rightarrow pi / 4}left(frac{1-tan x}{1-sqrt{2} sin x}right) ) is equal to
A.
B.
( c cdot-2 )
D.
11
214 The value of ( lim _{x rightarrow 0} frac{x}{5}left[frac{x}{2}right] ) (where ( [.] )
denotes the greatest integer function) is
A ( cdot frac{2}{5} )
в. ( -frac{2}{5} )
c. 0
( D cdot infty )
11
215 The value of ( lim _{boldsymbol{x} rightarrow boldsymbol{pi} / 2} frac{sin (boldsymbol{x} cos boldsymbol{x})}{cos (boldsymbol{x} sin boldsymbol{x})} ) is
equal to
A . 0
в. ( frac{pi}{2} )
( c . pi )
D . 2 ( pi )
11
216 ( lim _{x rightarrow infty}(sin sqrt{x+1}-sin sqrt{x})= )
A .
B. – –
( c cdot 0 )
D. None of these
11
217 Find the absolute maximum and
minimum values of the function f given by ( f(x)=cos ^{2} x+sin x, x in[0, pi] )
11
218 Apply the Imitsto given expression
( lim _{x rightarrow 0}(((x+1)(x+2)(x+3)(x+4)) )
11
219 34.
(2012)
If lim
(x²+x+1
-ax-b = 4, then
x oo x+1
(a) a=1, b=4
(6) a= 1, b=-4
© a=2, b=-3 (d) a=2, b=3
11
220 Evaluate the following limits. ( lim _{x rightarrow 0} frac{sqrt{a+x}-sqrt{a}}{x sqrt{a^{2}+a x}} )
A ( cdot frac{1}{2 sqrt{a}} )
B. ( frac{1}{2 a sqrt{a}} )
c. ( frac{1}{2 a} )
D. None of these
11
221 Evaluate ( lim _{x rightarrow 2} frac{7 x^{2}-11 x-6}{3 x^{2}-x-10} )
A ( cdot frac{17}{11} )
в. ( frac{11}{17} )
c. ( frac{17}{14} )
D. ( -frac{17}{11} )
11
222 ( lim _{x rightarrow 0} frac{e^{x^{2}}-cos x}{sin ^{2} x} ) is equal to :
( A cdot 2 )
B. ( frac{3}{2} )
( c cdot 3 )
( D cdot frac{5}{4} )
11
223 Find Limit for following Question ( l ) 11
224 ( operatorname{Let} f(x)=left{begin{array}{l}frac{sin [x]}{[x]} ;[x] neq 0 \ 0 ;[x]=0end{array}, ) then right.
( lim _{x rightarrow 0} f(x)= )
( mathbf{A} cdot mathbf{0} )
B. ( sin 1 )
( c cdot 2 )
D. does not exist
11
225 Evaluate the following question. ( lim _{x rightarrow a} frac{(x)^{3 / 2}-(a)^{3 / 2}}{x-a} ) 11
226 ( boldsymbol{L} boldsymbol{t}_{x rightarrow 0} frac{boldsymbol{x}}{sqrt{mathbf{1}+boldsymbol{x}}-mathbf{1}} ) 11
227 ( lim _{x rightarrow 2}left(left(frac{x^{3}-4 x}{x^{3}-8}right)^{-1}-left(frac{x+sqrt{2 x}}{x-2}-frac{ }{sqrt{x} x}right.right. )
is equal to
( A cdot frac{1}{2} )
B . 2
( c cdot 1 )
D. None of these
11
228 If ( lim _{x rightarrow 1} frac{x^{4}-1}{x-1}=lim _{x rightarrow k} frac{x^{3}-k^{3}}{x^{2}-k^{2}}, ) then find
the value of ( k )
( A cdot frac{5}{3} )
B. ( -frac{8}{3} )
( c cdot frac{8}{3} )
D. None of these
11
229 ( lim _{x rightarrow 0} frac{x tan 2 x-2 x tan x}{(1-cos 2 x)^{2}}= )
( A cdot 2 )
B. – 2
( c cdot frac{1}{2} )
D. ( -frac{1}{2} )
11
230 Evaluate : ( underset{n rightarrow infty}{L t} frac{1}{2^{n}} ) 11
231 ( lim _{x rightarrow 0} frac{a x+x cos x}{b sin x} ) 11
232 Evaluate the given limit:
( lim _{x rightarrow 0} frac{a x+x cos x}{b sin x} )
11
233 If ( f(x)=frac{4-7 x}{7 x+4}, lim _{x rightarrow 0} f(x)=l ) and
( lim _{x rightarrow infty} f(x)=m ) the quadratic equation
having roots as ( frac{1}{l} ) and ( frac{1}{m} ) is
A ( cdot x^{2}-1=0 )
B. ( x^{2}-x+1=0 )
c. ( x^{2}-frac{1}{2}=0 )
D. ( x^{3}-1=0 )
11
234 Evaluate:
( lim _{x rightarrow 0} frac{sin x}{sqrt{x^{2}}}= )
( mathbf{A} cdot mathbf{1} )
B. –
( c cdot 0 )
D. doesn’t exist
11
235 Find the points of local maxima or local minima and corresponding local maxima and local minimum value of
the following functions. Also, find the points of inflection, if any. ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}-boldsymbol{2} boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{a}^{2} boldsymbol{x}, boldsymbol{a}>boldsymbol{0}, boldsymbol{x} in boldsymbol{R} )
11
236 1.
lim
vi-cos 2x
v
is
[2002]
(a) 1
(c) zero
(b)-1
d) does not exist
11
237 The value of ( lim _{x rightarrow 2 a} frac{sqrt{x-2 a}+sqrt{x}-sqrt{2 a}}{sqrt{x^{2}-4 a^{2}}} )
is
A ( cdot frac{1}{sqrt{a}} )
B. ( frac{1}{2 sqrt{a}} )
c. ( frac{sqrt{a}}{2} )
D. ( 2 sqrt{a} )
11
238 Evaluate ( lim _{x rightarrow pi / 2} frac{cos x}{pi-2 x} ) 11
239 Evaluate: ( lim _{x rightarrow a} frac{x^{14}-a^{14}}{x^{-7}-a^{-7}} ) 11
240 ( lim _{x rightarrow 0} 4 x^{2}+3 x+2 ) is equal to
( A cdot 2 )
B ( cdot e^{2} )
c. 1
D.
11
241 Solve:
( lim _{x rightarrow 3} frac{x^{2}-9}{x-3} )
11
242 Assertion
If a and b are positive and [x] denotes
the greatest integer ( leq x ), then ( lim _{x rightarrow 0^{+}} frac{x}{a}left[frac{b}{x}right]=frac{b}{a} )
Reason
( lim _{x rightarrow infty} frac{{x}}{x}=0, ) where denotes fractional
part of ( mathbf{x} )
A. Both Assertion and Reason are correct and Reason is
the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
11
243 ( lim _{x rightarrow 0} frac{sin x^{0}}{x}= )
( mathbf{A} cdot mathbf{1} )
В . ( frac{pi}{180} )
c. 180
D.
11
244 Find the intervals in which the following functions are increasing or decreasing ( f(x)=10-6 x-2 x^{2} ) 11
245 Solve ( lim _{x rightarrow 1}left[frac{x-1}{x^{2}-x}-frac{1}{x^{3}-3 x^{2}+x}right] ) 11
246 Find the value of ( lim _{x rightarrow a} frac{x^{n}-a^{n}}{x-a} )
A ( cdot n a^{n} )
В ( cdot n a^{n-1} )
( c cdot n a )
D.
11
247 Evaluate ( lim _{x rightarrow 2} frac{x-2}{x^{2}+x-6} ) 11
248 ( underset{x rightarrow 1}{L t} frac{(2 x-1)(sqrt{x}-1)}{2 x^{2}+x-3} ) 11
249 Evaluate the following limit ( lim _{x rightarrow 0} frac{(1+x)^{6}-1}{x} ) 11
250 ( lim _{x rightarrow 0} frac{(1+x)^{4}-1}{(1+x)^{3}-1}=? )
A ( cdot frac{3}{4} )
B. ( frac{9}{16} )
( c cdot frac{4}{3} )
D. None of these
11
251 Find a
( lim _{boldsymbol{theta} rightarrow boldsymbol{pi}} frac{sqrt{(mathbf{2}-cos boldsymbol{theta})}-mathbf{1}}{(boldsymbol{pi}-boldsymbol{theta})^{2}} ) is ( =frac{mathbf{1}}{boldsymbol{a}} )
11
252 ( A B C ) is an isosceles triangle inscribed in a circle of radius ( r . ) If ( A B=A C ) and ( h )
is the altitude from ( A ) to BC. If the
triangle ( A B C ) has perimeter ( P ) and area
( Delta ) then ( lim _{h rightarrow 0} 512 r frac{Delta}{P^{3}} ) is equal to
11
253 Using the ( in-delta ) definition prove that
( lim _{x rightarrow-2}(3 x+8)=2 )
11
254 ( lim _{x rightarrow frac{pi}{2}}(1+3 cos x)^{sec x}= )
A ( cdot e^{2} )
B ( cdot e^{3} )
( c cdot e^{-2} )
D. ( e^{-3} )
11
255 ( lim _{x rightarrow frac{pi}{2}} frac{left(1-tan frac{x}{2}right)(1-sin x)}{left(1+tan frac{x}{2}right)(pi-2 x)^{3}} ) is
A . 0
B. ( frac{1}{32} )
( c cdot alpha )
( D )
11
256 (20015)
17. lim
*
(a)
sin(it cos2x)
2 equals
(b) a
(c) T2
(d)
1
11
257 If ( boldsymbol{a}>mathbf{0}, lim _{x rightarrow infty} frac{[boldsymbol{a} boldsymbol{x}+boldsymbol{b}]}{boldsymbol{x}} ) is where [
denotes G.I.F.
( mathbf{A} cdot mathbf{0} )
B.
c. ( [a] )
( D cdot[b] )
11
258 ( operatorname{Let} boldsymbol{f}(boldsymbol{x})=left{begin{array}{cc}boldsymbol{x}^{2} & boldsymbol{x}=mathbf{2} \ frac{boldsymbol{k}left(boldsymbol{x}^{2}-boldsymbol{4}right)}{mathbf{2}-boldsymbol{x}} & boldsymbol{x} in boldsymbol{Z}-mathbf{2}end{array}right. )
then ( lim _{x rightarrow 2} f(x) )
A. exists only when ( k=-1 )
B. exits for every real ( k )
C. Exits for every real k except ( k-1 )
D. does not exits
11
259 ( k=lim _{x rightarrow 2} frac{x^{2}-x}{x} ) find ( k ) 11
260 16.
, a>0.
4
Let L = lim –
x->0
If L is finite, then
(2009)
(2) a=2
) a=1 © L=
(a) L = 32
32
na
11
261 The value of ( lim _{n rightarrow infty} cos frac{x}{2} cos frac{x}{2^{2}} ldots cos frac{x}{2^{n}} )
is
A . 1
B. ( frac{sin x}{x} )
c. ( frac{x}{sin x} )
D. none of these
11
262 Value of ( operatorname{lt}_{x rightarrow 0}left(frac{1+tan x}{1+sin x}right)^{operatorname{cosec} x} ) equals
A . 1
B.
( c cdot frac{1}{e} )
D. None of these
11
263 ( lim _{x rightarrow 0^{-}}left{sin ^{-1}[tan x]right}=l ) then ( l ) is equal
to
where ( [text { and }{} ) denotes greatest the integer and fraction part function
( mathbf{A} cdot mathbf{0} )
B. ( -sin ^{-1}(1) )
( c cdot sin 1 )
D. ( 2-frac{pi}{2} )
11
264 Find ( b )
[
begin{array}{l}
lim _{x rightarrow 0}left[frac{sin (x+a)+sin (a-x)-2 sin a}{x sin x}right] \
=-b sin a
end{array}
]
11
265 Find the following limit:
( lim _{x rightarrow 5} frac{sqrt{x-1}-2}{x-5} )
11
266 ( lim _{x rightarrow 2}left[frac{x^{5}-32}{x^{3}-8}right] ) 11
267 ( lim _{x rightarrow-1} frac{x+1}{sqrt{x^{2}+3}-2}= )
A . -2
B.
( c cdot 2 )
D.
