Limits PYQs Set 2 (PDF)

Set 2 of 5 · Questions 31-60 of 130

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Limits · Set 2 (Q31-60)
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31. $lim x \to 0 (tan (\frac{π}{4} + x))^{\frac{1}{x}} is equal to:$ (1) $e^{2}$ (2) $1$ (3) $e$ (4) $2$ 32. $If lim_{x \to 0} \frac{cos ( 2 x ) + acos ( 4 x ) - b}{x ^{4}} is finite, then (a + b) is equal to:$ (1) $0$ (2) $\frac{3}{4}$ (3) $-1$ (4) $\frac{1}{2}$ 33. $lim_{x \to 1 -} \frac{π - 2 sin ^{- 1} x}{1 - x} is equal to:$ (1) $\frac{1}{2 π}$ (2) $\frac{2}{π}$ (3) $\frac{π}{2}$ (4) $π$ 34. $Let [x] denote the greatest integer less than or equal to x . Then : lim x \to 0 \frac{tan ( π sin ^{2} x ) + ( \| x \| - sin ( x [ x ] ) ) ^{2}}{x ^{2}}$ (1) $does not exist$ (2) $equals π$ (3) $equals π + 1$ (4) $equals 0$ 35. $lim_{x \to 0} \frac{\int _{0}^{x} t s i n ( 10 t ) d t}{x} is equal to$ (1) $1$ (2) $10$ (3) $5$ (4) $0$ 36. $lim y \to 0 \frac{1 + 1 + y ^{4} - 2}{y ^{4}}$ (1) $exists and equals \frac{1}{4 2}$ (2) $exists and equals \frac{1}{2 2 ( 2 + 1 )}$ (3) $exists and equals \frac{1}{2 2}$ (4) $does not exist$ 37. $If f : R \to R is a differentiable function and f (2) = 6, then lim_{x \to 2} \int_{6}^{f (x)} \frac{2 tdt}{( x - 2 )} is:$ (1) $24 f^{’} (2)$ (2) $2 f^{’} (2)$ (3) $0$ (4) $12 f^{’} (2)$ 38. $lim_{x \to 0^{+}} \frac{tan ( 5 ( x ) ^{\frac{1}{3}} ) log _{e} ( 1 + 3 x ^{2} )}{( tan ^{- 1} 3 x ) ^{2} ( e ^{5 (x)^{\frac{4}{3}}} - 1 )} is equal to$ (1) $\frac{5}{3}$ (2) $1$ (3) $\frac{1}{3}$ (4) $\frac{1}{15}$ 39. $If a = lim_{x \to 0} \frac{1 + 1 + x ^{4} - 2}{x ^{4}} and b = lim_{x \to 0} \frac{sin ^{2} x}{2 - 1 + cos x}, then the value of a b^{3} is :$ (1) $36$ (2) $32$ (3) $25$ (4) $30$ 40. $lim x \to 0 \frac{sin ^{2} x}{2 - 1 + cos x} equals :$ (1) $42$ (2) $2$ (3) $22$ (4) $4$ 41. $If α is the positive root of the equation p (x) = x^{2} - x - 2 = 0, then lim_{x \to α^{+}} \frac{1 - c o s ( p ( x ))}{x + α - 4} is equal to:$ (1) $\frac{1}{2}$ (2) $\frac{1}{2}$ (3) $\frac{3}{2}$ (4) $\frac{3}{2}$ 42. $lim_{x \to \infty} x^{3} (x^{2} + 1 + x^{4} - x 2) =$ (1) $\frac{1}{2}$ (2) $\frac{1}{4 2}$ (3) $\frac{- 1}{4 2}$ (4) $\frac{- 1}{2}$ 43. $lim_{n \to \infty} \frac{( 1 ^{2} - 1 ) ( n - 1 ) + ( 2 ^{2} - 2 ) ( n - 2 ) + \dots + ( ( n - 1 ) ^{2} - ( n - 1 ) ) \cdot 1}{( 1 ^{3} + 2 ^{3} + \dots \dots + n ^{3} ) - ( 1 ^{2} + 2 ^{2} + \dots \cdot\cdot+ n ^{2} )} is equal to :$ (1) $\frac{2}{3}$ (2) $\frac{1}{3}$ (3) $\frac{3}{4}$ (4) $\frac{1}{2}$ 44. $Let f (x) be a polynomial of degree 5 such that x = \pm 1 are its critical points. If lim x \to 0 (2 + \frac{f ( x )}{x ^{3}}) = 4, then which one of the following is not true ?$ (1) $f (x) has minima at x = 1 & maxima at x = - 1$ (2) $f (1) - 4 f (- 1) = 4$ (3) $f (x) is maxima at x = 1 and minima at x = - 1$ (4) $f (x) is odd.