Karnataka 1st PUC Mathematics Limits and Derivatives MCQ with Answers

GOVERNMENT OF KARNATAKA
DEPARTMENT OF SCHOOL EDUCATION (PRE-UNIVERSITY)
18TH CROSS, MALLESHWARAM, BENGALURU – 560 012
CHAPTERWISE MULTIPLE CHOICE QUESTIONS FOR COMPETATIVE EXAM
SUBJECT: MATHEMATICS – I PUC
NAME OF THE CHAPTER: LIMITS AND DERIVATIVES

 Limit of a Function:-
lim f ( x)  l . This means that when x is close to a , f (x) is close to l .
x a

 Left-Hand Limit (LHL):-
lim f ( x)  f (a  0)  l1 . This means that when x  a from the left, f (x) is close to l1 .
x a 

 Right-Hand Limit (RHL):-
lim f ( x)  f (a  0)  l 2 . This means that when x  a from the right, f (x) is close to l 2 .
xa 

NOTE: The limit exists when LHL = RHL = l .
 Algebra of Limits:-
If, lim f ( x)  f (a)  l and lim g ( x)  g (a)  m , where l and m are real numbers.
x a x a

 lim  f ( x)  g ( x)  f (a)  g (a)  l  m  If lim f ( x)   then lim f (1x )  0
x a xa x a

 lim  f ( x)  g ( x)  f (a)  g (a)  l  m g ( x)
 lim  f ( x)  lim f ( x)  lm
x a g ( x)
x a  x a 
 lim kf ( x)  kf (a)  k  l
x a
 lim f ( x)  lim f ( x)  l
 f ( x)  f (a) l x a xa
 If lim    , m  0
x a g ( x ) 
  g (a) m  If f ( x)  g ( x), x then lim f ( x)  lim g ( x) .
x a x a

 If f ( x)  g ( x)  h( x), x and lim f ( x)  lim h( x)  l then lim g ( x)  l .
x a x a x a

This result is known as the Sandwich (Squeeze) Theorem.

x a

xa

lim f ( g ( x))  f lim g ( x)  f (m) Provided f is continuous at g ( x)  m .

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MATHEMATICS CET MATERIAL

L-HOSPITAL’S Rule:-
If 𝑓(𝑥) and 𝑔(𝑥) are two continuous and differentiable functions at 𝑥 = 𝑎 and
𝐟(𝐱) 𝟎 ∞ 𝐟(𝐱) 𝐟 ′ (𝐱)
𝐥𝐢𝐦 𝐠(𝐱) = 𝟎 𝐨𝐫 ∞, Then 𝐥𝐢𝐦 𝐠(𝐱) = 𝐥𝐢𝐦 𝐠′ (𝐱)
𝐱→𝐚 𝐱→𝐚 𝐱→𝐚

[i.e., Differentiate the numerator and denominator separately and apply the rule
repeatedly until the indeterminate form is removed]
Limits at infinity:-

𝟏
As 𝑛 → ∞ , 𝒏 → 0  If 𝑎 > 𝑏 > 𝑐 > 0, lim (𝑎𝑛 + 𝑏 𝑛 + 𝑐 𝑛 )1/𝑛 = 𝑎.
𝑛→∞

0, ∣ 𝑎 ∣< 1, 𝑥 𝑛 −𝑎𝑛
 lim = 𝑛𝑎 𝑛−1 , 𝑛 ∈ Q.
 𝑛
lim 𝑎 = { 1, 𝑎 = 1, 𝑥→𝑎 𝑥−𝑎
𝑛→∞
∞, 𝑎 > 1. 𝑥 𝑚 −𝑎𝑚 𝑚
 lim 𝑥 𝑛−𝑎𝑛 = 𝑛 𝑎 𝑚−𝑛 .
1 𝑥→𝑎
 lim 𝑛𝑝 = 0, 𝑤ℎ𝑒𝑟𝑒 𝑝 > 0
𝑛→∞
1
 lim 𝑛𝑛 = 1.
𝑛→∞

 If 𝑎 > 0, 𝑏 > 0, lim (𝑎𝑛 + 𝑏 𝑛 )1/𝑛 = 𝑎.
𝑛→∞
𝑎0
, 𝑖𝑓 𝑛 = 𝑚,
𝑎0 𝑥𝑚 + 𝑏0
𝑎 𝑚−1
 lim
1𝑥 +..........+𝑎𝑚𝑛
= 0, 𝑚 < 𝑛,
𝑛→∞ 𝑏0 𝑥𝑛 + 𝑏 ∞, 𝑚 > 𝑛, 𝑎0 𝑏0 > 0,
1 𝑥𝑛−1 +..........+𝑏𝑛

