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Topper’s Package Mathematics - XI Limits
100
1. % TYPE BASED QUESTIONS
1.2
5xx 0
log(1 3x )limx(e 1)
is equal to :
(a)35 (b)
53
(c)35
(d)53
2. The value of x 2
2x 0
e log(1 x) (1 x)limx
is
equal to :(a) 0 (b) –3(c) –1 (d) infinity
3. 1
2
2x tan 3
tan x 2tanx 3limtan x 4tanx 3
equals :
(a) 1 (b) 2(c) 0 (d) 3
4.x 0
(2 x)sin(2 x) 2sin2limx
is equal to :
(a) sin 2 (b) cos 2(c) 1 (d) 2cos2 sin2
5. The value of 3x 6
x 2
e 1limsin(2 x)
is :
(a)32 (b) 3
(c) –3 (d) –1
6.x sinx
x 0
e elimx sinx
is equal to:
(a) –1 (b) 0(c) 1 (d) None of these
7.1
3x 0
sin x xlimx cos x
is equal to :
(a) 1/2 (b) 1/3(c) 1/6 (d) 1/12
8. 2x 0
8sin x x cos xlim3tan x x
is equal to :
(a) 3 (b) 2(c) –1 (d) 4
9. If 2 1/x
x 0lim[1 x log(1 b )]
= 22bsin , b > 0 and
( , ] , then the value of is :
(a) 4
(b) 3
(c) 6
(d) 2
10. The value of :
1 1 1 1
0
cosec (sec ) cot (tan ) cot cos(sin )lim
is(a) 0 (b) –1(c) –2 (d) 1
11. If f (4) = 4, f (4) = 1 then lim( )
x
f xx
4
22
is equal
to(a) 0 (b) 1(c) –1 (d) none of these
12. limlog
x
e xx
xx
RSTUVW0 2
1 1b g is equal to
(a)14
(b) 12
(c) 1 (d) none of these13. If ( ) 2, ( ) 1, ( ) 1, ( ) 2 f a f a g a g a then
( ) ( ) ( ) ( )lim
x a
g x f a g a f xx a is :
(a) –5 (b)15
(c) 5 (d) none of these14. If f(x), g(x) be differentiable functions and
f(1) = g (1) = 2 then
1
(1) ( ) ( ) (1) (1) (1)lim( ) ( )
x
f g x f x g f gg x f x is equal to
(a) 0 (b) 1(c) 2 (d) none of these
15. limx
x x
x x is equal to
0
3 24 3
(a) 1 (b) -1(c) 0 (d) none of these
LIMITSUnit11
LimitsTopper’s Package Mathematics - XI
101
16. limx
xx x e
x
0
2
2
1d i is equal to :
(a) 1 (b) 0
(c)12 (d) none of these
17. The value of limx0
2 2x x
x x
sin
sin is
(a) a log 2 - 2 (b) 1/2 log 2 - 2(c) log 2 - 2 (d) none of these
18. The value of Lim x xx xx
x
1 1 log is :
(a) 1 (b) -1(c) 2 (d) none of these
19. If Lim x a xxx
0 3
2sin sin is finite then the value of a
is :(a) –2 (b) –1(c) 2 (d) none of these
20. If Lim x ax ax a
9 9
9 then all possible values of a
can be :(a) 5 (b) 1(c) 3 (d) – 1
21. The value of Limx /2 (cosx)cosx is equal to :
(a) 1 (b) -1
(c) log 2 (d) none of these
22. Lt x x
xx
3 2
3 3
9....... :
(a) 1 (b)16
(c)13 (d) none of these
23. The value of lim ....x
nx x x nx
is
1
2
1:
(a) n (b)n 1
2
(c)n n( )1
2(d)
n n( )12
24. The value of limcos cos )
x
xx
0 4
1 1b g is :
(a) 1/8 (b) 1/2(c) 1/4 (d) none of these
25. The value of
