Let f x be a twice differentiable function and f0 5 then

Question

Level 1:

1. If α is a repeated root of ax² + bx + c = 0, then
lim_x→α tan (ax² + bx + c)/(x - α)² is

(a) a         (b) b
(c)  c         (d) 0

2. lim_x→0 (e^x + e^-x + 2cosx -4)/(x⁴) is equal to

(a) 0       (b) 1
(c) -6      (d) - 1/6

3. If lim_x→0 (729^x - 243^x - 81^x + 9^x + 3^x -1)/X³ = k(log 3)³, then

k=

(a) 4       (b) 5
(c) 6       (d) none of these

4. lim_x→0 1 - cos³x/xsinxcosx =

(a) 3/2     (b) -2
(c) 1         (d) none of these

5. If f(2), g (x) be differentiable functions and f (1) = g (1) = 2 then lim_x→1 (f(1)g(x) - f(x)g(1)- f(1) + g(1))/(g(x)- f (x)) is equal to

(a) 0        (b) 1
(c) 2         (d) none of these

6. lim_x→1 ∑x^r-1^r/(x-1) =

(a) 0    (b) n(n+1)/2

(c) 1    (d) none of these

7. Let f (x) be a twice differentiable function and f"(0) = 5, then lim_x→0 (3f (x) -4f (3x) + f (9x))/x² is equal to
(a) 30    (b) 120
(c) 40    (d) none of these

8. If (9) = 9 and f'(9) = 1, then lim_x→9 (3- √f(x))/(3 - √x) is equal to
(a) 0    (b) 1
(c) - 1   (d) none of these

9. lim_n→∞ (cosx/2 cosx/4 cosx/8.....cosx/2ⁿ) =

(a) x/sinx      (b) six/x

(c) 0              (d) none of these

10. If lim_x→0 (sin2x + asinx/x³) be finite, then the value of a and the limit are given by

(a) - 2, 1 (b) - 2, - 1
(c) 2, 1   (d) 2, - 1

11. The value of lim_n→∞ 1/n⁴[ 1(∑ⁿ_{k= 1}k) + 2(∑^n-1_{k =1} k ) + 3 [^n-2∑_k=1 k + . . . + n.1] will be
(a) 1/24   (b) 1/12
(c)1/6      (d) 1/3

12. If α and β be the roots of ax² + bx + c = 0, then

lim_x→α [ 1 + αx² + bx + c)^1/(x-α) is

(a) log |a (α - β)|  (b) e^a (α - β)
(c) e^a(β - α)          (d) none of these

13. lim_x→1(³√x² - 2³√(x+1))/(x-1)² is equal to

(a) 1/9     (b) 1/6

(c) 1/3      (d) (d) none of these

14. lim_n→∞ ∏ⁿ_r=3 (r³ -1/(r³ + 1))

(a) 1/3    (b) 6/7
(c) -2/3   (d) none of these

15. lim_x→5 (x²- 9x + 20)/ (x - [x]) =

(a) 1                     (b) 0
(c) does not exist (d) cannot be determined

16. If [x] denotes the integral part of x, then lim_n→∞1/n³ (∑ⁿ_{k= 1}[k²x]) =

(a) 0      (b) x/2
(c)x/3    (d) x/6

17. lim_n→∞ (tanθ + 1/2tanθ/2 + 1/2²tanθ/2² + .....+ 1/2ⁿtanθ/2n) =

(a) 1/θ       (b) 1/θ -2cot 2θ
(b) 2cot2θ  (d) none of these

18.  lim_n→∞ ((x) + 2(x) + (3x) + ...... +(nx))/n², where

{x} = x - [x] denotes the fractional part of x, is

(a) 1       (b) 0
(c) 1/2    (d) none of these

19. lim_x→a (sinx/sina)1/(x-a), a ≠ nΠ, n is an integer, equals

(a) e^{cot a} (b) e^{tan a}
(c) e^{sin a} (d) e^{cos a}

20. lim_x→0 (1 + tan x)/(1 + sin x)^1/sinx is equals to

(a) 0       (b) 1
(c) - 1     (d) none of these

21. lim_x→a (2 - x/a)^tanΠx/2a is equal to

(a)e^Π/2      (b) e^2/Π
(c) e^-2/Π    (d) e^-Π/2

22. lim_x→0 (cosx + a sinbx)a/x is equal to

(a) e^_-a2_b    (b) e^_ab2
(c) e^_a2_b      (d) e^_-b2_a

23. The value of lim_x→0(sinx/x)^{sinx/(x-xinx)} is
(a) 1            (b) - 1
(c) 0             (d) none of these

24. If A_i = x- a_i/|x- a_i|, i = 1, 2, ..., n and if a₁ < a₂ < a₃ <..... < a_n. Then lim_x→0 (A₁A₂...A_n), 1 ≤ m ≤ n

(a) is equal to (- 1)^m     (b) is equal to (- 1)^{m + 1}
(c) is equal to (- 1)^{m -1}  (d) does not exist