11
268 Evaluate ( lim _{x rightarrow 0} frac{x^{3}}{sin x^{2}} ) 11
269 ( lim _{h rightarrow 0} frac{sqrt{x+h}-sqrt{x}}{h} ) is equal to
A ( cdot sqrt{x} )
в. ( frac{1}{2 sqrt{x}} )
( c cdot 2 sqrt{x} )
D. ( frac{1}{sqrt{x}} )
11
270 Calculate the following limits. If ( lim _{x rightarrow-2 / 3} frac{6 x^{2}-5 x-6}{3 x^{2}-x-2} ) is k.Find ( 5 k ) 11
271 Evaluate ( : lim _{x rightarrow 3} frac{sqrt[4]{x}-sqrt[4]{3}}{sqrt[3]{x}-sqrt[3]{3}} ) 11
272 Find the following limit:
( lim _{x rightarrow 9}left(frac{3-sqrt{x}}{9-x}+frac{1}{3-sqrt{x}}-6 cdot frac{x^{2}+}{729}right. )
11
273 Evaluate the following limits. ( lim _{x rightarrow 2} frac{x^{3}-8}{x^{2}-4} ) 11
274 STATEMENT-1: If a and b are positive and ( [x] ) denotes the greatest integer
less than or equal to ( x, ) then ( lim _{x rightarrow 0^{+}} frac{x}{a}left[frac{b}{x}right]=frac{b}{a} )
STATEMENT-2: ( lim _{x rightarrow infty} frac{{x}}{x} rightarrow 0, ) where ( {x} )
denotes the fractional part of ( x )
A. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-
B. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
C. STATEMENT-1 is True, STATEMENT-2 is False
D. STATEMENT-1 is False, STATEMENT-2 is True
11
275 Evaluate the Given limit:
( lim _{x rightarrow 2} frac{3 x^{2}-x-10}{x^{2}-4} )
11
276 ( lim _{x rightarrow 1} frac{(log (1+x)-log 2)left(3.4^{x-1}-3 xright)}{1} )
( left.left.left{(7+x)_{3}^{1}right)-(1+3 x)^{frac{1}{2}}right)right} sin pi x )
11
277 ( lim _{x rightarrow 0} frac{(1-cos 2 x)(3+cos x)}{x tan 4 x} ) is equal to
( mathbf{A} cdot mathbf{4} )
B. 3
( c cdot 2 )
D.
11
278 For which value of ( a ) and ( b, lim _{x rightarrow 0} frac{sin 2 x}{x^{3}}+ )
( boldsymbol{a}+frac{boldsymbol{b}}{boldsymbol{x}^{2}}=mathbf{0} )
A ( quad b=-2 ) and ( a=frac{4}{3} )
B. ( b=2 ) and ( a=frac{4}{3} )
c. ( _{b}=-2 ) and ( a=frac{-4}{3} )
D. None of these
11
279 Evaluate: ( lim _{x rightarrow 1} frac{x^{frac{1}{3}}-1}{x^{-frac{2}{3}}-1} )
A ( cdot frac{-1}{2} )
B. ( frac{1}{3} )
( c cdot frac{2}{3} )
D. ( frac{3}{2} )
11
280 Evaluate :
( lim _{x rightarrow 2} frac{x^{2}-4}{x+3} )
( mathbf{A} cdot mathbf{0} )
B.
( c cdot-1 )
D. None of these
11
281 Evaluate the following limit:
( lim _{x rightarrow 0} frac{sqrt{(1+sin x)}-sqrt{(1-sin x)}}{x} )
11
282 ( lim _{x rightarrow 1} frac{sqrt{3+x}-sqrt{5-x}}{x^{2}-1} ) 11
283 If ( lim _{x rightarrow-2} frac{x^{p}+2^{p}}{x+2}=80 ) (where ( p ) is an odd
number), then ( p ) can be
( A cdot 3 )
B. 5
( c cdot 7 )
D. 9
11
284 Find ( lim _{x rightarrow 1} f(x), quad ) where ( quad f(x)= )
( left{x^{2}-1, quad x leq 1-x^{2}-1, quad x>1right} )
11
285 If ( f ) is differentiable at ( x=1 ) and
( lim _{h rightarrow 0} frac{1}{h} f(1+h)=5, f^{prime}(1)= )
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 3 )
D. 4
( E )
11
286 ( lim _{x rightarrow 0} frac{sin left(x^{g}right)}{x}(g=g r a d) )
A ( cdot frac{pi}{180} )
в. ( frac{pi}{90} )
c. ( frac{pi}{100} )
D. ( frac{pi}{200} )
11
287 Find the following limit:
( lim _{x rightarrow 1} frac{sqrt[3]{x}-1}{sqrt{x}-1} )
11
288 If ( f(x)=frac{x^{2}}{2}, 0 leq x<1 ) and ( f(x)= )
( frac{2 x^{2}-2 x+3}{2} ; 1 leq x leq 2, ) then
( lim _{x rightarrow 1} f(x)= )
A ( cdot frac{1}{2} )
B. ( frac{3}{2} )
c. does not exist
D. ( -frac{1}{2} )
11
289 The value of ( lim _{x rightarrow 2} frac{sqrt{1+sqrt{2+x}}-sqrt{3}}{x-2} ) is
( A cdot frac{1}{8 sqrt{3}} )
в. ( frac{1}{4 sqrt{3}} )
c. 0
D. None of these
11
290 ( lim _{y rightarrow 1}left(frac{1}{y^{2}-1}-frac{2}{y^{4}-1}right)= )
( A )
2
B. 3
( c cdot frac{1}{4} )
( D )
11
291 ( lim _{x rightarrow 0} frac{sqrt[3]{27+x}-3}{x}= )
A ( cdot frac{1}{9} )
в. ( frac{1}{27} )
( c cdot frac{1}{3} )
D.
11
292 Evaluate ( lim _{x rightarrow 0} frac{tan x}{x} ) 11
293 f ( l= )
( lim _{n rightarrow infty} sum_{r=2}^{n}left((r+1) sin frac{pi}{r+1}-r sin frac{pi}{r}right) )
then find ( {l} . ) (where {} denotes the
fractional part function
11
294 ( lim _{x rightarrow 0} frac{2 sin x-sin 2 x}{x^{3}} ) is equal to : 11
295 The value of ( lim _{x rightarrow 0} frac{tan 2 x}{sin 5 x} ) is
( mathbf{A} cdot mathbf{0} )
B. ( frac{2}{5} )
( c cdot 2 )
D. 5
11
296 Solve:
( lim _{x rightarrow 27} frac{x^{2 / 3}-9}{x-27} )
11
297 Evaluate the left hand limit of the function ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{l}frac{|boldsymbol{x}-mathbf{4}|}{boldsymbol{x}-mathbf{4}}, boldsymbol{x} neq mathbf{4} \ mathbf{0} quad, boldsymbol{x}=mathbf{4}end{array}right. ) 11
298 Evaluate ( lim _{x rightarrow 0} frac{sin x}{x} ) 11
299 ( lim _{x rightarrow 0} frac{xleft(e^{x}-1right)}{1-cos x} ) is equal to
( A cdot 0 )
B. ( infty )
( c cdot-2 )
D.
11
300 The value of
( lim _{x rightarrow a} frac{sqrt{x-b}-sqrt{a-b}}{x^{2}-a^{2}}(a>b) ) is
A ( cdot frac{1}{4 a} )
B. ( frac{1}{a sqrt{a-b}} )
c. ( frac{1}{2 a sqrt{a-b}} )
D. ( frac{1}{4 a sqrt{a-b}} )
11
301 The value of ( lim _{x rightarrow 0} frac{1-cos ^{3} x}{x sin x cos x} ) is?
A ( cdot frac{2}{5} )
B. ( frac{3}{5} )
( c cdot frac{3}{2} )
D.
11
302 If ( lim _{x rightarrow 1}left(1+a x+b x^{2}right)^{frac{c}{x-1}}=e^{3}, ) then
find conditions on a, b and c.
11
303 ( lim _{x rightarrow 0} frac{x^{3}+3 x^{2}-9 x-2}{x^{3}-x-6} ) 11
304 Evaluate the following limits. ( lim _{x rightarrow a} frac{sin sqrt{x}-sin sqrt{a}}{x-a} ) 11
305 For a certain value of ( c, lim _{x rightarrow infty}left[left(x^{5}+right.right. )
( left.left.7 x^{4}+2right)^{c}-xright] ) is finite ( & ) non zero. The
value of ( c ) and the value of the limit is
A ( cdot frac{1}{5}, frac{7}{5} )
B. 0,1
c. ( _{1,} frac{7}{5} )
D. none
11
306 Find the following limit:
( lim _{x rightarrow 1} frac{sqrt{1+x}-sqrt{1-x}}{sqrt[3]{1+x}-sqrt[3]{1-x}} )
11
307 Find the value of ( lim _{x rightarrow 0} frac{sqrt{2+x}-sqrt{2-x}}{x} )
( ^{mathrm{A}} cdot frac{1}{2 sqrt{2}} )
B. ( sqrt{2} )
c. ( frac{1}{sqrt{2}} )
D.
11
308 FHEMENT-1: ( left[lim _{x rightarrow 0} frac{sin x}{x}right]=0 )
STATEMENT-2: For ( boldsymbol{x} in(-boldsymbol{delta}, boldsymbol{delta}), ) where ( boldsymbol{delta} )
is positive and ( delta rightarrow 0, tan x>x )
A. STATEMENT-1 is True, STATEMENT-2 is True:
STATEMENT-2 is a correct explanation for STATEMENT
B. STATEMENT-1 is True, STATEMENT-2 is True:
STATEMENT-2 is not the correct explanation for STATEMENT-1
c. STATEMENT-1 is True, STATEMENT-2 is False
D. STATEMENT-1 is False, STATEMENT-2 is True
11
309 ( lim _{x rightarrow infty}left(frac{x^{2}+2 x-1}{2 x^{2}-3 x-2}right)^{frac{2 x+1}{2 x-1}} ) is equal
to
( mathbf{A} cdot mathbf{0} )
B. ( infty )
( c cdot frac{1}{2} )
D. None of these
11
310 ( lim _{x rightarrow infty} cos (sqrt{x+1})-cos (sqrt{x}) )
( mathbf{A} cdot mathbf{0} )
B.
c. Do not exist
D. 78
11
311 Let a function ( boldsymbol{f}(boldsymbol{x})= )
a relation between ( b ) and ( c ) such that
( lim _{x rightarrow 1} f(x) ) exists is
B. ( 3 b+2 c-1=0 )
c. ( 3 b-2 c-1=0 )
D. ( b-2 c+1=0 )
11
312 23. If a function f:[-2a, 2a] R is an odd function such that
f(x)=f(2a – x) for x e[a, 2a] and the left hand derivative at
x=a is then find the left hand derivative at x =-a.
(2003 – 2 Marks)
11
313 Calculate the following limits. ( lim _{x rightarrow 3} frac{x^{2}-5 x+6}{x-3} ) 11
314 ( lim _{x rightarrow-5} frac{2 x^{2}+9 x-5}{x+5} ) 11
315 ( f f(x)=left{begin{array}{ll}frac{x-|x|}{x}, & x neq 0 \ 2, & x=0end{array} ) show right.
that ( lim f(x) ) does not exist
11
316 Solve ( lim _{x rightarrow frac{pi}{4}} frac{f(x)-fleft(frac{pi}{4}right)}{x-frac{pi}{4}}, ) where
( f(x)=sin 2 x )
11
317 ( lim _{x rightarrow infty} sqrt{frac{x-sin x}{x+cos ^{2} x}}= )
( mathbf{A} cdot mathbf{1} )
B. 2
( c .3 )
D. None of these
11
318 Let ( a=m i nleft[x^{2}+2 xright], x in R ) and ( b= )
( lim _{x rightarrow 0} frac{sin x cdot cos x}{e^{x}+e^{-x}} . ) Then value of ( a+b ) is
A . -1
B . 2
( c .1 )
D.
11
319 The value of ( lim _{x rightarrow 3^{+}} frac{|x-3|}{x-3} ) equals
A . 1
B. – 1
( c .0 )
D. Does not exist
11
320 ( lim _{x rightarrow 0}left(1+frac{2}{x^{2}}right)^{x^{2}}= )
( A )
B. e
( c cdot e^{2} )
D. None of these
11
321 Find ( lim _{x rightarrow 2^{+}} f(x), ) where ( f(x)= )
( left{begin{array}{ll}4 x+9, & x2end{array}right. )
11
322 Solve:
( lim _{x rightarrow 2} frac{x^{2}-4}{sqrt{3 x-2}-sqrt{x+2}} )
11
323 ( f(x)=lim _{n rightarrow infty} frac{tan ^{2 n} x cdot sin x}{x} ) then
( lim _{x rightarrow 0} f(x)=1 )
If true enter 1 , else enter 0 .
11
324 Find the ( lim _{x rightarrow 0} frac{tan x sin x}{sin ^{3} x} ) 11
325 Solve:
( lim _{x rightarrow 8} frac{sqrt{1+sqrt{1+x}}-2}{x-8} )
11
326 ( frac{a cos x+b x sin x-5}{x^{4}} ) is finite, then
( boldsymbol{a}= )
A ( cdot frac{5}{2} )
B. 5
( c cdot frac{2}{5} )
D.
11
327 The value of ( lim _{x rightarrow 0} frac{2 sin ^{2} 3 x}{x^{2}} ) is 11
328 ( lim _{x rightarrow 0} frac{sin left(pi cos ^{2} xright)}{x^{2}} ) equals
A . ( -pi )
B.
( c cdot frac{pi}{2} )
( D )
11
329 If ( f(x)=e^{x}, ) then ( lim _{x rightarrow 0} f(f(x))^{frac{1}{(f(x))}} ) is
equal to (where ( {x} ) denotes fractional
part of ( boldsymbol{x} ) ).
A. ( f(1) )
B. ( f(0) )
( c .0 )
D. Does not exist
11
330 Evaluate the following limits. ( lim _{x rightarrow 0} 9 ) 11
331 The value of ( lim _{x rightarrow 0^{+}} frac{1}{3 x} ) is
A. ( -infty )
B. –
c. 0
( D cdot+infty )
11
332 ( lim _{x rightarrow 2} frac{x^{2}+5 x+6}{2 x^{2}-3 x} )
A . 10
B. ( infty )
( c cdot 2 )
D.
