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.

Limits And Derivatives Questions

List of limits and derivatives Questions

Question NoQuestionsClass
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
3If ( 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
5Evaluate ( : l t_{x rightarrow 0} x operatorname{cosec} x )11
6Evaluate :
[
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
8Evaluate:
( 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
10Find 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
11Find the value of
( lim _{x rightarrow 3} frac{sqrt{16 x^{2}+112}-16}{3 x^{2}-15 x+18} )
11
12solve the limit
( lim _{x rightarrow 3} frac{2}{x-3} )
( A cdot 2 )
B. 3
( c cdot 4 )
D. Does not exist
11
13The 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
14Solve
( lim _{x rightarrow 0}left(frac{1^{x}+2^{x}+3^{x}+ldots+n^{x}}{n}right)^{1 / 2}= )
11
15Solve ( lim _{x rightarrow 0} frac{sqrt{2+x}-sqrt{2}}{x} )11
16The 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
17If 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
19Evaluate the Given limit: ( lim _{x rightarrow 3}(x+3) )11
20The 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
22Evaluate: ( 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
23Solve ( : 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
25The 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
26Evaluate:
( lim _{x rightarrow infty} frac{2+cos ^{2} x}{x+2007} )
11
27Evaluate:
( lim _{x rightarrow frac{pi}{3}} frac{sin left(frac{pi}{3}-xright)}{2 cos x-1} )
11
28If ( 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
29Evaluate: ( 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
33Let ( 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
34Evaluate:
( 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
36Solve:
( 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
37State 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
39Solve:
( 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
41Evaluate the following limit ( lim _{x rightarrow 0} frac{(1+x)^{6}-1}{x} )11
42Evaluate :
( 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
44Evaluate :
( lim _{n rightarrow infty} frac{[1 . x]+[2 . x]+[3 . x]+ldots ldots+[n . x]}{n^{2}}, ) where
denotes the greatest integer function.
11
45Which 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
46Evaluate:-
( lim _{t rightarrow-3} frac{6+4 t}{t^{2}+1} )
11
47Evaluate :
( lim _{x rightarrow infty} frac{3 x^{2}+4 x+5}{4 x^{2}+7} )
11
48Show that ( lim _{x rightarrow 0} frac{e^{1 / x}-1}{e^{1 / x}+1} ) does not exist.11
49Prove that
( boldsymbol{L}=lim _{boldsymbol{n} rightarrow infty}left(1+frac{4}{n}right)^{3 n}=mathbf{1 2} )
11
5025.
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
5226. 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
53Find ( 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
563. 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
58Evaluate the following limits. ( lim _{x rightarrow 0}(cos x+sin x)^{1 / x} )11
59Evaluate the following question. ( lim _{x rightarrow a} frac{(x)^{3 / 2}-(a)^{3 / 2}}{x-a} )11
6023. 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
64Find ( 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
65V2 (1-сos 2x)
(1991 – 2 Marks)
The value of lim
x>0
(a) 1
(c) 0
(b)-1
(d) none of these
11
66Evaluate: ( 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
68If 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
71If ( 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
72Find the following limit:
( lim _{x rightarrow 4} frac{sqrt{1+2 x}-3}{sqrt{x}-2} )
11
73The value of ( lim _{x rightarrow 0}left[frac{tan x}{x}+frac{sin x}{x}right] ) is11
74Evaluate:
( lim _{x rightarrow infty}left(frac{2 x-3}{sqrt{x^{2}-1}}right) )
11
75Find 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
76If ( 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
78Evaluate: ( lim _{x rightarrow frac{pi}{2}}left(2 x tan x-frac{pi}{cos x}right) )11
79fig 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
81Evaluate 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
83Find 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
85Evaluate ( lim _{x rightarrow 0} frac{a^{sin x}-1}{sin x} )11
86The 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
87Solve 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
88Evaluate 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
91Evaluate ( : lim _{x rightarrow a} frac{x^{m}-a^{m}}{x^{n}-a^{n}}= )11
92Solve:
( 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
94What 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
95Evaluate:
( 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
99Solve:
( 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
101Evaluate ( lim _{x rightarrow 0} log frac{sin x}{x} )11
102The 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
104Evaluate 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
108Consider 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
109Evaluate the following limits. ( lim _{x rightarrow 0} frac{(1+x)^{6}-1}{x} )11
110Evaluate 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
111Evaluate:
( lim _{x rightarrow 0} tan ^{-1} frac{a}{x^{2}}, ) where ( a in mathbb{R} )
11
1121P +2P
3P +….. + n P
4.