25. lim_x→0 x^x is equal to

(a) 0     (b) 1
(c) - 1   (d) none of these

26. lim_x→0 tan([-Π²]x) - x²tan(-Π²)/sin²x equals, where [] denotes the greatest integer function

(a) 0              (b) 1

(c) tan10 -10  (d) ∞

27. lim_x→1 xsin{x}/(x-1), where {x} denotes the fractional part of x, is equal to

(a) -1     (b) 0

(c) 1       (d) does not exist

28. lim_n→∞ {7/10 + 29/10² + 133/10³ +.....+ (5ⁿ + 2ⁿ)/10ⁿ} is equal to

(a) 3/4     (b) 2
(c) 5/4      (d) 1/2

29. lim_x→-1 ((x⁴ + x² + x + 1)/(x² - x + 1))^{_{1 - cos(x+1)/(x + 1)}2} is equal to

(a) 1              (b) (2/3)^1/2

(c) (3/2)^1/2    (d) e^1/2

30. lim_n→∞(cosx/n)ⁿ is equal to

(a) e¹          (b) e^-1

(c) 1            (d) none of these

31. lim_x→0⁺ (b/x)[x/a]  where a > 0, b > 0 and [x] denotes greatest integer less than or equal to x is

(a) 1/a     (b) b

(c) b/a      (d) 0

32. If f(x) = {sin[x]/[x], [x] ≠ 0, where [x] denotes the greatest integer ≤ x, then lim_x→0 f(x) equals

                   { 0,            [x] = 0   

(a) 0            (c) -1
(c)  1           (d) none of these

33. lim_x→0(ln cosx/⁴√(1 + x² -1)) is

(a) log_a 6     (b) log_a3
(c) log_a 2     (d) none of these

34. The value of lim_x→3(log_a(x-3/(√(x + 6) -3)))) is
(a) log_a6       (b) log_a3
(c) log_a2       (b) none of these

35. lim_h→0(2[√3 sin(Π/6 + h) - cos(Π/6 + h)])/(√3h(√3 cosh -sinh) is equal to

(a) 4/3       (b) -4/3
(c) 2/3       (b) 3/4

36. Let f (x) = x - [x], where [x] denotes the greatest integer ≤ x and g(x) = lim_n→∞[f(x)]²ⁿ - 1/[f(x)]²ⁿ + 1, then g (x) =
(a) 0        (b) 1
(c) -1       (d) none of these

37. If f (x) = { tan^-1([x] + x)/[x] - 2x, [x] ≠ 0
                    {  0,                           [x] = 0
where [x] denotes the greatest integer less than or equal to x, then lim_x→0 f (x) is equal to

(a) -1/2     (b) 1

(c) Π/4       (d) does not exist

38. lim_x→2 (2^x - x²)/(x^x - 2²) is equal to
(a) log 2 -1/(log2+1)      (b) log 2 + 1/(log2 - 1)

(c) 1 (d) - 1

39. If lim_x→0 (xⁿ - sinxⁿ)/(x - sinⁿ x) is non-zero finite, then n may be equal to
(a) 1      (b) 2
(c) 3      (d) none of these

40. lim_x→Π/2 (sin x - (sin x)^{sin x} )/(1 - sin x + In sin x)
(a) 1      (b) 2
(c) 3       (d) 4

41. The value of lim_x→a √(a² - x²) cotΠ/2 √((a -x)/(a + x)) is
(a) 2a/Π      (b) - 2a/Π
(c) 4a/Π      (d) -4a/Π

42. lim_x→∞/2 f (x), where 2x- 3/x < f(x) < (2x² + 5x)/x² is

(a) 1     (b) 2
(c) -1    (d) -2

43. lim_x→0 (cosec³x.cotx - 2 cot³ x.cosec x + cot⁴x/sec x ) is equal to

(a) 1     (b) - 1
(c) 0      (d) none of these

Level 2:

44. The value of lim_x→0([100/sin x] + [99sinx/x]), where [.] represents greatest integer function, is
(a) 199     (b) 198
(c) 0          (d) none of these

45. If f (x) = sin x, x ≠ nΠ,
                =  2,      x = nΠ
where n ∈ Z and

g (x) = x² + 1,   x ≠ nΠ,
        = 3,            x = 2.
then lim_x→0 g[f (x)] is
(a) 1       (b) 0
(c) 3       (d) does not exist