11
333 Evaluate the following limits. ( lim _{x rightarrow 2}(3-x) ) 11
334 Evaluate: ( lim _{x rightarrow 1} frac{x^{frac{-2}{3}}-1}{x^{frac{-3}{4}}-1} )
A ( cdot frac{5}{9} )
B. ( frac{9}{5} )
( c cdot frac{8}{9} )
( D )
11
335 Examine the graph of ( y=f(x) ) as
shown and evaluate the following limits
(i) ( lim _{x rightarrow 1} f(x) )
(ii) ( lim _{x rightarrow 2} f(x) )
(iii) ( lim _{x rightarrow 3} f(x) )
(iv) ( lim _{x rightarrow 199} f(x) )
( (v) lim _{x rightarrow 3^{+}} f(x) )
11
336 ( lim _{boldsymbol{pi} atop boldsymbol{x} rightarrow frac{pi}{boldsymbol{4}}} frac{sec boldsymbol{x} cdot tan (boldsymbol{4} boldsymbol{x}-boldsymbol{pi})}{sin (boldsymbol{4} boldsymbol{x}-boldsymbol{pi})}= )
A ( cdot sqrt{2} )
B. ( frac{1}{sqrt{2}} )
( c cdot-sqrt{2} )
D. ( frac{-1}{sqrt{2}} )
11
337 ( lim _{x rightarrow 0} frac{1-cos x cos 2 x cos 3 x}{sin ^{2} 2 x} ) is equal to
( A cdot 7 / 2 )
в. ( 7 / 3 )
( c cdot 7 / 4 )
D. ( 7 / 5 )
11
338 24.
(a-n)nx – tan x) sin nx
If lim
= 0, where n is nonzero real
xo
x
number, then a is equal to
(2003)
(a) o (6) ht (c) n (d) + –
п
11
339 Evaluate
( lim _{x rightarrow 0} frac{1-sqrt{cos x}}{x^{2}} )
11
340 ( lim _{x rightarrow frac{1}{2}} frac{8 x-3}{2 x-1}-frac{4 x^{2}+1}{4 x^{2}-1} ) 11
341 If ( f(x)=frac{3 x^{2}+a x+a+1}{x^{2}+x-2}, ) then which
of the following can be correct? This question has multiple correct options
A ( cdot lim _{x rightarrow 1} f(x) ) exists ( Rightarrow a=-2 )
B. ( lim _{x rightarrow-2} f(x) ) exists ( Rightarrow a=13 )
( mathbf{c} cdot lim _{x rightarrow 1} f(x)=frac{4}{3} )
D. ( lim _{x rightarrow-2} f(x)=-frac{1}{3} )
11
342 Prove that:
( lim _{x rightarrow frac{pi}{2}} frac{tan 2 x}{x-frac{pi}{2}} )
11
343 Evaluate the following question. ( lim _{x rightarrow 0} sin x ) 11
344 Find ( lim _{x rightarrow pi / 4} frac{sin x-cos x}{x-pi} ) 11
345 If ( 0<x<pi, ) then
( frac{sin 8 x+7 sin 6 x+18 sin 4 x+12 sin 2 x}{sin 7 x+6 sin 5 x+12 sin 3 x} )
is equal to
( mathbf{A} cdot mathbf{6} )
B. 4
( c cdot 3 )
D. 2
E . 8
11
346 15 Ullo

Let f:
R
R
be a function. We say that f has
(h)
(0) exists and is finite, and
PROPERTY 1 if lim
50
h
PROPERTY 2 if limy
f(h)-f(0
exists and is finite
Then which of the following options is/are correct?
(JEE Adv. 2019)
(a) f(x) = x2/3 has PROPERTY1
(b) f(x)=sin x has PROPERTY2
© f(x)=x has PROPERTY 1
(d) f(x) = xx has PROPERTY 2
11
347 [
begin{aligned}
operatorname{Let} boldsymbol{f}(boldsymbol{x})=&{boldsymbol{x}+boldsymbol{lambda}, boldsymbol{x}<1\
& boldsymbol{2} boldsymbol{x}-boldsymbol{3}, boldsymbol{x} geq 1, text { if } lim _{boldsymbol{x} rightarrow 1} boldsymbol{f}(boldsymbol{x})
end{aligned}
]
exists, then find value of ( lambda )
11
348 Value of
( lim _{x rightarrow 0} frac{x+2 sin x}{sqrt{x^{2}+2 sin x+1}-sqrt{x-sin ^{2} x}} )
is
( A cdot 2 )
B.
( c cdot 6 )
D. – 2
11
349 If ( lim _{x rightarrow 1} frac{x^{4}-1}{x-1}=lim _{x rightarrow k} frac{x^{3}-k^{3}}{x^{2}-k^{2}} ) then ( k=? )
A ( cdot frac{2}{3} )
B. ( frac{4}{3} )
( c cdot frac{8}{3} )
D. none
11
350 ( lim _{x rightarrow a} frac{x^{m}-a^{m}}{x-a}= )
( mathbf{A} cdot m a^{m-1} )
B. ( m a^{m} )
c. ( frac{1}{m} a^{n} )
D. ( a^{m-1} )
11
351 ( lim _{x atop x rightarrow frac{pi}{4}} frac{1-tan x}{1-sqrt{2} sin x} ) 11
352 ( lim _{x rightarrow 0} frac{sin x-frac{x}{6}}{6 x}=? )
A ( cdot frac{5}{36} )
B. ( frac{2}{360} )
c. ( frac{7}{360} )
D. ( frac{11}{360} )
11
353 If ( lim _{x rightarrow-a} frac{x^{7}+a^{7}}{x+a}=7 )
then the value of a is
This question has multiple correct options
( A cdot 1 )
B. –
c. 7
D. none of these
11
354 Find it; ( lim _{x rightarrow 0}left(frac{x^{2}+x+1}{x+1}-a x-6right)=4 ) 11
355 If ( f(x)=frac{sin ([x] pi)}{x^{2}+x+1}, ) when
: ( ] ) denotes
the greatest integer function, then
11
356 If ( f(x)=a x^{2}+b x+c, ) show that ( lim _{h rightarrow 0} frac{f(x+h)-f(x)}{h}=2 a x+b ) 11
357 Write the value of ( lim _{x rightarrow 0} frac{sin x}{sqrt{1+x}-1} ) 11
358 If ( lim _{x rightarrow 0} frac{x^{a} sin ^{b} x}{left(sin ^{c} xright)}, ) where ( a, b, c in R-{0} )
exists & has non zero value then ( a+ )
( boldsymbol{b}=boldsymbol{c} )
If true enter 1 else enter 0
11
359 34
[JEE M 2015)
x-
lim (1 – cos2x) (3 + cosx) is equal to:
x tan 4x
(2) 2 (6) 2 (0) 4
(d) 3
11
360 Let ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{ll}a sin ^{2 n} x & text { for } x geq 0 text { and } \ b cos ^{2 m} x-1 & text { for } x<0 text { and }end{array}right. )
( n rightarrow infty )
( m rightarrow infty )
then –
A ( cdot fleft(0^{-}right) neq fleft(0^{+}right. )
в. ( fleft(0^{+}right) neq f(0) )
c. ( fleft(0^{-}right)=f(0) )
D. fis continuous at ( x=0 )
11
361 ( lim _{x rightarrow 0}(sin x)^{2 tan x}=? )
( A cdot 2 )
B.
( c cdot 0 )
D. Does not exist
11
362 If ( boldsymbol{a}=lim _{x rightarrow infty}(sqrt{x+sqrt{x}-} sqrt{x}) ) and ( b= )
( lim _{x rightarrow infty}(x-sqrt{x+x^{2}}) ) then
( mathbf{A} cdot a+b=1 )
B . ( a+b=0 )
( mathbf{c} cdot a=b )
D. ( a+b=2 )
11
363 ( lim _{x rightarrow pi / 4} frac{sqrt{2} cos x-1}{cot x-1} ) equals
A .
в. ( frac{1}{2} )
( c cdot frac{1}{sqrt{2}} )
( D cdot sqrt{2} )
11
364 im logx” -|*] neN. (x] denotes greatest integer less
5.
lim
[2002]
than or equal to x)
(a) has value-1
(c) has value 1
(b) has value 0
(d) does not exist
11
365 If ( f(x)=sqrt{frac{x-sin x}{x+cos ^{2} x} text { then } lim _{x rightarrow infty} f(x) text { is }} )
A.
B.
( c )
D. None of these
11
366 ( lim _{x rightarrow 0} frac{x^{4}left(cot ^{4} x-cot ^{2} x+1right)}{left(tan ^{4} x-tan ^{2} x+1right)} ) is equal to
A . 1
B.
( c cdot 2 )
D. None of these
11
367 20. For a R (the set of all real numbers), a -1,
(JEE Adv. 2013)
(1° +2° +…+na)
lim
n— (n+1)*-‘[(na+1)+na+2)+…+(na+n)] 60
Then a=
(a) 5
(b) 7
(c) -15
(d) -17
11
368 ( lim _{n rightarrow infty} frac{left(n^{2}+5 n+6right)}{(n+4)(n+5)} ) is equal to 11
369 ( lim _{x rightarrow 9} frac{2 x-7 sqrt{x}+3}{3 x-11 sqrt{x}+6} )
( mathbf{A} cdot frac{3}{4} )
B. ( frac{5}{3} )
( c cdot frac{5}{7} )
( D cdot frac{3}{7} )
11
370 If ( x ) is very large, then ( frac{2 x}{1+x} ) is
A. close to 0
B. arbitrarily large
c. lie between 2 and 3
D. close to 2
11
371 Let ( {x} ) denote the fractional part of ( x ) Then ( lim _{x rightarrow 0} frac{{x}}{tan {x}} ) is equal to
A . -1
B. 0
( c . )
D. Does not exist
11
372 ( f(x)left{begin{array}{cc}x^{4} & x^{2}<1 \ x & x^{2} geq 1end{array} ) Discuss the right.
existence of limit at ( x=1 ) and ( x=-1 )
A. Limit exist at both ( x=1 ) and ( x=-1 )
B. Limit does not exist at both ( x=1 ) and ( x=-1 )
c. limit exist at ( x=1 ) but not at ( x=-1 )
D. limit exist at ( x=-1 ) but not at ( x=1 )
11
373 ( lim _{x rightarrow infty}left(frac{x^{2}+5 x+3}{x^{2}+x+3}right)^{1 / x} )
A ( cdot e^{4} )
B. ( e^{2} )
( c cdot e^{3} )
( D )
11
374 ( lim _{n rightarrow infty} frac{n(2 n+1)^{2}}{(n+2)left(n^{2}+3 n-1right)} ) is equal to
( mathbf{A} cdot mathbf{0} )
B. 2
( c cdot 4 )
( D )
11
375 Evaluate the following limits. ( lim _{x rightarrow 3}left(frac{1}{x-3}-frac{3}{x^{2}-3 x}right) )
A ( cdot frac{1}{3} )
B. ( frac{1}{2} )
( c cdot-frac{1}{3} )
D. None of these
11
376 Evaluate the following question.
( lim _{x rightarrow 0} x^{2}-3 )
11
377 The value of ( lim _{x rightarrow infty}(x-sqrt{left(x^{2}-xright)}) )
is
A ( cdot frac{1}{2} )
в. ( -frac{1}{2} )
( c cdot 1 )
D.
11
378 1 – cos {2(x-2)}
[2011]
27.
