Limits JEE Main & Advanced Mathematics PYQs with Solutions

1

Let _ x 2 ( (x-2))(rx^2 + (p-2)x - 2p) (x-2)^2 = 5 for some r, p R . If the set of all possible values of q , such that the roots of the equation rx^2 - px + q = 0 lie in (0, 2) ,

JEE Main 2026 Solution View →

2

The value of _ x 0 ( x^2 ^2 x x^2 - ^2 x ) is:

JEE Main 2026 Solution View →

3

Let f(x) = _ y 0 (1 - (xy)) (xy) y^3 . Then the number of solutions of the equation f(x) = x , x R is :

JEE Main 2026 Solution View →

4

The product of all possible values of , for which _ x 0 ( 1 - ( x) (( +1)x) (( +2)x) ^2(( +1)x) ) = 2 , is:

JEE Main 2026 Solution View →

5

If _ x 2 (x^3 - 5x^2 + ax + b) ( x-1 - 1) _e(x-1) = m , then a + b + m is equal to :

JEE Main 2026 Solution View →

6

The value of _ x 0 _ e ( (e x) (e² x ) (e¹⁰ x ) ) e²-e^ 2 x is equal to

JEE Main 2026 Solution View →

7

If _ x 0 e ^ ( a -1) x +2 ~b x+( c -2) e ^ -x x x- _ e (1+x) =2 , then a ²+ b ²+ c ² is equal to :

JEE Main 2026 Solution View →

8

Let [ ] denote the greatest integer function and f(x)= _ n 1 n ³ _ k =1 ^ n [ k ² 3^ x ] . Then 12 _ j =1 ^ f( j ) is equal to _ _ _ _ .

JEE Main 2026 Solution View →

9

Let f: R (0, ) be a twice differentiable function such that f(3)=18, f^ (3)=0 and f^ (3)=4 . Then _ x 1 ( _ e ( f(2+x) f(3) )^ 18 (x-1)² ) is equal to :

JEE Main 2026 Solution View →

10

Let and be the real numbers such that _ x 0 1 x^3 ( 2 ₀^x 1 1-t^2 d t+ x x )=2 . Then the value of + is _______

JEE Advanced 2025 Solution View →

11

Let x₀ be the real number such that e^ x₀ +x₀=0 . For a given real number , define g(x)= 3 x e^x+3 x- e^x- x 3 (e^x+1 ) for all real numbers x . Then which one of the following sta

JEE Advanced 2025 Solution View →

12

Given below are two statements : Statement I : _ x 0 ( ⁻¹ x+ _e 1+x 1-x -2 x x^5 )= 2 5 Statement II : _ x 1 ( x ^ 2 1- x )= 1 e ^2 In the light of the above statements, choose the

JEE Main 2025 Solution View →

13

If the function f(x)= ( x)- ( x) x- x is continuous at x =0 , then f(0) is equal to ________

JEE Main 2025 Solution View →

14

_ x 0⁺ (5(x)^ 1 3 ) _e (1+3 x^2 ) ( ⁻¹ 3 x )^2 (e^ 5(x)^ 4 3 -1 ) is equal to

JEE Main 2025 Solution View →

15

If _ x 1⁺ (x-1)(6+ (x-1))+ (1-x) (x-1)^3 =-1 , where , R , then + is equal to

JEE Main 2025 Solution View →

16

If Lim _ x 0 ( x x )^ 1 x^2 =p , then 96 _e p is equal to ______

JEE Main 2025 Solution View →

17

If _ x 0 (2 x)+a (4 x)-b x^4 is finite, then (a+b) is equal to :

JEE Main 2025 Solution View →

18

For , , , R , if _ x 0 x^2 x+( -1) e^ x^2 2 x- x =3 , then + - is equal to:

JEE Main 2025 Solution View →

19

If _ t 0 ( ₀^1(3 x+5)^ t d x )^ 1 t = 5 e ( 8 5 )^ 2 3 , then is equal to ________

JEE Main 2025 Solution View →

20

Let [t] be the greatest integer less than or equal to t. Then the least value of ( p N ) for which ( _ x 0⁺ (x ( [ 1 x ]+ [ 2 x ]+ + [ p x ] )-x^2 ( [ 1 x^2 ]+ [ 2^2 x^2 ]+ + [ 9^2

JEE Main 2025 Solution View →

21

Let f(x)= _ n _ r =0 ^ n ( (x / 2^ r+1 )+ ^3 (x / 2^ r+1 ) 1- ^2 (x / 2^ r+1 ) ) . Then _ x 0 e ^x- e ^ f(x) (x-f(x)) is equal to