46. The value of lim_x→∞ (√x +√x + √x - √x) is
(a) 1/2           (b) 1

(c) 0               (d) none of these

47. The value of lim_x→∞[tan^-1 (x +1)/(x+2) - Π/4] is

(a) 1/2           (b) -1/2

(c) 1               (d) -1

48. lim_n→∞ cos(Π√(n² +n), n ∈ Z is equal to

(a) 0               (b) 1
(c) - 1             (d) None of these

49. lim_n→∞ (n^k sin²(n!))/(n +2), 0 < k < 1, is equal to

(a) ∞             (b) 1
(c) 0               (d) none of these

50. lim_x→1 √((1 - cos2(x -1))/(x -1)

(a) exists and it equals √2

(b) exists and it equals - √2

(c) does not exist because (x - 1) → 0,

(d) does not exist because left hand limit is not equal to right hand limit

51. The value of lim_x→∞ x⁵/5^x is

(a) 1              (b) - 1
(c) 0               (d) none of these

52. lim_x→0(cosx + sinx)1/x is equal to
(a) e               (b) e²
(c) e^-1            (d) 1

53. The value of lim_x→Π/4 ((2√2  - cosx + sinx)³)/(1 - sin2x) is

(a) 3/√2          (b) √2/3

(c) 1/√2           (d) √2

54. The value of lim_h→0 (ln( 1 + 2h) - 2ln(1+ h))/h²  is
(a) 1                (b) - 1
(c) 0                 (d) none of these

55. The value of lim_n→∞( 1/n + e^1/n/n + e^2/n/n + .....+ e^(n-1)/n/n _{) is}
(a)1                 (b) 0
(c) e - 1           (d) e + 1

56. lim_x→1 xsin(x - [x])/(x-1), where [.] denotes the greatest integer function, is equal to

(a) 1                 (b) -1
(c) ∞                (d) does not exist

57. If f(x) = ∫2sinx - sin2x/x³dx, x ≠ 0, then lim_x→0 f '(x) is
(a) 0                  (b)∞
(c) - 1                (d) 1

58. lim_x→Π/2 [x/2]/ln(sinx) (where [.] denotes the greatest integer function)

(a) does not exist     (b) equals 1
(c) equals 0              (d) equals - 1

59. lim_m→∞lim_n→∞ (1 + cos2m n!Πx) is equal to

(a) 2                     (b) 1
(c) 0                     (d) none of these
-3])]

60. lim_x→0 [sin([x-3])/[x-3]], where [ . ] represents greatest integer function, is

(a) 0                      (b) 1

(c) does not exist  (d) sin 1

61. The values of constants a and b so that

lim_x→∞ ((x²+1)/(x+1) - ax -b) =0 are

(a) a = 1, b = - 1        (b) a= -1, b = 1
(c) a = 0, b = 0           (d) a =2, b = -1

62. lim_x→∞ (1/1.2 + 1/2.3 + 1/3.4 + ...+ 1/n(n +1) is equal to

(a) 1            (b) - 1
(c) 0            (d) none of these

63. lim_x→∞  (log x)²/xⁿ, n > 0 is equal to

(a) 1             (b) 0
(c) - 1           (d) 1/2

64. If the r^th term, t_r, of a series is given by t_r = r⁴ + r² +1, then lim_n→∞ Σ_r=1ⁿ t_r, is

(a) 1              (b) -2
(c) 1/3           (d) none of these

65. lim_x→n(-1)[x], where [x] denotes the greatest integer less than or equal to x, is equal to

(a) (-1)ⁿ        (b) (-1)^n-1
(c) 0              (d) does not exist

66. lim_x→1 y³/x³ - y² - 1 as (x, y) → (1, 0) along the line

          y→o
y = x - 1 is given by
(a) 1                (b) ∞
(c) 0                 (d) none of these

67. lim_x→∞ (1 - 2 + 3 - 4 + 5 - 6 + ...-2n)/((√n² + 1) + √(4n² - 1)) is equal to

(a) 1/3                (b) -1/3
(c) -1/5               (d) none of these

68. The value of lim_x→-∞ [x⁴sin(1/x) + x²/(1 + |x|³)] is

(a) 1                (b) -1
(c) ∞               (d) none of these

69. lim_x→2 (2^x + 2^3-x - 6))/(2^-x/2 - 2^1-x) is equal to

(a) 8                (b) -1
(c) 2               (d) none of these

70. lim_x→0 8/x⁸(1- cosx²/2 - cosx⁴/4 + cosx²/2 cosx²/4) is equal to

(a) 1/16    (b) -1/16

(c) 1/32    (d) -1/32

71. lim_n→∞[log_n-1(n).log_n(n+1).log_n+1(n + 2) ......log_n^k-1(n^k) ] is equal to
(a) 0.        (b) n
(c) k          (d) none of these