lim
x 2
x-2
(b) equals – V2
(a) equals v2
(C) equals
(d) does not exist
11
379 Evaluate: ( lim _{x rightarrow-3} frac{x^{3}+27}{x+3} )
( A cdot 9 )
B . 27
( c .-27 )
( D )
11
380 ( lim _{x rightarrow infty} frac{1}{1+x^{2}}+frac{2}{4+x^{2}}+frac{3}{9+x^{2}}+ )
( cdots+frac{x}{x^{2}+x^{2}} )
11
381 Evaluate
( lim _{x rightarrow 0} frac{sqrt[k]{1+x}-1}{x}(mathrm{K} ) is a positive
integer ( ) )
( A cdot K )
B. – –
c. ( frac{1}{K} )
D. ( -frac{1}{K} )
11
382 ( lim _{x rightarrow o} cos (sqrt{1+x})-cos sqrt{x} ) 11
383 Evaluate: ( lim _{x rightarrow 2}left{frac{1}{x-2}-frac{4}{x^{2}-4}right} )
( mathbf{A} cdot mathbf{0} )
B. 4
( c cdot frac{1}{4} )
D. ( -frac{1}{4} )
11
384 ( lim _{n rightarrow infty}left(frac{1}{x^{2}}+frac{2}{x^{2}}+frac{3}{x^{2}}+ldots+frac{x}{x^{2}}right) ) 11
385 If ( {x} ) denotes fractional part of ( x, ) then ( lim _{x rightarrow 1} frac{x sin {x}}{x-1}= )
( mathbf{A} cdot mathbf{0} )
B. –
c. 1
D. does not exist
11
386 ( mathbf{A t} t=0, ) the function ( f(t)=frac{sin t}{t} ) has
This question has multiple correct options
A. A minimum
B. A discontinuity
c. A point of inflexion
D. A maximum
11
387 Evaluate the following limits. ( lim _{x rightarrow 3}left(frac{1}{x-3}-frac{2}{x^{2}-4 x+3}right) ) 11
388 Evaluate : ( quad operatorname{lit}_{x rightarrow 3} frac{|x-3|}{x-3} ) 11
389 If ( lim _{x rightarrow 0}left(frac{cos 4 x+a cos 2 x+b}{x^{4}}right) ) is finite
then the value of ( a, b ) respectively are
( mathbf{A} cdot 5-4 )
в. -5,-4
c. -4,3
D. 4,5
11
390 If ( lim _{x rightarrow 0} frac{x^{a} sin ^{b} x}{sin x^{c}} ) where ( a, b, c epsilon R-{0} )
exists and has non-zero value. Then
A ( cdot a+c=b )
B . ( a+b=c )
( mathbf{c} cdot a=b+c )
D. ( a+b+c=0 )
11
391 Evaluate ( lim _{x rightarrow 2} frac{f(x)-f(2)}{x-2} ) where ( f(x)= )
( x^{2}-4 )
( A cdot-1 )
B. 2
( c cdot 0 )
D. 4
11
392 ( lim _{x rightarrow 0^{+}} frac{sin ^{-1} 2 x}{tan ^{-1} 3 x}=? ) 11
393 If ( f(x) ) is differentiable function in the interval ( (0, infty) ) such that ( f(1)=1 ) and ( lim _{t rightarrow x} frac{t^{2} f(x)-x^{2} f(t)}{t-x}=1 ) for
( operatorname{each} x>0 ) then
11
394 Evaluate ( : lim _{x rightarrow infty} frac{5 x^{2}+1}{x^{2}+10} ) 11
395 Evaluate
( lim _{x rightarrow infty} 2^{-x} sin left(2^{x}right) )
11
396 ( boldsymbol{L} boldsymbol{t}_{boldsymbol{x} rightarrow mathbf{0}}=frac{sqrt{mathbf{1}+mathbf{x}}-sqrt{mathbf{1}+mathbf{x}^{2}}}{sqrt{mathbf{1 – x}^{2}}-sqrt{mathbf{1 – x}}}= )
( A )
B. –
( c cdot 0 )
( D )
11
397 If ( boldsymbol{A}_{boldsymbol{i}}=frac{boldsymbol{x}-boldsymbol{a}_{boldsymbol{i}}}{left|boldsymbol{x}-boldsymbol{a}_{boldsymbol{i}}right|} ) where ( boldsymbol{i}=mathbf{1}, boldsymbol{2}, boldsymbol{3} ) and
( boldsymbol{a}_{1}<boldsymbol{a}_{2}<boldsymbol{a}_{3} ) then ( lim _{boldsymbol{x} rightarrow boldsymbol{a}_{2}} boldsymbol{A}_{1} boldsymbol{A}_{2} boldsymbol{A}_{3}= )
A . -1
B.
( c cdot 0 )
D. does not exist
11
398 ( lim _{boldsymbol{n} rightarrow infty} frac{1}{n} sum_{boldsymbol{r}=1}^{boldsymbol{2} boldsymbol{n}} frac{boldsymbol{r}}{sqrt{boldsymbol{n}^{2}+boldsymbol{r}^{2}}} ) equals
( mathbf{A} cdot 1+sqrt{5} )
B ( .-1+sqrt{5} )
( mathrm{c} cdot-1+sqrt{2} )
D. ( 1+sqrt{2} )
11
399
24
5.
1, 8(a)=-1, g'(a)= 2, then the
If f(a) = 2, f'(a)=1, g(a)=-1, g'(a)=
value of lim 8(x)/(a)-g(a)f(x) is (1983 – 1 Mark)
x-a
(a) -5
(C) 5
(d) none of these
xa
11
400 Evaluate the following limits. ( lim _{x rightarrow 4} frac{x^{3}-64}{x^{2}-16} ) 11
401 If ( l=lim _{x rightarrow 0} frac{sin a x-sin x-x}{x^{3}} ) exists, and
is finite, then the values of ‘ ( l ) ‘ and ‘a’ are
respectively equal to
A ( cdot frac{7}{6}, 2 )
в. ( -frac{4}{3}, 2 )
c. ( -frac{7}{6},-2 )
D. ( -frac{7}{6}, 2 )
11
402 The value of ( lim _{boldsymbol{theta} rightarrow mathbf{0}} frac{mathbf{1}-cos mathbf{4} boldsymbol{theta}}{mathbf{1}-cos mathbf{6} boldsymbol{theta}} ) is
A. ( 9 / 4 )
B. 3/4
( c cdot 4 / 9 )
D. ( 9 / 3 )
11
403 If ( lim _{x rightarrow infty} f(x) ) exists and is finite and nonzero and if ( lim _{x rightarrow infty}left{f(x)+frac{3 f(x)-1}{f^{2}(x)}right}=3, ) then the
value of ( lim _{x rightarrow infty} f(x) ) is
A . 2
B.
( c cdot-1 )
D. 3
11
404 ( lim _{x rightarrow a} frac{sqrt{a+2 x}-sqrt{3 x}}{sqrt{3 a+x}-2 sqrt{x}} ) is equal to
A ( cdot frac{2}{sqrt{3}} )
B. ( -frac{1}{sqrt{3}} )
( c cdot frac{2}{3 sqrt{3}} )
D. ( frac{1}{sqrt{3}} )
11
405 Evaluate ( lim _{x rightarrow 0} frac{x tan x}{(1-cos x)} ) 11
406 Evaluate ( lim _{x rightarrow a} frac{sqrt{a+2 x}-sqrt{3 x}}{sqrt{3 a+x}-2 sqrt{x}},(a neq 0) )
A ( cdot frac{2}{3 sqrt{3}} )
B ( cdot frac{4}{3 sqrt{3}} )
c. ( frac{5}{3 sqrt{3}} )
D. ( frac{7}{3 sqrt{3}} )
11
407 Solve ( : lim _{x rightarrow 0} frac{sin 4 x}{sin x} ) 11
408 The value of
( lim x rightarrow infty x^{2} sin (operatorname{en} sqrt{cos frac{pi}{x}}) ) is
( A cdot-frac{pi^{2}}{2} )
B. ( -frac{pi}{1} )
c. ( frac{pi^{2}}{2} )
D. ( frac{pi^{2}}{4} )
11
409 ( lim _{x rightarrow 2} frac{2 x+5}{8-x^{3}} ) 11
410 ( lim _{x rightarrow 1+} frac{sqrt{x-1}}{sqrt{x^{2}-1}+sqrt{x^{3}-1}} ) 11
411 The value of ( lim _{x rightarrow 1} frac{x^{m}-1}{x^{n}-1} ) is?
( A cdot frac{m}{n} )
в. ( frac{n}{m} )
c. ( frac{n^{2}}{m^{2}} )
D. ( frac{m^{2}}{n^{2}} )
11
412 If ( lim _{x rightarrow 0}left(frac{sin 2 x}{x^{3}}+a+frac{b}{x^{2}}right)=0 ) then
the value of ( 3 a+b ) is
A . 2
B. -2
( c cdot-1 )
D.
11
413 State Yes or No. ( lim _{x rightarrow 3^{+}} frac{x}{[x]} ) is equal to ( lim _{x rightarrow 3^{-}} frac{x}{[x]} ) 11
414 If ( boldsymbol{f}(boldsymbol{x})=left|boldsymbol{x}^{2}-mathbf{5} boldsymbol{x}+boldsymbol{6}right|, ) then ( boldsymbol{f}^{prime}(boldsymbol{x}) )
equals
A. ( 2 x-5 ) for ( 2<x<3 )
B. ( 5-2 x ) for ( 2<x2 )
D. ( 5-2 x ) for ( x<3 )
11
415 Evaluate the following limits. ( lim _{x rightarrow pi / 2} frac{2^{-cos x}-1}{xleft(x-frac{pi}{2}right)} ) 11
416 Solve ( lim _{x rightarrow a} frac{x^{2}-(a+1) x+a}{x^{3}-a^{3}} )
A ( cdot frac{a-2}{a^{2}} )
в. ( frac{a-1}{3 a^{2}} )
c. ( frac{a+1}{3 a^{2}} )
D. None of these
11
417 ( operatorname{Let} boldsymbol{f}(boldsymbol{x})=left{begin{array}{l}sin boldsymbol{x}, boldsymbol{x} neq boldsymbol{n} boldsymbol{pi} \ boldsymbol{2}, boldsymbol{x}=boldsymbol{n} boldsymbol{pi}end{array}, text { where } boldsymbol{n} boldsymbol{epsilon} mathbb{Z}right. )
and
( boldsymbol{g}(boldsymbol{x})=left{begin{array}{l}boldsymbol{x}^{2}+mathbf{1}, boldsymbol{x} neq mathbf{2} \ boldsymbol{3}, boldsymbol{x}=mathbf{2}end{array}right. )
Then ( lim _{x rightarrow 0} g(f(x)) ) is
( A )
B.
( c cdot 3 )
D. none of these
11
418 The values of constants a and b so that ( lim _{x rightarrow infty}left(frac{x^{2}+1}{x+1}-a x-bright)=frac{1}{2}, ) are
A ( a=1, b=-frac{3}{2} )
в. ( a=-1, b=frac{3}{2} )
c. ( a=0, b=0 )
11
419 Solve ( lim _{x rightarrow-4} frac{x+4}{x^{2}-x-20} ) 11
420 Prove that ( operatorname{at} x=2, ) limits of function
does not exist ( boldsymbol{f}(boldsymbol{n})= )
( left{begin{array}{ccc}x^{2}+x+1 & text { if } & x geq \ n & text { if } & x<2end{array}right} )
11
421 ff ( (9)=0, f^{prime}(9)=4, ) then
( lim _{x rightarrow 9} frac{sqrt{f(x)}-3}{sqrt{x}-3}= )
( A cdot 9 )
B. 4
( c . ) 36
D. None of these
11
422 Find
( lim _{x rightarrow pi / 2} frac{sin (cos x) cos x}{sin x-operatorname{cosec} x} )
11
423 If ( boldsymbol{f}(boldsymbol{x})=-sqrt{25-boldsymbol{x}^{2}}, ) then find
( lim _{x rightarrow 1} frac{f(x)-f(1)}{x-1} )
11
424 ( lim _{x rightarrow frac{pi}{2}} frac{cot x-cos x}{(pi-2 x)^{3}} ) equals:
A ( cdot frac{1}{24} )
B. ( frac{1}{16} )
c. ( frac{1}{2} )
D.
11
425 ( lim _{x rightarrow 5} frac{x^{2}-25}{x-5}= )
A . 10
B. 1
( c cdot frac{2}{3} )
D.
11
426 32.
lim
(1-сos 2x)(3+cos x)
x tan 4x
is equal to
(JEEM 2013]
@
mo
() 2
(c) 1
(d)
2
11
427 ( lim _{x rightarrow 0} frac{(1-cos 2 x)(3+cos x)}{x tan 4 x} ) is equal to:
A ( cdot frac{1}{2} )
B. 1
c. 2
D. ( -frac{1}{4} )
11
428 ( lim _{x rightarrow 0} frac{sin ^{2} x}{x cos x} ) equals
( A )
B. 2
( c cdot 0 )
D.
11
429 ( lim _{x rightarrow 2} frac{x^{5}-32}{x^{3}-8}= )
( A cdot frac{3}{20} )
B. ( frac{20}{3} )
c. ( frac{10}{3} )
D. ( frac{3}{10} )
11
430 Evaluate the limit
( lim _{x rightarrow frac{pi}{6}} frac{cot ^{2} x-3}{cos e c x-2} )
11
431 The sum of an infinite geometric series whose first term is the limit of the
function ( boldsymbol{f}(boldsymbol{x})=frac{tan boldsymbol{x}-sin boldsymbol{x}}{sin ^{3} boldsymbol{x}} ) as ( boldsymbol{x} rightarrow mathbf{0} )
and whose common ratio is the limit of
the function ( g(x)=frac{1-sqrt{x}}{left(cos ^{-1} xright)^{2}} ) as ( x rightarrow )
1 is
A ( cdot frac{1}{3} )
в. ( frac{1}{4} )
c. ( frac{1}{2} )
D. ( frac{2}{3} )
11
432 Evaluate:-
( lim _{x rightarrow 36} frac{x-36}{sqrt{x}-6} )
11
433 ( lim _{x rightarrow 3} frac{x-3}{sqrt{x-2}-sqrt{4-x}} ) equals to
( mathbf{A} cdot mathbf{0} )
B. ( frac{3}{2} )
( c cdot frac{1}{4} )
D. none of these
11
434 Assertion
( boldsymbol{x}=mathbf{0} ) is point of minima of ( boldsymbol{f} )
Reason
( f^{prime}(0)=0 )
A. Both Assertion and Reason are correct and Reason is
the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
11
435 Evaluate the following limits. ( lim _{x rightarrow 0} frac{sqrt{2-x}-sqrt{2+x}}{x} )
A ( cdot frac{1}{sqrt{2}} )
B. ( -frac{1}{sqrt{3}} )
( c cdot-frac{1}{2} )
D. ( -frac{1}{sqrt{2}} )
11
436 ( lim _{x rightarrow sqrt{2}} frac{x^{2}-2}{x^{2}+sqrt{2 x}-4} ) 11
437 ( lim _{x rightarrow 0} frac{sqrt{1+x}-sqrt{1-x}}{2 x} ) equals 11
438 Solve ( lim _{x rightarrow a} frac{x sqrt{x}-a sqrt{x}}{x-a} ) 11
439 Evaluate ( : lim _{x rightarrow 0} frac{1-cos x}{x} ) 11
440 Solve:
( lim _{x rightarrow 0} frac{sqrt[3]{1+x}-sqrt[3]{1-x}}{x} )
11
441 ( lim _{x rightarrow 0} frac{sin x}{x} ) is equal to:
( A cdot 2 )
B. – –
( c )
D.