JEE Main 2025 Solution View →

22

_ x 0 cosec x ( 2 ^2 x+3 x - ^2 x+ x+4 ) is:

JEE Main 2025 Solution View →

23

_ x (2 x^2-3 x+5 )(3 x-1)^ x 2 (3 x^2+5 x+4 ) (3 x+2)^x is equal to :

JEE Main 2025 Solution View →

24

If _ x ( ( e 1- e ) ( 1 e - x 1+x ) )^x= , then the value of _ e 1+ _ e equals :

JEE Main 2025 Solution View →

25

Let k R . If _ x 0^+ ( ( k x)+ x+x)^ 2 x =e^6 , then the value of k is

JEE Advanced 2024 Solution View →

26

Let S be the set of all ( , ) R R such that _ x (x^2 ) ( _e x )^ ( 1 x^2 ) x^ ( _e(1+x) )^ =0 . Then which of the following is (are) correct?

JEE Advanced 2024 Solution View →

27

Let the function f:[1, ) R be defined by f(t)= array cc (-1)^ n+1 2, & if t=2 n-1, n N , (2 n+1-t) 2 f(2 n-1)+ (t-(2 n-1)) 2 f(2 n+1), & if 2 n-1 < t < 2 n+1, n N . array . Define

JEE Advanced 2024 Solution View →

28

Let f(x) be a continuously differentiable function on the interval (0, ) such that f(1)=2 and t x t¹⁰ f(x)-x¹⁰ f(t) t^9-x^9 =1 for each x>0 . Then, for all x>0, f(x) is equal to

JEE Advanced 2024 Solution View →

29

_ x 2 ( _ x^3 ^ ( / 2)^3 ( (2 t^ 1 / 3 )+ (t^ 1 / 3 ) ) d t (x- 2 )^2 ) is equal to

JEE Main 2024 Solution View →

30

_ x 0 e-(1+2 x)^ 1 2 x x is equal to

JEE Main 2024 Solution View →

31

Let _ n ( n n^4+1 - 2 n (n^2+1 ) n^4+1 + n n^4+16 - 8 n (n^2+4 ) n^4+16 . .+ + n n^4+n^4 - 2 n n^2 (n^2+n^2 ) n^4+n^4 ) be k , using only the principal values of the inverse trigon

JEE Main 2024 Solution View →

32

If = _ x 0⁺ ( e ^ x - e ^ x x - x ) and = _ x 0 (1+ x)^ 1 2 x are the roots of the quadratic equation a x^2+b x- e =0 , then 12 _ e ( a + b ) is equal to__________

JEE Main 2024 Solution View →

33

The value of _ x 0 2 ( 1- x 2 x [3] 3 x [10] 10 x x^2 ) is

JEE Main 2024 Solution View →

34

_ n (1^2-1 )(n-1)+ (2^2-2 )(n-2)+ + ((n-1)^2-(n-1) ) 1 (1^3+2^3+ +n^3 )- (1^2+2^2+ +n^2 ) is equal to :

JEE Main 2024 Solution View →

35

Let f:(- , )- 0 R be a differentiable function such that f^ (1)= _ a a^2 f ( 1 a ) . Then _ a a(a+1) 2 ⁻¹ ( 1 a )+a^2-2 _e a is equal to

JEE Main 2024 Solution View →

36

Let a >0 be a root of the equation 2 x^2+x-2=0 . If _ x 1 a 16 (1- (2+x-2 x^2 ) ) (1- a x)^2 = + 17 , where , Z , then + is equal to_______

JEE Main 2024 Solution View →

37

Let f be a differentiable function in the interval (0, ) such that f(1)=1 and _ t x t^2 f(x)-x^2 f(t) t-x =1 for each x>0 . Then 2 f(2)+3 f(3) is equal to _______

JEE Main 2024 Solution View →

38

Let f(x)= ₀^x (t+ (1-e^ ) ) d t, x R . Then, _ x 0 f(x) x^3 is equal to

JEE Main 2024 Solution View →

39

If _ x 1 (5 x+1)^ 1 / 3 -(x+5)^ 1 / 3 (2 x+3)^ 1 / 2 -(x+4)^ 1 / 2 = m 5 n (2 n )^ 2 / 3 , where gcd ( m , n )=1 , then 8 ~m +12 n is equal to______

JEE Main 2024 Solution View →

40

Let f x = x − 1 , x is even , 2 x , x is odd , x ∈ N . If for some a ∈ N , f f f a = 21 , then lim x → a − x 3 a − x a , where t denotes the greatest integer less than or equal to