{ −∞, 𝑚 > 𝑛, 𝑎0 𝑏0 < 0.
Trigonometric Limits:-

 lim
sin x
 lim
x
1 sin x 0 
x 0 
 lim 
x x 0 sin x x 0 x 180

 lim
tan x
 lim
x
1 1  cos x
x 0 x 0 tan x
 lim 0
x x 0 x
sin 1 x x 1  cos x 1
 lim  lim 1  1  lim 
x 0 x x 0 sin x x 0 x2 2
sin x  a  tan( x  a) 1  cos px p 2
 lim  lim 1  lim 
x a xa x  a xa x 0 x2 2
sin mx tan 1 mx sin 1 mx m 1  cos px p 2
 lim  lim  lim   lim  2
x 0 tan mx x 0 sin 1 nx x 0 tan 1 nx n x 0 1  cos qx q
sin x cos x
 lim  lim 0 cos(ax)  cos(bx) b 2  a 2
x  x x  x  lim 
x 0 x2 2
1 tan x  sin x 1
sin  
 x   lim x  sin  1   1  lim 
 lim   x 0 x3 2
x   1  x 
x
 
x

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MATHEMATICS CET MATERIAL

Logarithmic and Exponential Limits:-
log(1  x) ex 1 a x 1
 lim  lim log e x  lim 1  lim  log e a, 𝑎 > 0, 𝑎 ≠ 1
x 0 x x e x 0 x x 0 x

lim  f ( x)
x lim g ( X )log f ( x )
 1 
g ( x)
 e xa
lim 1    lim 1  x  x  e
1
 x a
x 
 x x 0

𝑥+𝛼 𝑥
 lim (𝑥+𝛽) = 𝑒 𝛼−𝛽
 
x

lim 1  x  x  lim 1    e 
1 𝑥→∞

x 0 x 
 x
Derivative of a Function:-
𝑑𝑦 𝑓(𝑥+ℎ)−𝑓(𝑥)
 Let 𝑦 = 𝑓(𝑥) is a real-valued function. Then at 𝒙 = 𝒄,, 𝑑𝑥 = lim (if the limit exists)
ℎ→0 ℎ
𝑑𝑦 𝑓(𝑥)−𝑓(𝑐)
 The derivative of a function is also defined as at 𝑥 = 𝑐, 𝑑𝑥
= lim 𝑥−𝑐
𝑥→𝑐

Algebra of Derivative of functions:-
Let 𝑢 and 𝑣 be differentiable functions of 𝑥 and let 𝑘 be a constant, then
𝑑 𝑑 𝑑𝑢
 (𝑘) = 0 and 𝑑𝑥 (𝑘𝑢) = 𝑘 𝑑𝑥 where 𝑘 is a constant.
𝑑𝑥
𝑑 𝑑𝑢 𝑑𝑣
 (𝑢 ± 𝑣) = 𝑑𝑥 ± 𝑑𝑥
𝑑𝑥
𝑑 𝑑𝑣 𝑑𝑢
 (𝑢 ∗ 𝑣) = 𝑢 𝑑𝑥 + 𝑣 𝑑𝑥 [Product rule]
𝑑𝑥
𝑑𝑢 𝑑𝑣
𝑑 𝑢 𝑣 −𝑢
 ( )= 𝑑𝑥 𝑑𝑥
, 𝑣 ≠ 0 [Quotient rule]
𝑑𝑥 𝑣 𝑣2

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MATHEMATICS CET MATERIAL

Think & Discuss MCQs
 1 1 
1. lim     Easy 
x 3 x  2
 x  3 
7 1 3
A) 6 B)5 C)4 D)7

log 1  ax 
2. lim
x 0
  Easy 
x
1 1
A)0 B) C)a D) 
a a

3. lim
1  x 2  1  x 2  (Average)
x 0 1  x 3  1  x 3

1 2 3
A)1 B)2 C)3 D) 2

x2  1
4. lim  (Average)
x 1 x  1

A)2 B)0 C)1 D) Doesn′t exist

5. The value of lim
x  KCET 2018 -  Easy 
x 0 x
A)1 B)-1 C)0 D) Doesn′ t exist

x4  x
6. lim  KCET 2025 - (Average)
x 1 x 1
1
A)0 B)2 C)7 D) Doesn′ t exist

3  y3  3
7. lim  KCET 2022 -  Easy 
y 0 y3
3 1
A) 2 3 B) C) 3 2 D) 2√3
2

e ax  e  bx
8. lim
x 0
  Easy 
x
𝑎
A) a  b B)𝑎 − 𝑏 C)𝑏 D) ab
3
8 x 2
9. lim  (Average)
x 0 x
1 −1
A) 0 B)1 C)12 D) 12