11
442 Evaluate ( lim _{x rightarrow o} frac{a x+sin x}{b sin x} ) 11
443 ( lim _{x rightarrow a} frac{sin x-sin a}{x-a} ) 11
444 ( lim _{x rightarrow infty}(sqrt{x^{2}+8 x+3}- )
( sqrt{x^{2}+4 x+3})= )
A . 0
B. ( infty )
( c cdot 2 )
D. ( frac{1}{2} )
11
445 Compute ( lim _{x rightarrow 2+}([x]+x) ) and ( lim _{x rightarrow 2-}([x]+ )
( boldsymbol{x}) )
11
446 39.
lim cotx – cos x
**(1-2x)3
equals:
[JEEM 2017
N
11
447 Evaluate ( : lim _{x rightarrow-2} frac{frac{1}{x}+frac{1}{2}}{x+2} ) 11
448 ( lim _{boldsymbol{x} rightarrow frac{pi}{2}} frac{operatorname{cosec} boldsymbol{x}-cot boldsymbol{x}}{boldsymbol{x}}= )
( mathbf{A} cdot 1 )
( B cdot frac{2}{pi} )
( c cdot frac{1}{2} )
( D cdot frac{1}{5} )
11
449 ( lim _{x rightarrow-infty} sqrt{x^{2}+1}-x ) 11
450 The value of the function
( lim _{x rightarrow 0} frac{8}{x^{8}}left(1-cos frac{x^{2}}{2}-cos frac{x^{2}}{4}+cos frac{x^{2}}{2}right. )
is
A ( cdot frac{1}{16} )
B. ( frac{1}{15} )
c. ( frac{1}{32} )
( D )
11
451 Find ( lim _{x rightarrow(2 n+1) pi^{+}} sin left([sin x] frac{pi}{6}right), ) where []
is a greatest integer function and ( n epsilon I )
A. 0
B. ( frac{1}{2} )
( c cdot-frac{1}{2} )
D. none of these
11
452 Evaluate ( int_{0}^{200 pi} sqrt{1+cos x} d x ) 11
453 What is the value of ( lim _{x rightarrow 0} frac{sin x}{tan 3 x} )
A ( cdot frac{1}{4} )
B. ( frac{1}{3} )
( c cdot frac{1}{2} )
D.
11
454 ( lim _{x rightarrow 0} frac{27^{x}-9^{x}-3^{x}+1}{sqrt{2}-sqrt{1+cos x}}= )
( mathbf{A} cdot mathbf{0} )
B . ( 8 sqrt{2}(log 3)^{2} )
c. ( 8(log 3)^{2} )
D.
11
455 If ( f(x)=a x^{2}+b x+c ) then show that
( lim _{h rightarrow 0} frac{f(x+h)-f(x)}{h}=2 a x+b )
11
456 Evaluate the Given limit:
( lim _{x rightarrow 1} frac{a x^{2}+b x+c}{c x^{2}+b x+a}, a+b+c neq 0 )
11
457 ( lim _{n rightarrow infty}left[frac{1}{2 n}+frac{1}{sqrt{4 n^{2}-1}}+frac{1}{sqrt{4 n^{2}-4}}+. .right. )
is equal to
A ( cdot frac{pi}{4} )
в. ( frac{pi}{2} )
c.
D.
11
458 Find the limits of ( frac{e^{m x}-e^{m a}}{x-a}, ) when ( x= )
( a )
11
459 Evaluate: ( lim _{x rightarrow 0} frac{6^{x}-1}{sqrt{3-x}-sqrt{3}} ) 11
460 If ( boldsymbol{m}, boldsymbol{n} in boldsymbol{N}, ) then the value of
( lim _{x rightarrow 0} frac{sin x^{n}}{(sin x)^{m}} ) is
This question has multiple correct options
( mathbf{A} cdot 1, ) if ( n=m )
B. ( 0, ) if ( n>m )
( c cdot infty, ) if ( n<m )
D. ( n / m, ) if ( n<m )
11
461 The largest value of the non-negative
integer ( a ) for which
( lim _{x rightarrow 1}left{frac{-a x+sin (x-1)+a}{x+sin (x-1)-1}right}^{frac{1-x}{1-sqrt{x}}}= )
( mathbf{1} )
( overline{4}^{text {is }} )
11
462 If ( f(x)=left{begin{array}{l}2 x+b(x<alpha) \ x+d(x geq alpha)end{array} ) is such right.
that
( lim _{x rightarrow alpha} f(x)=L, ) then ( L= )
( mathbf{A} cdot 2 d-b )
B. ( b-d )
c. ( d+b )
D. ( b-2 d )
11
463 Solve
( lim _{x rightarrow 3} frac{x^{3}-7 x^{2}+15 x-9}{x^{4}-5 x^{3}+27 x-27} )
11
464 ( lim _{x rightarrow 0} frac{e^{x}-e^{sin x}}{2(x-sin x)}= )
A. ( -frac{1}{2} )
в. ( frac{1}{2} )
c. 1
D.
11
465 Evaluate: ( lim _{x rightarrow 0} frac{e^{alpha x}-e^{beta x}}{sin alpha x-sin beta x} ) 11
466 The value of ( lim _{x rightarrow 2} frac{sqrt{1+sqrt{2+x}}-sqrt{3}}{x-2} ) is
equal to?
A. ( frac{1}{8 sqrt{3}} )
в. ( frac{1}{sqrt{3}} )
c. ( 8 sqrt{3} )
D. ( sqrt{3} )
11
467 ( operatorname{Let} f(x)=left{begin{array}{l}x+1, x>0 \ 2-x, x leq 0end{array} text { and } g(x)=right. )
( left{begin{array}{c}x+1, x<1 \ x^{2}-2 x-2,1 leq x<2 \ x-5, x geq 2end{array}right. )
Find the LHL and RHLof ( g(f(x)) ) at ( x=0 ) lim
and, hence, find ( _{boldsymbol{x} rightarrow 0} g(f(x)) )
11
468 27. Let f(x)=-
1-x(1+1-x)
11-x
–COS
for x = 1. Then
(JEE Ady. 2017)
(a) lim – -f(x) = 0
(6) lim 1-f(x) does not exist
(c) lim + f(x) = 0
(d) limu+f(x) does not exist
11
469 Given, ( boldsymbol{f}(boldsymbol{x})=mathbf{1}-|boldsymbol{x}-mathbf{2}| ) for ( mathbf{1} leq boldsymbol{x} leq mathbf{3} )
and ( f(3 x)=a f(x) ) for all other values
of ( x . ) If ( a=3 ), then ( lim _{x rightarrow 10^{+}} f(x)=? ) and ( lim _{x rightarrow 10^{-}} f(x)=? )
11
470 If ( lim _{x rightarrow 0} frac{sqrt{1+x}-sqrt{1-x}}{2 x}=frac{1}{a}, ) then ( a ) is
equal to
11
471 Prove that, ( [boldsymbol{x}]+[mathbf{5} boldsymbol{x}]+[mathbf{1 0 x}]+[mathbf{2 0 x}]= )
( 36 k+35 . k varepsilon I ) does not have any real
solution. Here [.] denotes greatest integer function.
11
472 If ( lim _{x rightarrow 2^{+}} frac{x-3}{x^{2}-4}=a ) then ( frac{1}{a} ) is equal
to
11
473 Evaluate ( lim _{x rightarrow infty} sqrt{x^{2}+x+}-sqrt{x^{2}+1} ) 11
474 ( lim _{x rightarrow 0} frac{a^{x}-b^{x}}{e^{x}-1} ) 11
475 If ( f(x)=frac{x^{2}+6 x}{sin x}, ) then ( lim _{x rightarrow 0^{-}} f(x)= )
A . 2
B. 4
( c cdot 6 )
( D )
11
476 ( lim _{x rightarrow 0} frac{sqrt{1+tan x}-sqrt{1-tan x}}{sin x}= )
( A cdot 0 )
B. 1
( c cdot 2 )
D.
11
477 ( lim _{x rightarrow 0} frac{2 x^{2}+3 x+4}{2}= )
A .2
B.
( c cdot 3 sqrt{5} )
D. ( 2 sqrt{5} )
11
478 ( lim _{x rightarrow frac{pi}{4}} frac{sin x}{cos ^{-1}left[frac{1}{4}(3 sin x-sin 3 x)right]} )
D denotes integer function, is-
A ( cdot frac{sqrt{2}}{pi} )
B. 1
( c cdot frac{4}{pi} )
D. does not exist
11
479 c
MCQs with One Correct Answer
1. Iff() = V4-sin, then lim f(x) is
(1979)
Y+COS
x

(a) o
(c) 1
(b)
(d) none of these
11
480 Calculate the following limits. ( lim _{x rightarrow 0} frac{sqrt{6+x}-sqrt{6-x}}{x} ) 11
481 ( f(x)=left{begin{aligned} 4 x, & x0 end{aligned}right. )
equals
A .
B.
( c cdot 3 )
D. does not exist
11
482 Solve :
( lim _{x rightarrow 2} frac{sqrt{1+sqrt{2+x}-sqrt{3}}}{x-2} )
11
483 Evaluate the following limits. ( lim _{x rightarrow 1} frac{x^{2}+1}{x+1} ) 11
484 Evaluate ( lim _{x rightarrow 0} frac{sin x}{|x|} )
A .
B. 0
c. -1
D. does not exist
11
485 Solve: ( lim _{x rightarrow 2} frac{a^{tan x}-a^{sin x}}{tan x-sin x}, a>0 ) 11
486 Consider the following statements:
( S_{1}: lim _{x rightarrow 0^{-}} frac{[x]}{x} ) is an indeterminate from
(where [] denotes greatest integer function).
( boldsymbol{S}_{2}: lim _{x rightarrow infty} frac{sin left(3^{x}right)}{3^{x}}=0 )
( S_{3}: lim _{x rightarrow infty} sqrt{frac{x-sin x}{x+cos ^{2} x}} ) does not exist.
( boldsymbol{S}_{4} )
( lim _{n rightarrow infty} frac{(n+2) !+(n+1) !}{(n+3) !}(n in N)=0 )
Which of the statements ( S_{1}, S_{2}, S_{3}, S_{4} )
are true or false:
A. ( F T F T )
в. ( F T T T )
c. ( F T F F )
D. ( T T F T )
11
487 The value of ( lim _{x rightarrow 0}left[frac{a}{x}-cot frac{x}{a}right] ) is
A . 0
B.
( c )
D. ( frac{a}{3} )
11
488 Solve:
( lim _{x rightarrow 27} frac{left(x^{1 / 3}+3right)left(x^{1 / 3}-3right)}{x-27} )
11
489 12. lim-2-
en+ nr=1 n 2 + 2
equals
(1997 – 2 Marks)
(a) 1+ √5 (6 – 1+√5 (c) -1+√2 (d) 1+√2
11
490 ( lim _{n rightarrow infty} frac{1}{n^{3}}left[left(1^{2}+2^{2}+ldots+n^{2}right)right]=? )
A . ( 1 / 3 )
B. 1/16
c. ( 1 / 12 )
D.
11
491 ( lim _{x rightarrow 0} frac{sqrt{2+x}-sqrt{2}}{x} ) 11
492 ( operatorname{lf} lim _{x rightarrow 0} frac{sin 2 x+cos 2 x}{x^{3}+5}=k ) then ( k= )
( A cdot 2 )
B . – –
( c cdot 1 / 5 )
D.
11
493 ( lim _{x rightarrow 3^{-}} frac{|x-3|}{x-3}= ) 11
494 Find ( c )
( lim _{x rightarrow 0} frac{log tan left(frac{pi}{4}+a xright)}{sin b x}=c frac{a}{b} )
11
495 16. Find lino {tan(Tt/4+x)}”/>
(1993 – 2 Marks)
11
496 Find: ( lim _{x rightarrow 0^{+}} frac{1}{x} )
( A cdot O )
B. ( -infty )
( C cdot infty )
D. does not exist
11
497 Defined ( boldsymbol{f}:left[-frac{1}{2}, inftyright) rightarrow boldsymbol{R} ) by ( boldsymbol{f}(boldsymbol{x})= )
( sqrt{mathbf{1}+mathbf{2} boldsymbol{x}}, boldsymbol{x} inleft[-frac{mathbf{1}}{mathbf{2}}, inftyright) . ) Then compute
( lim _{boldsymbol{x} rightarrowleft(frac{mathbf{1}}{mathbf{2}}right)} boldsymbol{f}(boldsymbol{x}) ). and also find ( lim _{boldsymbol{x} rightarrow frac{-mathbf{1}}{mathbf{2}}} boldsymbol{f}(boldsymbol{x}) )
11
498 ( int frac{sin ^{2} x cos ^{2} x}{left(sin ^{3} x+cos ^{3} xright)^{2}} ) 11
499 (2000)
16.