JEE Main 2024 Solution View →

41

Let x denote the fractional part of x and f x = cos − 1 1 − x 2 sin − 1 1 − x x − x 3 , x ≠ 0 . If L and R respectively denotes the left hand limit and the right hand limit of f x

JEE Main 2024 Solution View →

42

Let f : → R → 0 , ∞ be strictly increasing function such that lim x → ∞ f 7 x f x = 1 . Then, the value of lim x → ∞ f 5 x f x − 1 is equal to

JEE Main 2024 Solution View →

43

If lim x → 0 a x 2 e x − b log e 1 + x + c x e − x x 2 sin x = 1 , then 16 a 2 + b 2 + c 2 is equal to ______.

JEE Main 2024 Solution View →

44

lim x → 0 e 2 sin x - 2 sin x - 1 x 2

JEE Main 2024 Solution View →

45

Let a be the sum of all coefficients in the expansion of ( 1 – 2 x + 2 x 2 ) 2023 ( 3 - 4 x 2 + 2 x 3 ) 2024 and b = lim x → 0 ∫ 0 x log 1 + t t 2024 + 1 d t x 2 . If the equations

JEE Main 2024 Solution View →

46

Let f : - π 2 , π 2 → R be a differentiable function such that f 0 = 1 2 , If lim x → 0 x ∫ 0 x f ( t ) d t e x 2 - 1 = α , then 8 α 2 is equal to :

JEE Main 2024 Solution View →

47

Let the slope of the line 45 x + 5 y + 3 = 0 be 27 r 1 + 9 r 2 2 for some r 1 , r 2 ∈ R . Then lim x → 3 ∫ 3 x 8 t 2 3 r 2 x 2 - r 2 x 2 - r 1 x 3 - 3 x d t is equal to ______.

JEE Main 2024 Solution View →

48

lim x → π 2 1 x - π 2 2 ∫ x 3 π 2 3 cos 1 t 3 d t is equal to

JEE Main 2024 Solution View →

49

If lim x → 0 3 + α sin x + β cos x + log e ( 1 - x ) 3 tan 2 x = 1 3 , then 2 α - β is equal to :

JEE Main 2024 Solution View →

50

If a = lim x → 0 1 + 1 + x 4 - 2 x 4 and b = lim x → 0 sin 2 x 2 - 1 + cos x , then the value of a b 3 is :

JEE Main 2024 Solution View →

51

Let f : 0 , 1 → ℝ be the function defined as f x = n if x ∈ [ 1 n + 1 , 1 n ) where n ∈ ℕ . Let g : 0 , 1 → ℝ be a function such that &#87

JEE Advanced 2023 Solution View →

52

If lim x → 0 e a x - cos ( b x ) - c x e - c x 2 1 - cos ( 2 x ) = 17 , then 5 a 2 + b 2 is equal to

JEE Main 2023 Solution View →

53

If α > β > 0 are the roots of the equation a x 2 + b x + 1 = 0 , and lim x → 1 α 1 - cos x 2 + b x + a 2 ( 1 - α x ) 2 1 2 = 1 k 1 β - 1 &#94

JEE Main 2023 Solution View →

54

lim x → 0 1 - cos 2 ( 3 x ) cos 3 ( 4 x ) sin 3 ( 4 x ) log e 2 x + 1 5 is equal to

JEE Main 2023 Solution View →

55

lim n → ∞ 2 1 2 - 2 1 3 2 1 2 - 2 1 5 . . . . 2 1 2 - 2 1 2 n + 1 is equal to

JEE Main 2023 Solution View →

56

lim x → ∞ 3 x + 1 + 3 x - 1 6 + 3 x + 1 - 3 x - 1 6 x + x 2 - 1 6 + x - x 2 - 1 6 x 3

JEE Main 2023 Solution View →

57

Let f , g and h be the real valued functions defined on ℝ as f x = x x , x ≠ 0 1 , x = 0 , g x = sin x + 1 x + 1 , x ≠ - 1 1 , x = - 1 and h x = 2 x - f x , where

JEE Main 2023 Solution View →

58

Let x = 2 be a root of the equation x 2 + p x + q = 0 and f x = 1 - cos x 2 - 4 p x + q 2 + 8 q + 16 x - 2 p 4 , x ≠ 2 p 0 , x = 2 p . Then lim x → 2 p + f x where &#18