lim
n->00
nP+1
[2002]
1
1
1
p+1
1-P
Pp-1
P +2
11
113JEE 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
114Evaluate the following limits. ( lim _{x rightarrow 1} frac{sqrt{x+8}}{sqrt{x}} )11
115Evaluate the following limits. ( lim _{x rightarrow 0} frac{(a+x)^{2}-a^{2}}{x} )11
116Evaluate ( : lim _{x rightarrow a} frac{x-a}{|x-a|} )
( A )
B.
( c .-c )
D. Does not exist
11
117Evaluate: ( lim _{x rightarrow 2} frac{x-2}{sqrt{x+2}-2} )11
118Evaluate ( lim _{x rightarrow I}left(tan frac{pi x}{4}right)^{tan frac{pi x}{2}} )11
119Arrange 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
1202
lim
oo 11-n?
n
2 + …. +
1-₂2
1,2
1-n²s
is equal to
(1984 – 2 Marks)
(d) none of these
11
121Find 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
122Solve:
( 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
127The 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
132If ( 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
133Solve ( 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
134The 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 ) Find11
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
141The 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
143Solve ( 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
1461.
SuolellVETU
Find the derivative of sin (x2 + 1) with respect to x from first
principle.
(1978)
Tidalad
11
147The 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
152Find 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
153Evaluate: ( 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
156Evaluate ( : lim _{x rightarrow 0}left(frac{1-cos x}{x^{2}}right) )11
157Let ( 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
158If ( lim _{x rightarrow 2} frac{x^{n}-2^{n}}{x-2}=80 ) and ( n in N, ) find
( mathbf{n} )
11
159This 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
160Solve: ( lim _{x rightarrow 2} frac{left(1-3^{x}-4^{x}+12^{x}right)}{sqrt{(2 cos x+7)}-} )11
161For ( 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
162Evaluate: ( 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
164Let ( 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
166Evaluate ( boldsymbol{x} stackrel{lim }{boldsymbol{a}} frac{boldsymbol{x}^{boldsymbol{7}}-boldsymbol{a} boldsymbol{7}}{boldsymbol{x}-boldsymbol{a}} )11
167Evaluate: ( lim _{x rightarrow 1} frac{1-x^{frac{-1}{3}}}{1-x^{frac{-2}{3}}} )11
168The 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
172Evaluate: ( 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
173Solve:
( 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
17536. 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
176Solve ( 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
178If ( 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
179Prove 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
184Find :
( 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
186Let ( 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
188The fraction ( frac{sqrt{mathbf{3 x – a}}-sqrt{x+a}}{x-a} )
becomes ( frac{mathbf{0}}{mathbf{0}} ) when ( boldsymbol{x}=boldsymbol{a} )
11
189Evaluate ( lim _{x rightarrow a} frac{x^{7}-a^{7}}{x-a} )11
190Solve the following limit ( lim _{h rightarrow 0} frac{sqrt{h}}{sqrt{16+sqrt{h}-4}} )11
191Find 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
192Evaluate ( 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
193Evaluate: ( 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
197The 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
200Evaluate: ( 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
203Evaluate the following limit :
( lim _{x rightarrow 0} frac{sin ^{2} 3 x}{x^{2}} )
( A )
B. 3
( c cdot 9 )
D.
11
204If ( |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
205The 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
208Compute ( : 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
210Find 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
211Evaluate the following limits. ( lim _{x rightarrow sqrt{3}} frac{x^{4}-9}{x^{2}+4 sqrt{3} x-15} )11
212Evaluate:
( 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
214The 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
215The 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
217Find the absolute maximum and
minimum values of the function f given by ( f(x)=cos ^{2} x+sin x, x in[0, pi] )
11
218Apply the Imitsto given expression
( lim _{x rightarrow 0}(((x+1)(x+2)(x+3)(x+4)) )
11
21934.