For x ERlim
x
oo (x+2)
(a) e
(b) et
(c) es
(d) es
11
500 ( lim _{x rightarrow 0}left[frac{(1+x)^{frac{1}{8}}-(1-x)^{frac{1}{8}}}{x}right] ) 11
501 Evaluate ( : lim _{x rightarrow 1}left(frac{1}{1-x}-frac{3}{1-x^{3}}right) ) 11
502 Evaluate: ( lim _{x rightarrow 0} x^{2} ) 11
503 Evaluate ( : lim _{x rightarrow 1} frac{sqrt{5 x-4}-sqrt{x}}{x^{3}-1} ) 11
504 If ( lim _{x rightarrow 3} frac{x^{n}-3^{n}}{x-3}=108, ) find the value of
( mathbf{n} )
11
505 Find the value of ( lim _{x rightarrow 0} frac{sin 2 x}{2 x^{2}+x} ) 11
506 Find the limits of
( frac{sqrt{boldsymbol{x}}-sqrt{mathbf{2 a}}+sqrt{boldsymbol{x}-mathbf{2} boldsymbol{a}}}{sqrt{boldsymbol{x}^{2}-mathbf{4} boldsymbol{a}^{2}}}, ) when ( boldsymbol{x}=mathbf{2} boldsymbol{a} )
11
507 ( lim _{x rightarrow 0}left[frac{sin x}{tan (5 x)} frac{(1-cos 4 x)left(5^{x}-4^{x}right)}{x^{3}}right] )
equals to
A ( cdot frac{8}{3} ell nleft[frac{5}{2}right. )
в. ( frac{8}{5} ell nleft[frac{5}{2}right. )
c. ( frac{8}{5} ln left[frac{5}{4}right. )
D. None of these
11
508 ( lim _{x rightarrow 1} frac{(2 x-3)(sqrt{x}-1)}{2 x^{2}+x-3} cdot ) is ( frac{-1}{b} ) Find 11
509 ( lim _{x rightarrow 0} frac{3^{2 x}-2^{3 x}}{x} ) is equal to
A ( cdot log frac{3}{2} )
B.
c. ( log frac{9}{8} )
D.
11
510 The value of ( lim _{n rightarrow infty}left(frac{1}{sqrt{4 n^{2}-1}}+frac{1}{sqrt{4 n^{2}-4}}+dots+frac{ }{sqrt{3}}right. )
is.
A ( cdot frac{1}{4} )
B. ( frac{pi}{12} )
c.
D.
11
511 ( lim _{x rightarrow 0} frac{sqrt{x+6}-sqrt{6}}{x}= )
А. ( frac{1}{sqrt{6}} )
в. ( -frac{1}{sqrt{6}} )
с. ( frac{1}{2 sqrt{6}} )
D. ( -frac{1}{2 sqrt{6}} )
11
512 ( lim _{x rightarrow 1} frac{x^{4}-3 x^{3}+2}{x^{3}-5 x^{2}+3 x+1} ) 11
513 If ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) defined by ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{ll}frac{x-2}{x^{2}-3 x+2} & x in R-{1,2} \ 2 & x=1 \ 1 & x=2end{array}right. )
then ( lim _{x rightarrow 2} frac{f(x)-f(2)}{x-2} ) is
( A )
B.
c. 1
( D cdot-frac{1}{2} )
11
514 Calculate the following limits. ( lim _{n rightarrow infty} frac{frac{1}{2}+1+frac{3}{2}+ldots frac{n}{2}}{0.25 n^{2}+n+3}, n epsilon N ) 11
515 If ( lim _{n rightarrow infty} frac{n .3^{n}}{n(x-2)^{n}+n .3^{n+1}-3^{n}}=frac{1}{3} )
then the range of ( x text { is (When } n in N) )
A. [2,5)
()
B ( cdot(1,5) )
c. (-1,5)
( D cdot(-infty, infty) )
11
516 Answer the following question in one
word or one sentence or as per exact requirement of the question. Write the value of ( lim _{x rightarrow 0^{-}} frac{sin x}{sqrt{x}} )
11
517 If ( f(x)=left{begin{array}{l}x: x0end{array}right. )
A .
B.
( c cdot 2 )
D. does not exist
11
518 33. If lim [1+x fn(1+62)]’ * = 26 sin? O, b>0 and 0 €(-1,],
then the value of O is
(2011)
x=0
+
Bly
+
(b)
+
11
519 ff ( f(x)=2 x-3, a=2, l=1 ) and ( epsilon= )
0.001 then ( delta>0 ) satisfying ( 0<mid x- )
( boldsymbol{a}|<boldsymbol{delta},| boldsymbol{f}(boldsymbol{x})-boldsymbol{l} mid<boldsymbol{epsilon}, ) is:
A. 0.005
B. 0.0005
c. 0.001
D. 0.0001
11
520 If ( lim _{x rightarrow 2^{-}} frac{x-3}{x^{2}-4}=a ) then ( frac{1}{a} ) is equal
to
11
521 Evaluate ( lim _{x rightarrow 0} frac{sin x}{x} ) 11
522 Evaluate the following question.
( lim _{x rightarrow 2} 2 x+5 )
11
523 Assertion
( lim _{boldsymbol{x} rightarrow mathbf{0}} frac{sqrt{1-cos 2 x}}{boldsymbol{x}} ) does not exist.
Reason ( |sin x|=left{begin{array}{cc}sin x ; & 0<x<frac{pi}{2} \ -sin x ; & -frac{pi}{2}<x<0end{array}right. )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is
not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
11
524 If
( lim _{n rightarrow infty} frac{1^{a}+2^{a}+ldots . .+n^{a}}{(n+1)^{a-1}[(n a+1)+(n a+2)+} )
for some positive real number
( overline{mathbf{6 0}} )
a, then a is equal to
A ( cdot frac{15}{2} )
B. 8
c. ( frac{17}{2} )
D.
11
525 ( $ $ ) lbegin ( {text { matrix }} lim || x ) ( mathrm{~ | x r i g h t a r r o w l q u a d ~ 1 ~ l e n d { m a t r i x } $ $ ~} )
( left(frac{1}{x-1}-frac{2}{x^{2}-1}right) )
11
526 8.
(1985 – 2 Marks)
If f(x) = sin|x][x]+0
= 0 [x]=0
Where [x] denotes the greatest integer less than o
to x. then lim f(x) equals –
le greatest integer less than or equal
3-
0
(a)
(c)
1
-1
(b) 0
(d) none of these
Tot
11
527 ( int_{0}^{1} frac{boldsymbol{d} boldsymbol{x}}{sqrt{1+boldsymbol{x}-sqrt{boldsymbol{x}}}}= )
( A cdot frac{2 sqrt{2}}{3} )
B. ( frac{4 sqrt{2}}{3} )
( c cdot frac{8 sqrt{2}}{3} )
D. None of these
11
528 ( lim _{x rightarrow 0} frac{1-cos left(x^{2}right)}{x^{3}left(4^{x}-1right)} ) is equal to
This question has multiple correct options
A ( frac{1}{4 log 2} )
B. ( frac{1}{2 log 4} )
( mathbf{c} cdot log 4 )
D ( quad 1-frac{1}{2} log left(frac{e^{2}}{4}right) )
11
529 Solve:
( lim _{x rightarrow 0}left(left[x^{2}right]-[x]^{2}right) )
11
530 Evaluate :
( lim _{x rightarrow infty} x^{2}(sqrt{frac{x+2}{x}}-sqrt[3]{frac{x+3}{x}}) )
11
531 Let ( boldsymbol{alpha}, boldsymbol{beta} in boldsymbol{R} ) be such that
( lim _{x rightarrow 0} frac{x^{2} sin (beta x)}{a x-sin x}=1 . ) Then ( 6(alpha+beta) )
equals
A . 5
B. 7
( c cdot 8 )
( D )
11
532 s)
2.
Find the derivative of
x-1
ex²_7x+5
when x #1
50) = 2*745 when xa1
when x = 1
atx=1
(1979)
11
533 Evaluate the following limits. ( lim _{x rightarrow 0} frac{a x+b}{c x+d}, d neq 0 )
A ( cdot frac{a}{c} )
B. ( a )
( c cdot frac{b}{d} )
D. None of these
11
534 Find the limits of the following ( operatorname{expression} frac{1-x^{2}}{2 x^{3}-1} div frac{1-x}{2 x^{2}},(1) ) when
(2) when ( boldsymbol{x}=mathbf{0} )
( boldsymbol{x}=infty )
11
535 ( lim _{x rightarrow 0}left(frac{a}{b}+frac{cos x}{b}right) ) 11
536 The value of ( lim _{x rightarrow 1}left(frac{x^{n}+x^{n-1}+x^{n-2}+ldots .+x^{2}+x}{x-1}right. )
is
A ( cdot frac{n(n+1)}{2} )
B.
( c cdot 1 )
( D )
11
537 Solve:
( lim _{x rightarrow 0} frac{(1-cos 3 x)}{x sin 2 x} )
11
538 If the expression ( frac{x-p}{x^{2}-3 x+2} ) takes all
real values for ( x in R ) then find the
limits for ‘p’.
11
539 Evaluate the following limits. ( lim _{x rightarrow 2} frac{sqrt{1+4 x}-sqrt{5+2 x}}{x-2} )
A ( cdot frac{1}{2} )
B.
( c cdot frac{1}{4} )
D.
11
540 ( lim _{x rightarrow infty} frac{sqrt{x^{2}+1}+sqrt[3]{x^{2}+1}}{sqrt[4]{x^{4}+1}+sqrt[5]{x^{4}-1}} ) is equal to
A.
B. –
c. 0
D. – 2
11
541 ( lim _{x rightarrow 1} frac{sqrt[3]{x^{2}}-2 sqrt[3]{x}+1}{(x-1)^{2}} ) is equal to
A ( cdot frac{1}{9} )
B.
( c cdot frac{1}{3} )
D. none of these
11
542 The value of
( lim _{x rightarrow a} frac{sqrt{x-b}-sqrt{a-b}}{x^{2}-a^{2}}(a>b) ) is
A ( cdot frac{1}{4 a} )
B. ( frac{1}{a sqrt{a-b}} )
c. ( frac{1}{2 a sqrt{a-b}} )
D. ( frac{1}{4 a sqrt{a-b}} )
11
543 ( boldsymbol{L} boldsymbol{t}_{n rightarrow infty} boldsymbol{n} cos left(frac{boldsymbol{pi}}{boldsymbol{n}}right) sin left(frac{boldsymbol{2} boldsymbol{pi}}{mathbf{3} boldsymbol{n}}right)= )
A ( cdot frac{2 pi}{3} )
B.
( c . pi )
D.
11
544 16.
If lim1
x-00
then the values of a and b, are
[2004]
(a) a= 1 and b=2
© a ER,b=2
b) a=1,5 ER
(d) a ER, ER
11
545 Evaluate ( lim _{x rightarrow 0} f(x), ) where ( f(x)= )
( left{begin{array}{ll}|x|, & x neq 0 \ 0, & x=0end{array}right. )
11
546 Evaluate
( lim _{x rightarrow 0} frac{sqrt{2}-sqrt{1+cos x}}{sin ^{2} x} )
11
547 Find the limit ( lim _{x rightarrow 0} frac{x}{sqrt{1+x}-1} ) 11
548 Evaluate the following limits. ( lim _{x rightarrow 1} frac{1+(x-1)^{2}}{1+x^{2}} ) 11
549 ( lim _{x rightarrow 0} tan left(frac{pi}{4}+xright) ) 11
550 [2002]
(a) et
(b) e²
(c) e
(d 1
o.
11
551 1+ 24 +34 +…n
1+23 +33 +…n
10.
lim
n->
– lim
n->00
[2003]

can
be () Zero(a)
11
552 5. Use
the
formula
lima’ – 1 – Ina
to
find
x->0
x
2* 1
*-0 (1+x)/2-1
lim

(1982-2 Marks)
11
553 sec? tdt
12.

is
The value of limo
x
[2003]
xsin x
(a) o
(b) 3
(C) 2
(d) 1
11
554 Solve ( lim _{x rightarrow 3} frac{x^{2}-9}{x^{3}-6 x^{2}+9 x+1} ) 11
555 Find the value of ( lim _{x rightarrow 1} e(1+sin pi x)^{cot pi x} ) 11
556 The set of all values of ‘a’ for which
( lim _{x rightarrow a}[x] ) does not exist is ( ([x] )
denotes greatest integer function)
A. a is any integer
B. a is a positive rational number
C. a is a negative rational integer
D. a is complex number
11
557 Solve ( lim _{x rightarrow 0} frac{sin 4 x}{sin 2 x} ) 11
558 Evaluate the following limits. ( lim _{x rightarrow 3} frac{x^{2}-4 x+3}{x^{2}-2 x-3} )
A ( cdot frac{1}{2} )
B. ( frac{2}{3} )
( c cdot frac{1}{3} )
D.
11
559 ( lim _{n rightarrow infty} frac{1^{3}+2^{3}+ldots+n^{3}}{n^{4}}=frac{1}{4} l . ) Find 11
560 Solve
( lim _{x rightarrow 0} frac{tan 8 x}{sin 2 x} )
11
561 If ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{cl}frac{sin {cos boldsymbol{x}}}{boldsymbol{x}-frac{pi}{2}}, & boldsymbol{x} neq frac{boldsymbol{pi}}{2} \ mathbf{1}, & boldsymbol{x}=frac{boldsymbol{pi}}{2}end{array}right. )
(.) represents the fractional part function,
then ( lim _{x rightarrow pi / 2} f(x) ) is?
( A )
B.
c. Does not exist
D.