JEE Main 2023 Solution View →

59

The value of lim n → ∞ 1 + 2 - 3 + 4 + 5 - 6 + … + ( 3 n - 2 ) + ( 3 n - 1 ) - 3 n 2 n 4 + 4 n + 3 - n 4 + 5 n + 4 is

JEE Main 2023 Solution View →

60

The set of values of a for which lim x → a x - 5 - 2 x + 2 = 0 , where, ζ denotes the greatest integer less than or equal to ζ is equal to

JEE Main 2023 Solution View →

61

lim t → 0 1 1 sin 2 t + 2 1 sin 2 t + 3 1 sin 2 t . . . . . . n 1 sin 2 t sin 2 t is equal to

JEE Main 2023 Solution View →

62

If β = lim x → 0 e x 3 - 1 - x 3 1 3 + 1 - x 2 1 2 - 1 sin x x sin 2 x , then the value of 6 β is

JEE Advanced 2022 Solution View →

63

Let α be a positive real number. Let f : ℝ → ℝ and g : α , ∞ → ℝ be the functions defined by f x = sin π x 12 and g x = 2 log e x

JEE Advanced 2022 Solution View →

64

If lim x → 0 α e x + β e - x + γ sin x x sin 2 x = 2 3 , where α , β , γ ∈ R , then which of the following is NOT correct?

JEE Main 2022 Solution View →

65

lim x → 0 x + 2 cos x 3 + 2 x + 2 cos x 2 + 3 sin x + 2 cos x x + 2 3 + 2 x + 2 2 + 3 sin x + 2 100 x is equal to

JEE Main 2022 Solution View →

66

Let f : ℝ → ℝ be a function defined as f x = a sin π x 2 + 2 - x , a ∈ ℝ , where t is the greatest integer less than or equal to t . If lim x &#8

JEE Main 2022 Solution View →

67

Let β = lim x → 0 α x - e 3 x - 1 α x e 3 x - 1 for some α ∈ ℝ . Then the value of α + β is:

JEE Main 2022 Solution View →

68

lim x → π 4 8 2 - cos x + sin x 7 2 - 2 sin 2 x is equal to

JEE Main 2022 Solution View →

69

If lim n → ∞ n 2 - n - 1 + n α + β = 0 then 8 α + β is equal to

JEE Main 2022 Solution View →

70

The value of lim x → 1 x 2 - 1 sin 2 π x x 4 - 2 x 3 + 2 x - 1 is equal to:

JEE Main 2022 Solution View →

71

If lim x → 1 sin 3 x 2 - 4 x + 1 - x 2 + 1 2 x 3 - 7 x 2 + a x + b = - 2 , then the value of a - b is equal to

JEE Main 2022 Solution View →

72

Let t denote the greatest integer ≤ t and t denote the fractional part of t . Then integral value of α for which the left hand limit of the function f x = 1 + x + α

JEE Main 2022 Solution View →

73

Let a be an integer such that lim x → 7 18 - 1 - x x - 3 a exists, where t is greatest integer ≤ t . Then a is equal to

JEE Main 2022 Solution View →

74

lim x → 0 cos ( sin x ) - cos x x 4 is equal to

JEE Main 2022 Solution View →

75

lim x → 1 2 sin cos - 1 x - x 1 - tan cos - 1 x is equal to

JEE Main 2022 Solution View →

76

lim x → π 2 tan 2 x 2 sin 2 x + 3 sin x + 4 1 2 - sin 2 x + 6 sin x + 2 1 2 is equal to

JEE Main 2022 Solution View →

77

Let f x be a polynomial function such that f x + f ' x + f ' ' x = x 5 + 64 . Then, the value of lim x → 1 f x x - 1 is equal to

JEE Main 2022 Solution View →

78

Let f ( x ) = x 6 + 2 x 4 + x 3 + 2 x + 3 , x ∈ R . Then the natural number n for which lim x → 1 x n f ( 1 ) - f ( x ) x - 1 = 44 is _____ .