(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
220Evaluate 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
221Evaluate ( 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
223Find 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
225Evaluate 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
228If ( 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
230Evaluate : ( 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
232Evaluate the given limit:
( lim _{x rightarrow 0} frac{a x+x cos x}{b sin x} )
11
233If ( 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
234Evaluate:
( lim _{x rightarrow 0} frac{sin x}{sqrt{x^{2}}}= )
( mathbf{A} cdot mathbf{1} )
B. –
( c cdot 0 )
D. doesn’t exist
11
235Find 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
2361.
lim
vi-cos 2x
v
is
[2002]
(a) 1
(c) zero
(b)-1
d) does not exist
11
237The 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
238Evaluate ( lim _{x rightarrow pi / 2} frac{cos x}{pi-2 x} )11
239Evaluate: ( 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
241Solve:
( lim _{x rightarrow 3} frac{x^{2}-9}{x-3} )
11
242Assertion
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
244Find the intervals in which the following functions are increasing or decreasing ( f(x)=10-6 x-2 x^{2} )11
245Solve ( lim _{x rightarrow 1}left[frac{x-1}{x^{2}-x}-frac{1}{x^{3}-3 x^{2}+x}right] )11
246Find 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
247Evaluate ( 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
249Evaluate 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
251Find 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
253Using 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
257If ( 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
26016.
, 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
261The 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
262Value 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
264Find ( 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
265Find 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
268Evaluate ( 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
270Calculate 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
271Evaluate ( : lim _{x rightarrow 3} frac{sqrt[4]{x}-sqrt[4]{3}}{sqrt[3]{x}-sqrt[3]{3}} )11
272Find 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
273Evaluate the following limits. ( lim _{x rightarrow 2} frac{x^{3}-8}{x^{2}-4} )11
274STATEMENT-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
275Evaluate 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
278For 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
279Evaluate: ( 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
280Evaluate :
( lim _{x rightarrow 2} frac{x^{2}-4}{x+3} )
( mathbf{A} cdot mathbf{0} )
B.
( c cdot-1 )
D. None of these
11
281Evaluate 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
283If ( 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
284Find ( 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
285If ( 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
287Find the following limit:
( lim _{x rightarrow 1} frac{sqrt[3]{x}-1}{sqrt{x}-1} )
11
288If ( 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
289The 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
292Evaluate ( lim _{x rightarrow 0} frac{tan x}{x} )11
293f ( 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
295The 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
296Solve:
( lim _{x rightarrow 27} frac{x^{2 / 3}-9}{x-27} )
11
297Evaluate 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
298Evaluate ( 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
300The 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
301The 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
302If ( 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
304Evaluate the following limits. ( lim _{x rightarrow a} frac{sin sqrt{x}-sin sqrt{a}}{x-a} )11
305For 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
306Find the following limit:
( lim _{x rightarrow 1} frac{sqrt{1+x}-sqrt{1-x}}{sqrt[3]{1+x}-sqrt[3]{1-x}} )
11
307Find 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
308FHEMENT-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
311Let 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
31223. 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
313Calculate 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
316Solve ( 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
318Let ( 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
319The 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
321Find ( lim _{x rightarrow 2^{+}} f(x), ) where ( f(x)= )
( left{begin{array}{ll}4 x+9, & x2end{array}right. )
11
322Solve:
( 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
324Find the ( lim _{x rightarrow 0} frac{tan x sin x}{sin ^{3} x} )11
325Solve:
( 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
327The value of ( lim _{x rightarrow 0} frac{2 sin ^{2} 3 x}{x^{2}} ) is11
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
329If ( 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
330Evaluate the following limits. ( lim _{x rightarrow 0} 9 )11
331The 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
333Evaluate the following limits. ( lim _{x rightarrow 2}(3-x) )11
334Evaluate: ( 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
335Examine 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
33824.