11
562 ( lim _{x rightarrow 3} x^{2}-3 x+1 ) 11
563 Let ( k=lim _{x rightarrow 0} frac{x sin x}{x^{2}}, ) then the value of
( frac{5 k+1}{k} ) is
11
564 ( lim _{x rightarrow a}(x-a)left(frac{1}{x-a}-frac{1}{x^{2}-(a+b) x+}right. )
( = )
A ( cdot frac{a-b-1}{a-b} )
B.
( c )
D. ( frac{a+b}{a-b} )
11
565 Evaluate ( lim _{x rightarrow 3}left(4 x^{2}+3right) )
A . 36
B . 39
c. 40
D. None of these
11
566 ( lim _{x rightarrow 0}left(x^{-3} sin 3 x+a x^{-2}+bright) ) exists and
is equal to ( 0, ) then
A ( cdot a=-3 ) and ( b=frac{9}{2} )
B. ( a=3 ) and ( b=frac{9}{2} )
c. ( _{a=-3 text { and } b=-frac{9}{2}} )
D. ( a=3 ) and ( b=-frac{9}{2} )
11
567 Find the value of the limit:
( lim _{x rightarrow 1} frac{x^{2}-6 x+5}{x^{2}+3 x-4} )
A . -1.25
в. -0.80
c. 0.80
D. 1.25
E. The limit does not exists
11
568 3.
Evaluate: lim (a +
(a+h)? sin(a+h)-a-sina
Lh 0
h
(1980)
11
569 Evaluate: ( lim _{x rightarrow 3}left[frac{1}{x-3}+frac{9 x}{27-x^{3}}right] ) 11
570 ( lim _{n rightarrow infty}left(frac{2 n^{3}}{2 n^{2}+3}+frac{1-5 n^{2}}{5 n+1}right) ) is equal to.
A . 0
B. 1
( c cdot 1 / 5 )
D.
11
571 The value of ( lim _{x rightarrow 2} int_{2}^{x} frac{3 t^{2}}{x-2} d t ) is
A . 10
B. 12
( c cdot 8 )
D. 16
11
572 ( lim _{x rightarrow 1}left(frac{1+x}{2+x}right)^{frac{(1-sqrt{x})}{(1-x)}}= )
( mathbf{A} cdot mathbf{1} )
в. ( ln 2 )
( c cdot sqrt{frac{2}{3}} )
D. Does not exist
11
573 Calculate the following limits. ( lim _{x rightarrow-3} frac{x+3}{sqrt{x+4}-1} ) 11
574 Find the value of ( lim _{x rightarrow infty} frac{sqrt{x}}{sqrt{x}+3} ) 11
575 Find the following limit:
( lim _{x rightarrow 0} x cot 3 x )
11
576 ( lim _{x rightarrow a} frac{x^{2}-a^{2}}{x-a} ) 11
577 ( lim _{x rightarrow 0} frac{sqrt{2}-sqrt{1+cos x}}{sin ^{2} x} ) equals
A ( cdot sqrt{2} )
B. ( frac{sqrt{2}}{8} )
c.
D. none of these
11
578 Let ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{ll}x^{2}+k, & text { when } x geq 0 \ -x^{2}-k, & text { when } x<0end{array} . ) If the right.
function ( f(x) ) be continous at ( x=0 )
then ( boldsymbol{k}= )
( mathbf{A} cdot mathbf{0} )
B. 1
c. 2
D. –
11
579 The value of
( lim _{x rightarrow infty}left{frac{x}{x+frac{sqrt[3]{x}}{x+frac{sqrt[3]{x}}{x+sqrt[3]{x}}} cdots}right} )
( A )
B
( c )
( D )
11
580 The value of the limit ( lim _{x rightarrow 1} frac{sin left(e^{x-1}-1right)}{log x} )
is
( mathbf{A} cdot mathbf{0} )
B. ( e )
( c cdot frac{1}{e} )
D.
11
581 What is the value of
[
begin{array}{l}
lim _{x rightarrow frac{pi}{2}}[cos x] ? \
x
end{array}
]
11
582 ( underset{x rightarrow 0}{operatorname{Lt} frac{(1+x)^{6}-1}{(1+x)^{5}-1}}= )
( A cdot frac{1}{2} )
B. 6 5
c. 1
( D )
11
583 Find ( lim _{x rightarrow 1} f(x), ) where ( f(x)= )
( left{begin{array}{ll}x^{2}-1, & x leq 1 \ -x^{2}-1, & x>1end{array}right. )
11
584 Evaluate ( : lim _{x rightarrow 2} frac{3 x^{2}-x-10}{x^{2}-4} ) 11
585 – is
[2003]
an
(a) oo
(c) 0
32
11
586 Check the existance of the limit ( lim _{x rightarrow 0} frac{|x|}{x} ) 11
587 ( operatorname{Let} mathbf{L}=lim _{mathbf{x} rightarrow 0}= )
( frac{boldsymbol{a}-sqrt{boldsymbol{a}^{2}-boldsymbol{x}^{2}}-frac{boldsymbol{x}^{2}}{boldsymbol{4}}}{boldsymbol{x}^{4}}, boldsymbol{a}>0 ) If ( boldsymbol{L} ) is finite
then
This question has multiple correct options
( mathbf{A} cdot a=2 )
B. ( a=1 )
( c cdot_{L}=frac{1}{64} )
D. ( L=frac{1}{32} )
11
588 The value of ( lim _{x rightarrow 0} frac{sqrt{x^{2}+1}-1}{sqrt{x^{2}+9}-3} ) is?
( A cdot 3 )
B. 4
( c cdot 1 )
D. 2
11
589 Evaluate: ( lim _{x rightarrow 0} frac{xleft(e^{sin x}-1right)}{1-cos x} )
A ( cdot frac{1}{2} )
B . 2
( c cdot 0 )
D.
11
590 Solve;
( lim _{x rightarrow 0} frac{(1+x)^{6}-1}{(1+x)^{2}-1} )
11
591 The limit of the following is ( lim _{x rightarrow 3} frac{sqrt{1-cos left(x^{2}-10+21right)}}{x-3} )
( A cdot_{-}(2)^{frac{3}{2}} )
B・(2)
c. ( (2)^{-frac{3}{2}} )
D.
11
592 Evaluate the following limits. ( lim _{x rightarrow 2}left(frac{1}{x-2}-frac{4}{x^{3}-2 x^{2}}right) ) 11
593 Evaluate the following limits. ( lim _{x rightarrow 1} frac{sqrt{5 x-4}-sqrt{x}}{x-1} ) 11
594 ا
log
(
3
+
x
)

log
(
3

3
)
=
k
,
the value of kis ا م
(2003)
r0
(b) 0
| سا
11
595 Show that the function ( boldsymbol{f}(boldsymbol{x})= ) ( sin (2 x+pi / 4) ) is decreasing on
( (3 pi / 8,5 pi / 8) )
11
596 4. If G(x) = -V25 – x? then lima
G(x)-G() has the value
x1 x 1
(1983 – 1 Mark)
(c) -√24
(b) 5
(d) none of these
11
597 Compute ( operatorname{Lim} frac{3^{x}-2^{x}}{x} ) 11
598 ( lim _{x rightarrow a}left(frac{x+1}{2 x+1}right)^{x^{2}}= ) 11
599 If ( lim _{x rightarrow 0^{+}} frac{2}{x^{1 / 5}}=a ) then ( frac{1}{a} ) is equal
to
11
600 ( lim _{n rightarrow infty} frac{(2 n-1)(3 n+5)}{(n-1)(3 n+1)}= )
A . 2
B. ( infty )
( c cdot 0 )
D. None of these
11
601 ( lim _{x rightarrow 0} frac{(27+x)^{1 / 3}-3}{9-(27+x)^{2 / 3}} ) equals:
A . ( -1 / 6 )
в. ( 1 / 6 )
( c cdot 1 / 3 )
D. ( -1 / 3 )
11
602 Find the maximum value of ( 2 x^{3}- )
( 24 x+107 ) in the interval ( [1,3] . ) Find the
maximum value of the same function in
[-3,-1]
11
603 the value of ( boldsymbol{x} stackrel{lim }{longrightarrow} mathbf{0} frac{sin boldsymbol{x}^{mathbf{0}}}{boldsymbol{x}} ) 11
604 ( lim _{x rightarrow pi / 4} frac{1-cot ^{3} x}{2-cot x-cot ^{3} x} ) is
A ( cdot frac{11}{4} )
B. ( frac{3}{4} )
( c cdot frac{1}{2} )
D. ( -frac{1}{2} )
11
605 ✓ f(x)-1
6.
Iff(1) =1,f|(1)=2, then lim v
i s
[2002]
x-1
VX-1
(a) 2
(6) 4
(c) 1
(d) 1/2
11
606 ( lim _{x rightarrow pi / 2} frac{cot x-cos x}{(pi-2 x)^{3}} ) equals
A. ( frac{1}{16} )
в.
( c cdot frac{1}{4} )
D.
11
607 Evaluate ( lim _{x rightarrow-2^{+}} frac{x^{2}-1}{2 x+4} ) 11
608 Calculate ( : lim _{x rightarrow 0}left(frac{1}{x^{2}}-frac{1}{sin ^{2} x}right) )
A. ( infty )
B. ( frac{1}{3} )
( c cdot-frac{1}{3} )
D. does not exist
11
609 The value of ( lim _{x rightarrow b} frac{sqrt{x-a}-sqrt{b-a}}{x^{2}-b^{2}} ) for
( b>a ) is
A ( cdot frac{1}{4 b sqrt{a-b}} )
В. ( frac{1}{4 b sqrt{b-a}} )
c. ( frac{1}{4 a sqrt{a-b}} )
D. ( frac{1}{b sqrt{b-a}} )
11
610 The value of ( lim _{x rightarrow 0} frac{(tan ({x}-1)) sin {x}}{{x}({x}-1)} )
is given by :
where ( {x} ) denotes the fractional part
function
( A cdot 1 )
B. ( tan 1 )
( c cdot sin 1 )
D. Does not exist
11
611 Approximately, what is ( lim _{x rightarrow infty} frac{sqrt{7} x^{2}+3 x-2}{x^{2}+5} ? )
A .2 .32
в. 2.43
( c .2 .54 )
D. 2.65
E . 2.76
11
612 ( lim _{x rightarrow 0} frac{1-cos ^{2} x}{x sin x cos x} ) is equal to
A . 1
B. ( frac{3}{5} )
( c cdot frac{3}{2} )
D.
11
613 Find the value of ( k ) so that the function ( f )
is contiuous at the indicated point. ( f(x)=begin{array}{ll}k x+1 & , x leq pi \ cos x & , x>piend{array} ) at ( x=pi )
11
614 ( lim _{x rightarrow infty} frac{(3 x-1)(2 x+5)}{(x-3)(3 x+7)} ) is equal to
( A cdot frac{1}{2} )
B. 2
( c cdot 0 )
D. none of these
11
615 Evaluate: ( lim _{x rightarrow 0} frac{(1+x)^{4}-1}{(1+x)^{3}-1} )
A ( cdot-frac{4}{3} )
B.
( c cdot frac{4}{3} )
D.
11
616 Find the following limit:
( lim _{x rightarrow a} frac{cos x-cos a}{x-a} )
11
617 Find ( mathbf{v} )
( lim _{x rightarrow 0} frac{x cos x-sin x}{x^{2} sin x} cdot ) is ( =-frac{1}{v} )
11
618 ( lim _{x rightarrow 0} frac{x e^{x}-log (1+x)}{x^{2}} ) equals 11
619 ( lim _{x rightarrow 0} frac{sin x^{n}}{(sin x)^{m}}(m<n) ) is equal to
A . 1
B.
c. ( frac{n}{m} )
D. None of he above
11
620 If ( f(x)=left{begin{array}{cc}x^{2} sin frac{1}{x} & x neq 0 \ 0 & x=0end{array} ) then R.H.D right.
is
( mathbf{A} cdot mathbf{1} )
B . –
( c )
D. None
11
621 The value of ( lim _{alpha rightarrow beta}left[frac{sin ^{2} alpha-sin ^{2} beta}{alpha^{2}-beta^{2}}right] ) is:
( mathbf{A} cdot mathbf{0} )
B.
c. ( frac{sin beta}{beta} )
D. ( frac{sin 2 beta}{2 beta} )
11
622 x →0
29. The value of lim ((sin x)/* + (1+x)sin x), where x > 0 is
(a) o (6) 1 (1) 1
(2006 – 3M, -1)
(d) 2
11
623 ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ccc}|boldsymbol{x}-mathbf{2}|+mathbf{1} & boldsymbol{i} boldsymbol{f} & boldsymbol{x}mathbf{2}end{array}right} )
If ( lim _{x rightarrow a} f(x) ) exists then
A ( . a=2 )
( mathbf{B} cdot a epsilon R )
( mathbf{c} cdot a epsilon R-{2} )
D ( cdot a epsilon R-{1,2} )
11
624 ( lim _{x rightarrow 0} frac{1-cos x}{x log (1+x)}= )
A .
B.
c. -1
D.
11
625 13. Let f(a) = g(a)=k and their nth derivatives
f”(a),8″(a) exist and are not equal for some n. Further if
tim f(a)g(x)-f(a)-g(a)f(x)+ f(a) – 4
a g(x) – f(x)
then the value of k is
[2003]
(a) 0 (6) 4 (c) 2
d) 1
11
626 The value of ( lim _{x rightarrow infty} frac{sqrt{1+x^{4}}-left(1+x^{2}right)}{x^{2}} ) is
A ( .-1 )
B.