72. lim_n→∞[1/1.3 + 1/3.5 + 1/5.7 + .......+ 1/(2n+1)(2n+3) is equal to

(a) 1         (b) 1/2
(c) -1/2     (d) none of these

73. The value of lim_x→∞[1^1/x + 2^1/x + 3^1/x +......+ n^1/x ] is

(a) n!         (b)  1/2

(c) -1/2       (d) none of these

74. lim_n→∞ (1 +x)(1 +x²)(1 +x⁴) ........ (1 + x²ⁿ), |x| < 1 is equal to
(a) 1/(x-1)     (b)1/(1-x)
(c) 1 - x         (d) x - 1

75. lim_x→∞ xⁿ/e^x= 0, (n integer), for

(a) no value of n

(b) all values of n

(c) only negative values of n

(d) only positive values of n

76. The value of is lim_x→1  (xⁿ + x^n-1 + x^n-2 + ...+ x² + x - n)/(x - 1) is

(a) n(n+1)/2        (b) 0

(c) 1                     (d) n

77. If   t_r  =  (1² + 2² + 3² +...+r²)/(1³ +2³ +3³ +...+r³) and S_n = ∑ⁿ_r=1(-1)^r, then lim_n→∞, is given by

(a) 2/3                  (b) -2/3
(c) 1/3                   (d) -1/3

78. If lim_x→0 ((1 + a³) + 8e^1/x)/( 1 + (1 - b³)e^1/x) = 2 then

(a) a = 1, b = (- 3)^1/3         (b) a = 1, b = 3^1/3
(c) a = - 1, b = - (3)1/3       (d) none of these

79. If a = min {x² + 4x + 5, x ∈ R} and b =lim_θ→01 - cos2θ/θ² then the value of ∑ⁿ_r=0a^r.b^n-r is

(a) (2ⁿ⁺¹ -1)/4.2ⁿ        (b) 2^{n + 1} - 1
(c) (2ⁿ⁺¹ - 1)/3.2ⁿ      (d) none of these

80. lim_n→∞ ((1.2+2.3+3.4+...+n(n+1))/n³ is equal to
(a) 1                (b) - 1
(c) 1/3             (d) none of these

81. lim_x→0 log(1 + x + x²) + log(1 - x + x²) is is equal to

(a) 1                  (b) - 1
(c) 0                   (d) ∞

82. lim_x→e lim (ln x -1)/|x -e| is equal to
(a) 1/e               (b)-1/e
(c) e                   (d) does not exist

83. If x₁ = 3 and x_n+1 = √(2 + x_n), n ≥ 1, then lim_n→∞ X_n is equal to

(a) -1                  (b) 2
(c) √5                 (d) 3

84. The value of lim_x→∞ (3^x+1 - 5^x+1)/(3^x -5^x) is
(a) 5                   (b) -5
(c) - 5                 (d) none of these

85. lim_n→∞ 1/n ( 1+ e^1/n + e^2/n + .....+ e^n-1/n) is equal to

(a) e                  (b) - e
(c) e - 1             (d) 1 - e

86. lim_x→∞√(x + sinx)/(x - cosx) =
(a) 0                  (b) 1
(c) - 1                (d) none of these

87. If S_n = ∑ⁿ_i=1a_i and lim_n→∞ a_n = a, then lim_n→∞ (S_n+1 -S_n)/(√∑ⁿ_i=1i) is equal to

(a) 0                   (b) a

(c) √2a                (d) none of these

88. The value of  lim_n→∞ [³√(n² - n³) +n] is
(a) 1/3             (b) -1/3
(c) 2/3              (d) -2/3

89. The value of lim_n→∞ (⁴√(n⁵ + 2) - ³√(n² + 1)/ (⁵√(n⁴ + 2) - ²√(n³ + 1) is

(a) 1                (b) 0

(c) -1               (d) ∞

90. The integer n for which lim_x→0 ((cos x -1) (cos x - e^x))/xⁿ is a finite non-zero number, is

(a) 1               (b) 2
(c) 3                (d) 4

91. The value of lim_x→∞ (2√x +3³√x + 5⁵√x)/(√3x - 2 + ³√2x -3) is

(a) 2/√3           (b) √3        

(c) 1/√3            (d) none of these

92. lim_x→0 (x³√(z² - (z-x)²))/(³√(8xz - 4x²) + 3√(8xz)⁴ is equal to

(a) z/2^11/3       (b)  1/2^23/3.z

(c) 2^21/3z         (d) none of these

93. In a circle of radius r, an isosceles triangle ABC is inscribed with AB = AC. If the ΔABC has perimeter P = 2[√(2 hr - h²) + √(2 hr) and area A = h√(2 hr - h²), where h is the altitude from A to BC, then lim_h→0⁺ A/P³ is equal to

(a) 128 r                 (b) 1/128r

(c) 1/64r                 (d) none of these

94. lim_x→2(√(1 - cos{2(x-2)}))/(x-2) 

(a) equals 1/√2        (b) does not exist

(c) equals √2            (d) equals - √2