(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
339Evaluate
( 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
341If ( 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
342Prove that:
( lim _{x rightarrow frac{pi}{2}} frac{tan 2 x}{x-frac{pi}{2}} )
11
343Evaluate the following question. ( lim _{x rightarrow 0} sin x )11
344Find ( lim _{x rightarrow pi / 4} frac{sin x-cos x}{x-pi} )11
345If ( 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
34615 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
348Value 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
349If ( 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
353If ( 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
354Find it; ( lim _{x rightarrow 0}left(frac{x^{2}+x+1}{x+1}-a x-6right)=4 )11
355If ( f(x)=frac{sin ([x] pi)}{x^{2}+x+1}, ) when
: ( ] ) denotes
the greatest integer function, then
11
356If ( 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
357Write the value of ( lim _{x rightarrow 0} frac{sin x}{sqrt{1+x}-1} )11
358If ( 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
35934
[JEE M 2015)
x-
lim (1 – cos2x) (3 + cosx) is equal to:
x tan 4x
(2) 2 (6) 2 (0) 4
(d) 3
11
360Let ( 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
362If ( 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
364im 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
365If ( 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
36720. 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 to11
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
370If ( 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
371Let ( {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
375Evaluate 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
376Evaluate the following question.
( lim _{x rightarrow 0} x^{2}-3 )
11
377The 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
3781 – cos {2(x-2)}
[2011]
27.
lim
x 2
x-2
(b) equals – V2
(a) equals v2
(C) equals
(d) does not exist
11
379Evaluate: ( 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
381Evaluate
( 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
383Evaluate: ( 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
385If ( {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
387Evaluate the following limits. ( lim _{x rightarrow 3}left(frac{1}{x-3}-frac{2}{x^{2}-4 x+3}right) )11
388Evaluate : ( quad operatorname{lit}_{x rightarrow 3} frac{|x-3|}{x-3} )11
389If ( 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
390If ( 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
391Evaluate ( 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
393If ( 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
394Evaluate ( : lim _{x rightarrow infty} frac{5 x^{2}+1}{x^{2}+10} )11
395Evaluate
( 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
397If ( 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
400Evaluate the following limits. ( lim _{x rightarrow 4} frac{x^{3}-64}{x^{2}-16} )11
401If ( 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
402The 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
403If ( 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
405Evaluate ( lim _{x rightarrow 0} frac{x tan x}{(1-cos x)} )11
406Evaluate ( 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
407Solve ( : lim _{x rightarrow 0} frac{sin 4 x}{sin x} )11
408The 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
411The 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
412If ( 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
413State Yes or No. ( lim _{x rightarrow 3^{+}} frac{x}{[x]} ) is equal to ( lim _{x rightarrow 3^{-}} frac{x}{[x]} )11
414If ( 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
415Evaluate the following limits. ( lim _{x rightarrow pi / 2} frac{2^{-cos x}-1}{xleft(x-frac{pi}{2}right)} )11
416Solve ( 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
418The 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
419Solve ( lim _{x rightarrow-4} frac{x+4}{x^{2}-x-20} )11
420Prove 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
421ff ( (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
422Find
( lim _{x rightarrow pi / 2} frac{sin (cos x) cos x}{sin x-operatorname{cosec} x} )
11
423If ( 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
42632.
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
430Evaluate the limit
( lim _{x rightarrow frac{pi}{6}} frac{cot ^{2} x-3}{cos e c x-2} )
11
431The 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
432Evaluate:-
( 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
434Assertion
( 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
435Evaluate 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} ) equals11
438Solve ( lim _{x rightarrow a} frac{x sqrt{x}-a sqrt{x}}{x-a} )11
439Evaluate ( : lim _{x rightarrow 0} frac{1-cos x}{x} )11
440Solve:
( 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
442Evaluate ( 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
445Compute ( lim _{x rightarrow 2+}([x]+x) ) and ( lim _{x rightarrow 2-}([x]+ )
( boldsymbol{x}) )
11
44639.
lim cotx – cos x
**(1-2x)3
equals:
[JEEM 2017
N
11
447Evaluate ( : 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
450The 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
451Find ( 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
452Evaluate ( int_{0}^{200 pi} sqrt{1+cos x} d x )11
453What 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
455If ( 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
456Evaluate 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
458Find the limits of ( frac{e^{m x}-e^{m a}}{x-a}, ) when ( x= )
( a )
11
459Evaluate: ( lim _{x rightarrow 0} frac{6^{x}-1}{sqrt{3-x}-sqrt{3}} )11
460If ( 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
461The 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
462If ( 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
463Solve
( 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
465Evaluate: ( lim _{x rightarrow 0} frac{e^{alpha x}-e^{beta x}}{sin alpha x-sin beta x} )11
466The 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
46827. 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
469Given, ( 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
470If ( lim _{x rightarrow 0} frac{sqrt{1+x}-sqrt{1-x}}{2 x}=frac{1}{a}, ) then ( a ) is
equal to
11
471Prove 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
472If ( lim _{x rightarrow 2^{+}} frac{x-3}{x^{2}-4}=a ) then ( frac{1}{a} ) is equal
to
11
473Evaluate ( 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
475If ( 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
479c
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
480Calculate 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
482Solve :
( lim _{x rightarrow 2} frac{sqrt{1+sqrt{2+x}-sqrt{3}}}{x-2} )
11
483Evaluate the following limits. ( lim _{x rightarrow 1} frac{x^{2}+1}{x+1} )11
484Evaluate ( lim _{x rightarrow 0} frac{sin x}{|x|} )
A .