JEE Main 2021 Solution View →

79

If α = lim x → π / 4 tan 3 x - tan x cos x + π 4 and β = lim x → 0 cos x cot x are the roots of the equation, a x 2 + b x - 4 = 0 , then the ordered

JEE Main 2021 Solution View →

80

lim x → 0 sin 2 π cos 4 x x 4 is equal to :

JEE Main 2021 Solution View →

81

If lim x → ∞ x 2 - x + 1 - a x = b , then the ordered pair ( a , b ) is:

JEE Main 2021 Solution View →

82

If α , β are the distinct roots of x 2 + b x + c = 0 , then lim x → β e 2 x 2 + b x + c - 1 - 2 x 2 + b x + c ( x - β ) 2 is equal to

JEE Main 2021 Solution View →

83

lim x → 2 ∑ n = 1 9 x n ( n + 1 ) x 2 + 2 ( 2 n + 1 ) x + 4 is equal to :

JEE Main 2021 Solution View →

84

The value of lim x → 0 x 1 - sin x 8 - 1 + sin x 8 is equal to :

JEE Main 2021 Solution View →

85

Let f : R → R be a function such that f 2 = 4 and f ' 2 = 1 . Then, the value of lim x → 2 x 2 f 2 - 4 f x x - 2 is equal to:

JEE Main 2021 Solution View →

86

If lim x → 0 α x e x - β log e ( 1 + x ) + γ x 2 e - x x sin 2 x = 10 ,   α ,   β ,   γ ∈ R , then the value of α + &#

JEE Main 2021 Solution View →

87

If the value of lim x → 0 2 - cos x cos 2 x x + 2 x 2 is equal to e a , then a is equal to_____.

JEE Main 2021 Solution View →

88

If lim x → 0 sin - 1 x - tan - 1 x 3 x 3 is equal to L , then the value of ( 6 L + 1 ) is

JEE Main 2021 Solution View →

89

The value of lim n → ∞ r + 2 r + . . . + n r n 2 , where r is non-zero real number and r denotes the greatest integer less than or equal to r , is equal to :

JEE Main 2021 Solution View →

90

The value of the limit lim θ → 0 tan π cos 2 θ sin 2 π sin 2 θ is equal to :

JEE Main 2021 Solution View →

91

The value of lim x → 0 + cos - 1 x - [ x ] 2 · sin - 1 x - [ x ] 2 x - x 3 , where x denotes the greatest integer ≤ x is:

JEE Main 2021 Solution View →

92

If lim x → 0 a e x - b cos x + c e - x x sin x = 2 , then a + b + c is equal to ________.

JEE Main 2021 Solution View →

93

Let f x be a differentiable function at x = a with f ' a = 2 and f a = 4 . Then lim x → a x f a - a f x x - a equals:

JEE Main 2021 Solution View →

94

The value of lim h → 0 3 sin π 6 + h - cos π 6 + h 3 h 3 cos h - sin h is :

JEE Main 2021 Solution View →

95

If lim x → 0 a x - e 4 x - 1 a x e 4 x - 1 exists and is equal to b , then the value of a - 2 b is ___ .

JEE Main 2021 Solution View →

96

lim n → ∞ 1 + 1 + 1 2 + … … + 1 n n 2 n is equal to

JEE Main 2021 Solution View →

97

The value of the limit lim x → π 2 4 2 sin 3 x + sin x 2 sin 2 x sin 3 x 2 + cos 5 x 2 - 2 + 2 cos 2 x + cos 3 x 2 is _________

JEE Advanced 2020 Solution View →

98

Let e denote the base of the natural logarithm. The value of the real number a for which the right-hand limit lim x → 0 + 1 − x 1 χ − e − 1 x a is equa

JEE Advanced 2020 Solution View →

99

lim x → 0 x e 1 + x 2 + x 4 - 1 / x - 1 1 + x 2 + x 4 - 1

JEE Main 2020 Solution View →

100

If α is the positive root of the equation, p x = x 2 − x − 2 = 0 , then lim x → α + 1 − cos p x x + α − 4 is equal to

JEE Main 2020 Solution View →

101

lim x → a a + 2 x 1 3 - 3 x 1 3 3 a + x 1 3 - 4 x 1 3 a ≠ 0 is equal to:

JEE Main 2020 Solution View →

102

Let t denote the greatest integer ≤ t . If λ ε R − 0 , 1 , lim x → 0 1 − x + x λ − x + x = L , then L is equal to

JEE Main 2020 Solution View →

103

If lim x → 0 1 x 8 1 − cos x 2 2 − cos x 2 4 + cos x 2 2 cos x 2 4 = 2 − k then the value of k is

JEE Main 2020 Solution View →

104

lim x → 0 tan π 4 + x 1 / x is equal to

JEE Main 2020 Solution View →

105

If lim x → 1 x + x 2 + x 3 + . . . + x n - n x - 1 = 820 , n ∈ N then the value of n is equal to....

JEE Main 2020 Solution View →

106

lim x → 0 3 x 2 + 2 7 x 2 + 2 1 x 2 is equal to

JEE Main 2020 Solution View →

107