1  x  x 2 1
10. lim  (Average)
x0 x
1
A)0 B)1 C)2 D) 2

2026-27 MATHEMATICS CET MATERIAL Page 4 of 13

1  x3  1  x3
11. lim  (Average)
x0 x3
1
A)0 B)1 C)2 D)
2

3
1 x  3 1 x
12. lim  (Average)
x 0 x
1 2 3
A)1 B)2 C)3 D) 2

x
13. lim  (Average)
x 0 x4 2
1 −1
A)4 B) 2 C)3 D) 6

1 1 x  2
14. lim  (Difficult)
x 8 x 8
3 1 1
A)2 B)4 C)24 D) ∞

log(1  x)
15. lim  KCET 2013 -  Easy 
x 0 3x 1
A)log 𝑒 3 B)0 C)log 3 𝑒 D)1
x2 x  x
16. lim  KCET 2012 - (Difficult)
x 0 1  cos x

1 1
A)2log 𝑒 2 B)log 2 C)2 log 2 D)2

a  2 x  3x
17. lim  KCET 2012 -  Easy 
x 0
3a  x  2 x
2 1 3√3 2
A)3 B) C) 2 D) 3√3
√3

3.2 n1  4.5 n1
18. lim  KCET 2009 - (Average)
n  5.2 n  7.5 n

3 4 20
A)5 B)− 7 C)− 7 D) 0

sin x
19. lim  (Average)
x  x 
A)1 B)2 C)-1 D)−2
1
20. lim x sin    (Average)
x 0
x
1
A)0 B)1 C)2 D) Doesn’t exist

2026-27 MATHEMATICS CET MATERIAL Page 5 of 13

sin x
21. lim

  Easy 
x x
2

2  1
A)1 B) C) D)
 2 2
sin x 0
22. lim  (Average)
x 0 x
  180
A)1 B) C) D)
2 180 
2
23. lim x sin    KCET 2008 -  Easy 
x 0
x
1
A)  B)2 C)2 D)0

1  sin x
24. lim
 cos x
  Easy 
x
2

A)0 B)-1 C)1 D) Doesn’t exist
1  cos x
25. lim   Easy 
x x2
2

2 2 4 4
A) B) C) D)
2 4  2

1  cos 4
26. lim  (Average)
 0 1  cos 6

4 1 1
A)9 B)2 C)− 2 D)1

cos ecx  cot x
27. lim  (Average)
x 0 x
1 1
A)− 2 B)1 C)2 D)1

sec 2 x  2
28. lim
 tan x  1
  Easy 
x
4

A)3 B)1 C)0 D)2
tan( x 2  1)
29. lim  KCET 2007 -  Easy 
x 1  x  1
1 1
A)2 B) C)-2 D) 
2 2
sin x
30. lim
x 0 x 1  cos x 
  Easy 
1
A)0 B)2 C)1 D)-1

2026-27 MATHEMATICS CET MATERIAL Page 6 of 13

xe x  sin x
31. lim  KCET 2016 -  Easy 
x 0 x
A)3 B)1 C)0 D)2
x
sin 2  
32. lim 3  (Average)
x 0 x
1 1
A)3 B)9 C)9 D)0

sin 2 sin 3
33. lim  (Average)
 0 3 tan 4

2 3 1
A)3 B)2 C)2 D)2

sin 2  x   sin 2  x 
34. lim  A cos B then the values of A and B respectively
x0 x
A)1,2 B)2,1 C)1,1 D)2,2 (Average)
tan 2 x  x
35. lim  (Average)
x 0 3 x  sin x

1 1 1
A)2 B)2 C)− 2 D)4

cos 5  cos 7
36. lim   Easy 
 0 2
1 1
A)6 B)12 C)12 D)6

x 2 cos x
37. lim  (Average)
x 0 1  cos x

3 3
A)2 B)2 C)− 2 D)1

2 cos x  1
38. lim  KCET 2024 -  Easy 
x
 cot x  1
4

1 1
A)2 B)2 C) D)√2
√2

sin x
39. lim  (Average)
x 0 x 1  1 x
A)2 B)0 C)1 D)  1
sin x
40. lim  (Average)
x 0 x
A)1 B)-1 C) 0 D) Doesn’t exist