B. 0
c. -1
D. does not exist
11
485Solve: ( lim _{x rightarrow 2} frac{a^{tan x}-a^{sin x}}{tan x-sin x}, a>0 )11
486Consider 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
487The 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
488Solve:
( lim _{x rightarrow 27} frac{left(x^{1 / 3}+3right)left(x^{1 / 3}-3right)}{x-27} )
11
48912. 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
494Find ( c )
( lim _{x rightarrow 0} frac{log tan left(frac{pi}{4}+a xright)}{sin b x}=c frac{a}{b} )
11
49516. Find lino {tan(Tt/4+x)}”/>
(1993 – 2 Marks)
11
496Find: ( lim _{x rightarrow 0^{+}} frac{1}{x} )
( A cdot O )
B. ( -infty )
( C cdot infty )
D. does not exist
11
497Defined ( 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
501Evaluate ( : lim _{x rightarrow 1}left(frac{1}{1-x}-frac{3}{1-x^{3}}right) )11
502Evaluate: ( lim _{x rightarrow 0} x^{2} )11
503Evaluate ( : lim _{x rightarrow 1} frac{sqrt{5 x-4}-sqrt{x}}{x^{3}-1} )11
504If ( lim _{x rightarrow 3} frac{x^{n}-3^{n}}{x-3}=108, ) find the value of
( mathbf{n} )
11
505Find the value of ( lim _{x rightarrow 0} frac{sin 2 x}{2 x^{2}+x} )11
506Find 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} ) Find11
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
510The 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
513If ( 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
514Calculate 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
515If ( 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
516Answer 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
517If ( f(x)=left{begin{array}{l}x: x0end{array}right. )
A .
B.
( c cdot 2 )
D. does not exist
11
51833. 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
519ff ( 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
520If ( lim _{x rightarrow 2^{-}} frac{x-3}{x^{2}-4}=a ) then ( frac{1}{a} ) is equal
to
11
521Evaluate ( lim _{x rightarrow 0} frac{sin x}{x} )11
522Evaluate the following question.
( lim _{x rightarrow 2} 2 x+5 )
11
523Assertion
( 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
524If
( 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
5268.
(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
529Solve:
( lim _{x rightarrow 0}left(left[x^{2}right]-[x]^{2}right) )
11
530Evaluate :
( lim _{x rightarrow infty} x^{2}(sqrt{frac{x+2}{x}}-sqrt[3]{frac{x+3}{x}}) )
11
531Let ( 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
532s)
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
533Evaluate 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
534Find 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
536The 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
537Solve:
( lim _{x rightarrow 0} frac{(1-cos 3 x)}{x sin 2 x} )
11
538If 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
539Evaluate 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
542The 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
54416.
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
545Evaluate ( lim _{x rightarrow 0} f(x), ) where ( f(x)= )
( left{begin{array}{ll}|x|, & x neq 0 \ 0, & x=0end{array}right. )
11
546Evaluate
( lim _{x rightarrow 0} frac{sqrt{2}-sqrt{1+cos x}}{sin ^{2} x} )
11
547Find the limit ( lim _{x rightarrow 0} frac{x}{sqrt{1+x}-1} )11
548Evaluate 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
5511+ 24 +34 +…n
1+23 +33 +…n
10.
lim
n->
– lim
n->00
[2003]

can
be () Zero(a)
11
5525. Use
the
formula
lima’ – 1 – Ina
to
find
x->0
x
2* 1
*-0 (1+x)/2-1
lim

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

is
The value of limo
x
[2003]
xsin x
(a) o
(b) 3
(C) 2
(d) 1
11
554Solve ( lim _{x rightarrow 3} frac{x^{2}-9}{x^{3}-6 x^{2}+9 x+1} )11
555Find the value of ( lim _{x rightarrow 1} e(1+sin pi x)^{cot pi x} )11
556The 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
557Solve ( lim _{x rightarrow 0} frac{sin 4 x}{sin 2 x} )11
558Evaluate 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 . ) Find11
560Solve
( lim _{x rightarrow 0} frac{tan 8 x}{sin 2 x} )
11
561If ( 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
563Let ( 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
565Evaluate ( 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
567Find 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
5683.