40
cos sin ) coslimx
x xx
is
(a) 1/5 (b) 1/6(c) 1/4 (d) 1/2
26. The value of limln( )
h
In h hh
0 2
1 2 2 1b g is :
(a) 1 (b) –1(c) 0 (d) none of these
27. If limx a
x a
x aa xx a
1, then the value of a is
(a) 1 (b) 0(c) e (d) none of these
28. The integer n, for which0
(cos 1)(cos )limx
nx
x x ex
is a finite non zero number-(a) 1 (b) 2(c) 3 (d) 4
29.1/2 1/3
20
(cos ) (cos )limsinx
x xx
is :
(a) 1/6 (b) –1/12(c) 2/3 (d) 1/3
2. /, – , 0 × , 0°, °BASED QUESTIONS
30. If [x] denotes the greatest integer less than orequal to x for any real number x. Then,
n
[n 2]limn
is equal to :
(a) 0 (b) 2(c) 2 (d) 1
31.32 3
54 4 4n
x 1 x 1limx 1 x 1
equals
(a) 1 (b) 0(c) –1 (d) none of these
32. 2nlim [ x 2x 1 x]
is equal to :
(a) (b)12
(c) 4 (d) 1
33.2x
2x 1limx 2x 1
is equal to :
(a) 2 (b) –2(c) 1 (d) –1
34.n 2
x
2 (n 5n 6)lim(n 4)(n 5)
is equal to :
(a) 0 (b) 1(c) (d)
Topper’s Package Mathematics - XI Limits
102
35. If f(x)= 2x sinx
x cos x
, then x
lim f(x) is euqal to:
(a) 0 (b) (c) 1 (d) None of these
36. For x > 0 , sin x
1/xx
1lim (sinx)x
is equal to
(a) 0 (b) –1(c) 1 (d) 2
37. limn
n n
n na ba b
, where a > b > 1, is equal to :
(a) –1 (b) 1(c) 0 (d) none of these
38. The value of lim . .tan/ /
/ /x
x x
x xe ee e
xx
is
1 1
1 11
:
(a) 1 (b) 0(c) –1 (d) none of these
39. limsin !
, ,n
pn nn
p is equal to
2
10 1b g
:
(a) 0 (b) (c) 1 (d) none of these
40. The value of Limx
x x xx
4 2
31
1sin( / )
| |
is :
(a) 0 (b) 1(c) –1 (d) does not exist
41. The value of Limx0
1x
xFHG
IKJ
tan
is equal to :
(a) 0 (b) 1(c) e (d) none of these
42. The value of the Limx [13 + 23+ 33 + ......+n3]/n4
is equal to :
(a)13 (b) 1
(c)12 (d) none of these
43. If f(x) = x x
x x
sincos2 then Lim
x f(x) is equal to
(a) 0 (b) (c) 1 (d) none of these
44. If limx
xx
ax b
FHG
IKJ
3
211
2b g then
(a) a = 1, b = 1 (b) a = 1, b = 2(c) a = 1, b = –2 (d) none of these
45. The value of limsin
| |x
xx
x
x
FHG
IKJ
F
H
GGGG
I
K
JJJJ
2 1
1 is :
(a) 0 (b) 1(c) –1 (d) none of these
46. The value of lim /
x
xx equals
1:
(a) 0 (b) 1(c) e (d) e–1
47. The value of lim cos sinx
xx x
is
FHG
IKJ
FHG
IKJ
4 4 :
(a)2
(b)4
(c) 1 (d) none of these
48.3
4 4 4
1 8lim ...1 1 1x
nn n n
is :
(a) 1/4 (b) 1/8(c) 1/2 (d) none of these
49. The value of4 4 4 3 3 3
5 5x n
1 2 3 .... n 1 2 3 ... nlim – limn n
is
(a) zero (b)14
(c)15 (d)
130
3. 1 BASED QUESTIONS
50. The value of 1/ 1/ 1/
1 2 ......
nxx x xn
x
a a aLtn =...