2026-27 MATHEMATICS CET MATERIAL Page 7 of 13

x
sin 2  
4 
41. lim  Easy 
x 0 x2
1 1
A)4 B)4 C)16 D)16

3sin x  sin 3 x
42. lim  (Average)
x 0 x3
A)4 B)-4 C)8 D)  8
tan x  sin x
43. lim   Easy 
x 0 x2
1 1
A)0 B)1 C)2 D)− 2

tan x  sin x
44. lim  (Average)
x 0 x3
1
A)0 B)1 C)2 D)2

tan 3 x  sin 3 x
45. lim  (Average)
x 0 x5
5 3 3 2
A)2 B)2 C)5 D)5

1  sin 
46. lim   Easy 
  
2 cos   
2 
1
A)1 B)-1 C)0 D)2

sin 2 x  a sin x
47. lim is (Average)
x 0 x3
1
A)2 B)-2 C)−1 D)2

48. lim 1  sin  x 
cot  x
 (Average)
x 1

1
A)0 B)1 C)e D)𝑒

 x 2  ax  3b 
49. lim    5 then a  b  KCET 2026 - (Average)
x 3
 x 3 

A)-2 B)2 C)3 D)4
log(1  ax)  log(1  bx )
50. The function f ( x)  is not defined at 𝒙 = 𝟎. The value which should
x
be assigned to f at 𝒙 = 𝟎 so that it is continuous at 𝒙 = 𝟎 is KCET 2009 -  Easy 

A) a  b B) a  b C)0 D) log a  log b

2026-27 MATHEMATICS CET MATERIAL Page 8 of 13

1  x n  1 is equal to
51. lim
x 0
 Easy 
x
A) n B)1 C)  n D) 0
sin x  
52. lim
x 0
 Average 
x
A)1 B)-1 C)0 D) Doesn' texist
53. If x  is the greatest integer function, then lim x  is  Easy 
x 0

A)0 B)2 C)-1 D) Doesn' texist
54. lim x  1 , where [. ] is the greatest integer function, is equal to  Easy 
x 1

A)1 B)2 C)0 D) Doesn' texist
xm 1
55. lim n
x 1 x  1
is equal to  Easy 

m m m2
A)1 B) C D)
n n n2

56. lim 1  2 x  x is
1

x 0
 Easy 
A)e B) e 2 C) e 3 D)  e

57. lim
 x  12 x  3 is (Average)
x 1 2x2  x  3
1 −1
A)10 B) 10 C)1 D)  1

log(3  x)  log(3  x)
58. lim  k Then k  (Average)
x 0 x
1 2 2

A)0 B) 3 C) 3 D) 3
ax  bx
59. lim
x 0
is  Easy 
x
a log a
A)0 B)1 C) log   D)
b log b
x
 3
60. lim 1   is  Easy 
x 
 x

D)  e
2 6
A) 1 B) e C) e
x b
 xa
61. lim   is (Average)
x  x  b
 

A)1 B) e ba C) e a b D) eb
2026-27 MATHEMATICS CET MATERIAL Page 9 of 13

x
 x 
62. lim 
x  1  x
  (Average)
 
A)0 B) e 1 C) e D) e 2
1
63. If 0  p  q then lim  q  p   n n n
(Average)
x 

A) e B) p C) q D) 0

12  22  32  ___  n 2
64. lim   Easy 
x  n3
1 1 1
A)1 B) C) D)
2 3 4
n(13  23  33  ___  n3 )
65. lim  (Average)
n 
1  2  3  ____  n 
2 2 2 2 3

27 16 4
A)16 B) C) 0 D)
27 9
13  23  33  ___  n3
66. lim   Easy 
x  n4
1 1 1
A)1 B) C) D)
2 3 4
 n n n 1
67. lim  2 2  2  2 2  ____    KCET 2024 - (Difficult)
n  n  1 n 2 n 3
 5n 
2

 
A) B) 𝑡𝑎𝑛−1 3 C) D) 𝑡𝑎𝑛−1 2
4 2

12  22  32  42  ___  n2   n  n
1

68. lim  (Average)
n   n  1 n  10  n  100 
1 2
A)0 B) C) D) 
3 3
1  2  3  ___  n
69. lim n  N is equal to  Easy 
n  n2
1 1
A)0 B)2 C) 1 D)
4
n
 13 2 3 33 n3 
70.  2r  1  x then lim  2  2  2  ____  2   KCET 2019 - (Difficult)
n  x
r 1  x x x 