Evaluate: lim (a +
(a+h)? sin(a+h)-a-sina
Lh 0
h
(1980)
11
569Evaluate: ( 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
571The 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
573Calculate the following limits. ( lim _{x rightarrow-3} frac{x+3}{sqrt{x+4}-1} )11
574Find the value of ( lim _{x rightarrow infty} frac{sqrt{x}}{sqrt{x}+3} )11
575Find 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
578Let ( 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
579The 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
580The 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
581What 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
583Find ( 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
584Evaluate ( : lim _{x rightarrow 2} frac{3 x^{2}-x-10}{x^{2}-4} )11
585– is
[2003]
an
(a) oo
(c) 0
32
11
586Check 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
588The 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
589Evaluate: ( 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
590Solve;
( lim _{x rightarrow 0} frac{(1+x)^{6}-1}{(1+x)^{2}-1} )
11
591The 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
592Evaluate the following limits. ( lim _{x rightarrow 2}left(frac{1}{x-2}-frac{4}{x^{3}-2 x^{2}}right) )11
593Evaluate the following limits. ( lim _{x rightarrow 1} frac{sqrt{5 x-4}-sqrt{x}}{x-1} )11
594ا
log
(
3
+
x
)

log
(
3

3
)
=
k
,
the value of kis ا م
(2003)
r0
(b) 0
| سا
11
595Show that the function ( boldsymbol{f}(boldsymbol{x})= ) ( sin (2 x+pi / 4) ) is decreasing on
( (3 pi / 8,5 pi / 8) )
11
5964. 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
597Compute ( 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
599If ( 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
602Find 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
603the 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
607Evaluate ( lim _{x rightarrow-2^{+}} frac{x^{2}-1}{2 x+4} )11
608Calculate ( : 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
609The 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
610The 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
611Approximately, 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
613Find 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
615Evaluate: ( 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
616Find the following limit:
( lim _{x rightarrow a} frac{cos x-cos a}{x-a} )
11
617Find ( 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}} ) equals11
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
620If ( 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
621The 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
622x →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
62513. 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
626The 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
627Find ( : lim _{x rightarrow a} sqrt{x} )11
628The 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
630Find 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
63114.
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
632ЭТАТЕМЕNT-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
633Solve ( 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 equal11
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
638The 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
639If ( 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
640The value of
( left(497 e^{-2}right) lim _{x rightarrow 0}left(tan left(frac{pi}{4}+xright)right)^{1 / x} )
11
641Prove 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
64337.
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
644Evaluate: ( lim _{x rightarrow 20} frac{sqrt{x+5}+5}{sqrt{x+5}-5} )
( A )
B. 2
( c cdot 4 )
D. ( infty )
11
645If ( 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
646If ( 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
648Solve:
( lim _{x rightarrow 0} frac{x}{3-sqrt{x+9}} )
11
649Evaluate the following limits. ( lim _{x rightarrow-5} frac{2 x^{2}+9 x-5}{x+5} )11
650Find:
( 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
652If ( 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
654within what respective limits must
( A / 2 ) lie when ( 2 sin A / 2= )
( +sqrt{1+sin A}-sqrt{1-sin A} )
11
655Evaluate 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
658If ( 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
659Evaluate ( lim _{x rightarrow 0} frac{sin ^{-1}[cos x]}{1+[cos x]} )
denotes the greatest integer function)
A . 0
B. –
( c . )
D. ( infty )
11
660If ( 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
662Assertion
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
668Solve: ( lim _{x rightarrow 2} frac{e^{x^{2}}-cos x}{x^{2}} )11
66929. 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
670Assertion
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
672Let ( 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
675If ( 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
676The 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
678Find 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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