(a) 0 (b) 1(c) a1 a2 a3........an (d) none of these
51. If 0 < x < y, then lim( ) /
n
n n ny x
1 is equal to
(a) e (b) x(c) y (d) none of these
52. If limx
x
x xe
F
HGIKJ 1 2
22 then :
(a) 1 2,(b) 2 1,(c) 1, tanany real cons t(d) 1
53. The value of Limx
2 32 5
2
2
8 32
xx
x
FHG
IKJ
is equal to :
(a) e–2 (b) e–4
LimitsTopper’s Package Mathematics - XI
103
(c) e–8 (d) 1
54. If Lim x Kx
x
11( sin )cot
then value of k is :(a) –1 (b) 0(c) 1 (d) none of these
55. The value of Limx0
1/tan
4
xx
is :
(a) e (b) e2
(c) e3 (d) 156. If [x] denotes the greatest integer less than or
equal to x, then,
lim.....
n
x x x n x
n
1 2 32 2 2 2
3 equals :
(a)x2
(b)x3
(c)x6 (d) 0
57. The value of lim tan
x
xx
122b g
is equal to :
(a) e2/ (b) e1/
(c) e2/ (d) e1/
58. If a, b, c, d are positive then limx
c dx
a bx
FHG
IKJ 1 1
(a) ed b/ (b) ec a/
(c) e c d a b b g/ (d) e
59. The value of lim sinsinx a
x axa
is
FHG
IKJ
1
:
(a) e asin (b) e atan
(c) e acot (d) 1
60.
21/
20
2limxx x
x
e ex
is :
(a) e1/2 (b) e1/4
(c) e1/3 (d) e1/12
61. If f(x) = x xx x
x2
25 3
2
FHG
IKJ then lim
x f(x) is equal to
(a) e4 (b) e3
(c) e2 (d) none of these
4. LHL AND RHL NEWTON LEEBNIT’Z RULESERIES BASED QUESTIONS
62. For a R (the set of all real numbes), a 1,a a a
a 1n
(1 2 ... n )lim(n 1) (na 1) (na 2) ... (na n)
=
160
than, a is equal to :(a) 5 (b) 7
(c) 152 (d)
172
63.[x]
x 0lim( 1)
, where [] denotes the greatest inte-ger function is equal to :(a) 0 (b) 1(c) –1 (d) does not exit
64. If : R R is defined by f(x) [x 3] [x 4] forx R , then
–x 3lim f(x) is equal to where [ ]
denote the greatest integer function:
(a) – 2 (b) –1(c) 0 (d) 1
65.
2sec x
22
2x4
f(t)dt
limx
16
equals :
(a)8 f(2)
(b)2 f(2)
(c)2 1f
2
(d) 4f(2)
66. If f : R R be a differentiable function having
f(2)= 6 1f (2)48
. Then
f(x)3
6x 2
4t dt
limx 2
is equal
to :(a) 18 (b) 12(c) 36 (d) 24
67. The value of x
2 3x 0
(1 e )sin xlimx x
is equal to :
(a) –1 (b) 0(c) 1 (d) 2
68. If f (x) =
sin[x] [x] 0[x]0, [x] 0
where, [x] denotes the greatest integer lessthan or equal to x, then
x 0lim f(x)
is equal to :
(a) 1 (b) 0(c) –1 (d) None of these
69. The value of Limx0
(1 cos2 )2
xx
is :
(a) 1 (b) –1(c) 0 (d) does not exist
Topper’s Package Mathematics - XI Limits
104
70. If Limx0
ae b x cex x
x x cossin
= 2 then :
(a) a = 1, b = 1, c = 1 (b) a = 1, b = 2, c = 1,(c) a = 1, b = 2, c = 2 (d) a = 2, b = 1, c = 1
71. The value of limsin
x
x x x
x
F
H
GGG
I
K
JJJ0
3
56 is
(a) 0 (b) 1
(c)160 (d)
1120
72. The value of lim | |sinx
xx
:
(a) is equal to –1 (b) is equal to 1(c) is equal to (d) does not exist
73. The value of lim
/
x
xx e ex
x
0
1
2
1 12
b g
(a) 1124e
(b) 1124e
(c) 24e
(d) none of these
74. lim sin sin
x
t t
x y
a
y
a
xe dt e dt
F
HIKzz0
1 2 2
is equal to (where a
is a contant.)