( c cdot 2 )
D. none of these
11
627 Find ( : lim _{x rightarrow a} sqrt{x} ) 11
628 The value of ( lim _{x rightarrow infty} frac{x+cos x}{x+sin x} ) is
A . -1
B. 0
c.
D. None of these
11
629 ( lim _{x rightarrow pi / 2} frac{2 x-pi}{cos x} )
( A cdot-2 )
B. – –
( c )
D.
11
630 Find the value of ( k ) so that the function ( f )
is continuous at the indicated point. ( f(x)=begin{array}{cc}k x^{2} & , x leq 2 \ 3 & , x>2end{array} ) at ( x=2 )
A ( cdot k=frac{1}{4} )
B. ( _{k=frac{1}{2}} )
c. ( _{k=frac{3}{4}} )
D. ( k=1 )
11
631 14.
lim VI -cos 2(x-1)
(1998 – 2 Marks)
* 1 X-1
(a) exists and it equals 2
(b) exists and it equals –
c) does not exist because x-1 0
(d) does not exist because the left hand limit is not equal
to the right hand limit.
11
632 ЭТАТЕМЕNT-1 ( : lim _{x rightarrow alpha} frac{sin (f(x))}{x-alpha}, ) where
( f(x)=a x^{2}+b x+c, ) is finite and non-
zero, then ( lim _{x rightarrow alpha} frac{frac{e^{frac{1}{f(x)}}-1}{frac{1}{f(x)}} text { does not }}{e^{frac{1}{f(x)}+1}} )
exist. STATEMENT-2 : ( lim _{x rightarrow alpha} frac{f(x)}{x-alpha} ) can take
finite value only when it takes ( frac{0}{0} ) form.
A. STATEMENT-1 is True, STATEMENT-2 is True STATEMENT-2 is a correct explanation for STATEMEN
B. STATEMENT-1 is True, STATEMENT-2 is True STATEMENT-2 is NOT a correct explanation fo STATEMENT-1
C. STATEMENT-1 is True, STATEMENT-2 is False
D. STATEMENT-1 is False, STATEMENT-2 is True
11
633 Solve ( lim _{x rightarrow frac{pi}{2}} frac{2^{-cos x-1}}{xleft(x-frac{pi}{2}right)} ) 11
634 ( lim _{x rightarrow infty} 2 x(sqrt{x^{2}+1}-x)= )
( mathbf{A} cdot mathbf{1} )
B. ( 1 / 2 )
( c cdot 0 )
D. –
11
635 ( lim _{x rightarrow frac{pi}{2}} frac{left(frac{pi}{2}-xright) sec x}{operatorname{cosec} x}= )
( A )
B.
( c )
( D cdot underline{1} )
11
636 ( lim _{x rightarrow 0} frac{sqrt{1-cos 2 x}}{sqrt{2} x} ) is equal 11
637 ( lim _{x rightarrow 0} frac{sin left(x^{3}right)}{x} ) is equal to
A
B. 3
( c cdot 0 )
D. None of these
11
638 The value of ( lim _{x rightarrow 0} frac{(1-cos 2 x) sin 5 x}{x^{2} sin 3 x} ) is
A . ( 10 / 3 )
B. 3/10
c. ( 6 / 5 )
D. ( 5 / 6 )
11
639 If ( f(x)=frac{tan x}{tan x} ) then
A ( cdot lim _{x rightarrow 0} f(x) ) does not exists
B ( cdot lim _{x rightarrow 0} f(x) ) exists
c. ( f(x) ) has fundamental period ( pi )
D. all of these
11
640 The value of
( left(497 e^{-2}right) lim _{x rightarrow 0}left(tan left(frac{pi}{4}+xright)right)^{1 / x} )
11
641 Prove that:
( lim _{x rightarrow 0} frac{x^{2} sin frac{1}{x}}{sin x}=0 )
11
642 ( lim _{x rightarrow k}(x-[x]), ) where ( k ) is an integer
is equal to (where [.] denotes greatest integer function).
( mathbf{A} cdot mathbf{1} )
B.
( c cdot-1 )
D. Does not exist
11
643 37.
37. – ((n + >60 x2), con sequal M 2016
lim
(n+1)(n+2)…3n
n2n
is equal to:
[JEE M 2016]
n->
(6) 3 log 3-2
2
11
644 Evaluate: ( lim _{x rightarrow 20} frac{sqrt{x+5}+5}{sqrt{x+5}-5} )
( A )
B. 2
( c cdot 4 )
D. ( infty )
11
645 If ( f(x)=left|begin{array}{ccc}cos x & x & 1 \ 2 sin x & x^{2} & 2 x \ tan x & x & 1end{array}right|, ) then
( lim _{x rightarrow 0} frac{f^{prime}(x)}{x} )
A. Exists and is equal to -2
B. Does not exist
C. Exist and is equal to 0
D. Exists and is equal to 2
11
646 If ( boldsymbol{x}_{1}=sqrt{mathbf{3}} ) and ( boldsymbol{x}_{boldsymbol{n}+1}=frac{boldsymbol{x}_{boldsymbol{n}}}{mathbf{1}+sqrt{mathbf{1}+boldsymbol{x}_{boldsymbol{n}}^{2}}} )
( forall n epsilon N, ) then ( lim _{n rightarrow infty} 2^{n} x_{n} ) equal to
A ( cdot frac{3}{2 pi} )
в. ( frac{2}{3 pi} )
c. ( frac{2 pi}{3} )
D. ( frac{3 pi}{2} )
11
647 ( lim _{x rightarrow 2} frac{x^{8}-256}{x-2} ) 11
648 Solve:
( lim _{x rightarrow 0} frac{x}{3-sqrt{x+9}} )
11
649 Evaluate the following limits. ( lim _{x rightarrow-5} frac{2 x^{2}+9 x-5}{x+5} ) 11
650 Find:
( lim _{x rightarrow 0} frac{sin x-x+frac{1}{6} x^{3}}{x^{3}} )
11
651 ( lim _{x rightarrow 0} frac{3^{x}-2^{x}}{4^{x}-3^{x}} ) is equal to
( mathbf{A} cdot mathbf{1} )
B . – –
( c cdot 0 )
D. none of these
11
652 If ( lim _{x rightarrow frac{pi}{2}-} tan x=a ) then ( frac{1}{a} ) is equal
to
11
653 ( lim _{x rightarrow 0} frac{x-sin x}{x+cos ^{2} x}= )
( mathbf{A} cdot mathbf{0} )
B.
( c cdot-1 )
D.
11
654 within what respective limits must
( A / 2 ) lie when ( 2 sin A / 2= )
( +sqrt{1+sin A}-sqrt{1-sin A} )
11
655 Evaluate the Given limit:
( lim _{x rightarrow 0} frac{sin a x}{sin b x}, a, b neq 0 )
11
656 ( lim _{mathbf{x} rightarrow 2}left(frac{sqrt{1-cos {2(mathbf{x}-mathbf{2})}}}{mathbf{x}-mathbf{2}}right) )
A. does not exist
B. equals ( sqrt{2} )
c. equals ( -sqrt{2} )
D. equals ( frac{1}{sqrt{2}} )
11
657 ( lim _{x rightarrow frac{pi}{4}} frac{sqrt{1-sqrt{sin 2 x}}}{pi-4 x}=-frac{1}{m} . ) Find ( n ) 11
658 If ( cdot ) donotes fractional part of ( x, ) then
( lim _{x rightarrow 3+}{x}^{2} frac{sin (x-3)}{(x-3)} ) is equal?
( mathbf{A} cdot mathbf{0} )
в. ( frac{1}{2} )
( c cdot frac{1}{3} )
( D )
11
659 Evaluate ( lim _{x rightarrow 0} frac{sin ^{-1}[cos x]}{1+[cos x]} )
denotes the greatest integer function)
A . 0
B. –
( c . )
D. ( infty )
11
660 If ( lim _{x rightarrow infty}{(sqrt{x^{4}+a x^{3}+3 x^{2}+b x+2}-sqrt{a} )
is finite, then the value of ( a ) is?
A . 3
B. 5
( c cdot 2 )
D. Any real number
11
661 ( lim _{x rightarrow 0} frac{sin ^{-1} x-tan ^{-1} x}{x^{3}}= )
( A cdot 2 )
B.
( c cdot-1 )
D.
11
662 Assertion
If ( lim _{boldsymbol{x} rightarrow mathbf{0}} boldsymbol{f}(boldsymbol{x}) ) and ( lim _{boldsymbol{x} rightarrow mathbf{0}} boldsymbol{g}(boldsymbol{x}) ) exists finitely
( operatorname{then} lim _{x rightarrow 0} f(x) cdot g(x) ) exists finitely
Reason
If ( lim _{x rightarrow 0} f(x) cdot g(x) ) exists finitely then
( lim _{x rightarrow 0} f(x) cdot g(x)=lim _{x rightarrow 0} f(x) cdot lim _{x rightarrow 0} g(x) )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Assertion is incorrect and Reason is correct
11
663 ( lim _{x rightarrow 0}left(frac{x-sin x}{x}right) sin left(frac{1}{x}right) ) 11
664 ( lim _{x rightarrow infty} x sin left(frac{2}{x}right) ) is equal to
A ( cdot infty )
B.
( c cdot 2 )
D. ( frac{1}{2} )
11
665 ( lim _{x rightarrow 5} frac{sin ^{2}(x-5) tan (x-5)}{left(x^{2}-25right)(x-5)} ) is equal to
A. 1
B. ( frac{1}{10} )
( c cdot 0 )
( D )
11
666 ( lim _{x rightarrow pi / 2} frac{left(1-tan frac{x}{2}right)}{left(1+tan frac{x}{2}right)} frac{(1-sin x)}{(pi-2 x)^{3}} ) is equal
to –
A. 0
в. ( frac{1}{32} )
( c cdot alpha )
D.
11
667 ( boldsymbol{L}=lim _{boldsymbol{x} rightarrow 2} frac{boldsymbol{x}^{4}-boldsymbol{8} boldsymbol{x}}{sqrt{boldsymbol{x}^{2}+mathbf{2 1}-mathbf{5}}} ) 11
668 Solve: ( lim _{x rightarrow 2} frac{e^{x^{2}}-cos x}{x^{2}} ) 11
669 29. Let f: R
→ [0,-) be such that lim
f(x) exists and
5
im (f(x)) – 9
= 0. Then lim f(x) equals:
** Vr – 51
(a) 0 (6) 1 (c) 2 (d) 3
11
670 Assertion
If ( lim _{x rightarrow 0}left{f(x)+frac{sin x}{x}right} ) does not exist
then ( lim _{x rightarrow 0} f(x) ) does not exist.
Reason
( lim _{x rightarrow 0} frac{sin x}{x} ) exists and has value 1
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
11
671 ( lim _{x rightarrow infty} sqrt{x+sqrt{x+sqrt{x}}}-sqrt{x} ) is equal to
A . 0
B.
( c cdot log 2 )
D. ( e^{4} )
11
672 Let ( f(x)=left[begin{array}{cc}e^{x}, & x leq 0 \ |1-x| & x>0end{array}right], ) then which
one of the following statement is incorrect ?
A. continuous and differentiable at ( x=0 )
B. ( lim _{x rightarrow-infty} f(x)=0 )
c. one local maxima and one local minima
D. Decreasing function in (0,1)
11
673 ( lim _{x rightarrow 1}left{-x-frac{1}{x}right}, ) where ( {.} ) denotes the
fraction part function
( A ). is equal to 1
B. is equal to 0
c. Does not exist
D. None of these
11
674 ( lim _{n rightarrow infty}(1+x)left(1+x^{2}right)left(1+x^{4}right) dots(1+ )
( left.x^{2 n}right),|x|<1, ) is equal to
A ( cdot frac{1}{x-1} )
в. ( frac{1}{1-x} )
c. ( 1-x )
( D )
11
675 If ( l_{1}=frac{d}{d x}left(e^{sin x}right), l_{2}= )
( lim _{h rightarrow 0} frac{e^{sin (x+h)}-e^{sin x}}{h} ) and ( l_{3}= )
( int e^{sin x} cos x d x, ) then which one of the
following is correct?
A. ( l_{1} neq l_{2} )
в. ( frac{d}{d x}left(l_{3}right)=l_{2} )
( mathbf{c} cdot int l_{3} d x=l_{2} )
D. ( l_{2}=l_{3} )
11
676 The value of:
( lim _{n rightarrow 0} cos left(frac{x}{2}right) cos left(frac{x}{4}right) cos left(frac{x}{8}right) dots dots cos left(frac{x}{2^{n}}right) )
is
11
677 ( lim _{x rightarrow 0} frac{sin 3 x tan 4 x}{x sin 5 x}= )
( A . )
B. ( frac{5}{12} )
c. 0
D. ( frac{12}{5} )
11
678 Find the limits of
[
begin{array}{l}
frac{sqrt{boldsymbol{a}^{2}+boldsymbol{a} boldsymbol{x}+boldsymbol{x}^{2}}-sqrt{boldsymbol{a}^{2}-boldsymbol{a} boldsymbol{x}+boldsymbol{x}^{2}}}{sqrt{boldsymbol{a}+boldsymbol{x}}-sqrt{boldsymbol{a}-boldsymbol{x}}}, text { when } \
boldsymbol{x}=mathbf{0}
end{array}
]
11

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