$ 45. $If lim_{x \to 1^{+}} \frac{( x - 1 ) ( 6 + λcos ( x - 1 ) ) + μsin ( 1 - x )}{( x - 1 ) ^{3}} = - 1, where λ, μ \in R, then λ + μ is equal to$ (1) $20$ (2) $19$ (3) $18$ (4) $17$ 46. $lim t \to 0 (1^{\frac{1}{sin ^{2} t}} + 2^{\frac{1}{sin ^{2} t}} + \dots + n^{\frac{1}{sin ^{2} t}})^{sin^{2} t} is equal to$ (1) $n^{2} + n$ (2) $n$ (3) $\frac{n ( n + 1 )}{2}$ (4) $n^{2}$ 47. $For α, β, γ \in R, if lim x \to 0 \frac{x ^{2} sin αx + ( γ - 1 ) e ^{x^{2}}}{sin 2 x - βx} = 3, then β + γ - α is equal to:$ (1) $-1$ (2) $4$ (3) $6$ (4) $7$ 48. $Let x = 2 be a root of the equation x^{2} + px + 1 = 0 and f (x) = (\frac{1 - cos ( x ^{2} - 4 px + q ^{2} + 8 q + 16 )}{( x - 2 p ) ^{4}}, x \neq = 2 p 0, x = 2 p) The lim x \to 2 p^{2} (f (x)) Where (.) denotes greatest integer function, is$ (1) $2$ (2) $1$ (3) $0$ (4) $- 1$ 49. $If x = log_{e} (cot (\frac{π}{4} + θ)), then lim_{θ \to 0} \frac{θ}{( sin hx ) ( cos hx )} =$ (1) $0$ (2) $- \frac{1}{2}$ (3) $-2$ (4) $1$ 50. $If f (x) = (\frac{sin 3 x}{e ^{2 x} - 1}; x \neq = 0 k - 2; x = 0) is continuous at x = 0, then k =$ (1) $\frac{3}{2}$ (2) $\frac{1}{2}$ (3) $\frac{9}{5}$ (4) $\frac{7}{2}$ 51. $Let f : R \to R be a continuous function satisfying f (0) = 1 and f (2 x) - f (x) = x for all x \in R . If lim_{n \to \infty} (f (x) - f (\frac{x}{2 ^{n}})) = G (x), then \sum_{r = 1}^{10} G (r^{2}) is equal to$ (1) $215$ (2) $420$ (3) $385$ (4) $540$ 52. $lim_{x \to 0} \frac{x cot ( 4 x )}{sin ^{2} x cot ^{2} ( 2 x )} is equal to :$ (1) $0$ (2) $2$ (3) $4$ (4) $1$ 53. $If lim x \to 0 \frac{sin ^{- 1} x - tan ^{- 1} x}{3 x ^{3}} is equal to L, then the value of (6 L + 1) is$ (1) $\frac{1}{6}$ (2) $\frac{1}{2}$ (3) $6$ (4) $2$ 54. $Let f : R \to (0, \infty) be a twice differentiable function such that f (3) = 18, f^{’} (3) = 0 and f^{’ ’} (3) = 4 . Then lim_{x \to 1} (log_{e} (\frac{f ( 2 + x )}{f ( 3 )})^{\frac{18}{( x - 1 ) ^{2}}}) is equal to :$ (1) $1$ (2) $18$ (3) $2$ (4) $9$ 55. $The value of lim h \to 0 (\frac{3 sin ( \frac{π}{6} + h ) - cos ( \frac{π}{6} + h )}{3 h ( 3 cosh - sinh )}) is:$ (1) $\frac{3}{4}$ (2) $\frac{2}{3}$ (3) $\frac{4}{3}$ (4) $\frac{2}{3}$ 56. $lim_{x \to 1} \frac{2 ^{2 x - 2} - 2 ^{x} + 1}{sin ^{2} ( x - 1 )} =$ (1) $2 log 2$ (2) $2 (log 2)^{2}$ (3) $\frac{1}{2} (log 2)^{2}$ (4) $(log 2)^{2}$ 57. $Image of (1, 2, 3) w.r.t a plane is (\frac{- 7}{3}, \frac{- 4}{3}, \frac{- 1}{3}) then which of the following points lie on the plane$ (1) $(- 1, 1, - 1)$ (2) $(- 1, - 1, - 1)$ (3) $(- 1, - 1, 1)$ (4) $(1, 1, - 1)$ 58. $Let [t] denote the greatest integer \leq t . If for some λ \in R - {0, 1}, lim_{x \to 0} \frac{1 - x + ∣ x ∣}{λ - x + [ x ]} = L, then L is equal to:$ (1) $2$ (2) $\frac{1}{2}$ (3) $0$ (4) $1$ 59. $Let f : \to R \to (0, \infty) be strictly increasing function such that lim_{x \to \infty} \frac{f ( 7 x )}{f ( x )} = 1 . Then, the value of lim_{x \to \infty} (\frac{f ( 5 x )}{f ( x )} - 1) is equal to$ (1) $4$ (2) $0$ (3) $7/5$ (4) $1$ 60. $If lim_{x \to 0} (\frac{tan x}{x})^{\frac{1}{x ^{2}}} = p, then 96 log_{e} p is equal to_______.$