(a) e ysin2(b) sin2y e ysin2
(c) 0 (d) none of these
75.
0
3 | |lim7 5| |x
x xx x
is :
(a) 2 (b)16
(c) 0 (d) does not exist
76. Let ( ) sgn(sgn(sgn )).f x x Then 0limx ( )f x is
(a) 1 (b) 2(c) 0 (d) none of these
77. Let and be the distinct roots of ax2 + bx + c
= 0, then 2
2x 0
1– cos(ax bx c)lim(x – )
is equal to:
(a) 0 (b)2
2a ( )2
(c) 21 ( )2 (d)
22– a ( )
2
78. If 2x
2x 0
a blim 1x x
= e2, then the values of a
and b, are :(a) aR, b = 2 (b) a = 1, bR(c) aR, bR (d) a = 1 and b = 2
79. Let f (a) = g(a) = k and their nth derivativesnf (a) , ng (a) exist and are not equal for some n.
Further if
( ) ( ) ( ) ( ) ( ) ( )lim( ) ( )x a
f a g x f a g a f x g ag x f x = 4, then
the value of k is :(a) 2 (b) 1(c) 0 (d) 4
INTEGER TYPE QUESTIONS
1. 2 2lim ( 8 3 4 3)x
x x x x
2.0
sinlim1 1x
xx x
is equal to
3.
lim cosm
m
xm
4. Let , R be such that 2
0
sin( )lim 1sinm
x xx x
.
Then 6( + ) equals
5. If 2 3
sin cos tan
( )2 1 1
x x x
f x x x xx
, then 20
( )limx
f xx
is
6. The value of 2
20
1 1lim
9 3x
x
x
is
7. 20
cos sinlimtanx
x x xx x
is equal to
8.cosec
0
1 tanlim1 sin
x
x
xx
is equal to
9. The value of lim1
n
nn
xx
where x < –1 is
10. The value of 2
00
coslim
x
n
tdt
x
is
LimitsTopper’s Package Mathematics - XI
105
1. (A), 2. (B), 3. (B), 4. (D), 5. (C), 6. (C), 7. (C),
8. (A), 9. (D), 10. (C), 11. (B), 12. (D), 13. (C), 14. (C),
15. (D), 16. (C), 17. (D), 18. (C), 19. (A), 20. (B), 21. (D),
22. (B), 23. (C), 24. (A), 25. (B), 26. (B), 27. (A), 28. (C),
29. (B), 30. (C), 31. (B), 32. (D), 33. (B), 34. (A), 35. (C),
36. (C), 37. (B), 38. (A), 39. (A), 40. (B), 41. (C), 42. (D),
43. (C), 44. (C), 45. (A), 46. (C), 47. (B), 48. (D), 49. (C),
50. (C), 51. (C), 52. (C), 53. (C), 54. (D), 55. (B), 56. (B),
57. (C), 58. (A), 59. (C), 60. (D), 61. (A), 62. (B), 63. (D),
64. (C), 65. (A), 66. (A), 67. (C), 68. (D), 69. (D), 70. (B),
71. (D), 72. (D), 73. (D), 74. (A), 75. (D), 76. (D), 77. (B),
78. (B), 79. (D)
1. (2) 2. (1) 3. (1) 4. (7) 5. (1) 6. (3) 7. (2)
8. (1) 9. (1) 10. (1)
INTEGER TYPE QUESTIONS