1 1
A)4 B)4 C) D) 11
2

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 2 2 
71. lim  n.sin cos  KCET 2010 -  Easy 
n 
 3n 3n 
 2 
A) B) C) 1 D
6 3 3
f 2  f x 
72. If f ( x)  9  x then lim
2
is  Easy 
x 2 x2
A)−2 B)2 C) 4 D)  4
f x   2
73. If the function f (x) satisfies lim   then lim f ( x)   Easy 
x1 x2  1 x 1

A)2 B)3 C)1 D)0
 x 2  1, if x  2
74. If f ( x)   then lim f ( x)  lim f ( x)  KCET 2026 - (Average)
 x  1, if x  2 x 1 x 2

A)3 B)5 C)7 D)9
 x 2  1, if 0  x  2
75. Let f ( x )   the quadratic equation whose roots are xlim f ( x) and
 2 x  3, if 2  x  3  2

lim f ( x) (Average)
x  2

A) x 2  6 x  9  0 B) x 2  7 x  8  0
C) x 2  14 x  49  0 D) x 2  10 x  21  0

 sin x 
 , x   0
76. If f ( x)   x  , where 
. denotes the greatest integer function then
 0 , x   0

lim f ( x) is equal to (Average)
x 0

A)1 B)0 C)−1 D) Doesn' texist

77. If f : R  R is continuous such that f ( x  y )  f ( x) f ( y ) x, y  R and f (1)  2
then f 100    Easy 
A)100 B)50 C)200 D) 2100

78. Let the function satisfy the equation f ( x  y )  f ( x) f ( y ) x, y  R where f (0)  0. If
f (5)  3 and f ' (0)  2, then f ' (5)  KCET 2026 - (Difficult)
A)1 B)0 C)−1 D) 6

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 ax 2  bx  c 
79. Statement 1: lim  2
x 1 cx  bx  a 
 1 where a  b  c  0
 

1 1
   1
Statement 2: lim  x 2   KCET 2021 - (Average)
 x2  4
x 2

 
A) Only Statement 2 is true B) Only Statement 1 is true
C) Both Statements1and2 are true D) Both Statements 1 and 2 are false
sin x cos x tan x
f ( x)
80. If f ( x)  x 3 x2 x then lim is KCET 2012 - (Difficult)
x0 x2
2x 1 x

A)0 B)3 C)2 D)1
cos x 1 0
81. If f ( x)  0 2 cos x 3 then lim f ( x) is KCET 2021- (Average)
x 
0 1 2 cos x

A)0 B)3 C)−1 D)1

1  
82. If f ( x)  x sin x then f   is equal to  Easy 
2
1
A)0 B)1 C)−1 D)2

1 dy
83. If y  x  , then at x  1 is  Easy 
x dx
1 1
A)1 B) C) D) 0
√2 2
x4
84. If f ( x)  then f 1 1 is  Easy 
2 x
5 4
A)4 B)5 C) 1 D) 0

1
1
85. If y  x 2 then dy is  Easy 
1 dx
1 2
x
 4x  4x 1 x2 4x
A) B) C) D)
x  1
2 2
x2  1  4x  
x2 1

sin x  cos x dy
86. If y  , then at x  0 is  Easy 
sin x  cos x dx
1
A)-2 B)0 C) D) Doesn' t exist
2

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sin( x  9) dy
87. If y  , then at x  0 is  Easy 
cos x dx
A) cos9 B) sin 9 C) 0 D) 1
x2 x100
88. If f ( x)  1  x   ______  , then f ' (1) is equal to  Easy 
2 100
1
A) B)100 C) 0 D) Doesn' t exist
100
89. If f ( x)  x100  x 99  ______  x  1, then f ' (1) is equal to  Easy 
A) 5050 B) 5049 C) 5051 D)50051
90. If f ( x)  1  x  x 2  x 3  x 4  _______  x 99  x100 , then f ' (1) is equal to (Average)
A) 150 B)  50 C)  150 D)50
THINK & DISCUSS MCQS
1 2 3 4 5 6 7 8 9 10
A C C D D C D A C D
11 12 13 14 15 16 17 18 19 20
B C A C C A B C C A
21 22 23 24 25 26 27 28 29 30
B C D A C A C D A B
31 32 33 34 35 36 37 38 39 40
C D D D B C A B C D
41 42 43 44 45 46 47 48 49 50
D A C D B C C C C A
51 52 53 54 55 56 57 58 59 60
A D D D B B B C C C
61 62 63 64 65 66 67 68 69 70
C B C C C D D A B A
71 72 73 74 75 76 77 78 79 80
B C A B D D D D D B
81 82 83 84 85 86 87 88 89 90
C B D A A A A B A D

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