Limits PYQs Set 3 (PDF)

Set 3 of 5 · Questions 61-90 of 130

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Limits · Set 3 (Q61-90)
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61. $lim x \to a \frac{( a + 2 x ) ^{\frac{1}{3}} - ( 3 x ) ^{\frac{1}{3}}}{( 3 a + x ) ^{\frac{1}{3}} - ( 4 x ) ^{\frac{1}{3}}} (a \neq = 0) is equal to:$ (1) $(\frac{2}{9}) (\frac{2}{3})^{\frac{1}{3}}$ (2) $(\frac{2}{3}) (\frac{2}{9})^{\frac{1}{3}}$ (3) $(\frac{2}{3})^{\frac{4}{3}}$ (4) $(\frac{2}{9})^{\frac{4}{3}}$ 62. $lim_{x \to 0} \frac{x + 2 sin x}{x ^{2} + 2 sin x + 1 - sin ^{2} x - x + 1} is$ (1) $6$ (2) $2$ (3) $3$ (4) $1$ 63. $Let f (x) = lim θ \to 0 (\frac{cos πx - x ^{(\frac{2}{θ})} sin ( x - 1 )}{1 + x ^{(\frac{2}{θ})} ( x - 1 )}), x \in R . Consider the following two statements: (I) f (x) is discontinous at x = 1 . (II) f (x) is continous at x = - 1 . Then,$ (1) $Only (II) is True$ (2) $Neither (I) nor (II) is True$ (3) $Both (I) and (II) are True$ (4) $Only (I) is True$ 64. $If lim_{x \to 0} \frac{e ^{(a - 1) x} + 2 cosbx + ( c - 2 ) e ^{- x}}{xcosx - log _{e} ( 1 + x )} = 2, then a^{2} + b^{2} + c^{2} is equal to :$ (1) $5$ (2) $9$ (3) $3$ (4) $7$ 65. $Let f : (- \infty, \infty) - 0 \to R be a differentiable function such that f^{’} (1) = lim_{a \to \infty} a^{2} f (\frac{1}{a}) . Then lim_{a \to \infty} \frac{a ( a + 1 )}{2} tan^{- 1} (\frac{1}{a}) + a^{2} - 2 log_{e} a is equal to$ (1) $\frac{3}{2} + \frac{π}{4}$ (2) $\frac{3}{4} + \frac{π}{8}$ (3) $\frac{3}{8} + \frac{π}{4}$ (4) $\frac{5}{2} + \frac{π}{8}$ 66. $lim x \to 3^{-} \frac{x ^{3} - 3 x ^{2} - 4 x + 12}{2 x ^{3} - 7 x ^{2} + 2 x + 3} =$ (1) $0$ (2) $\infty$ (3) $\frac{5}{14}$ (4) $\frac{6}{13}$ 67. $If lim_{x \to 1} \frac{x ^{2} - ax + b}{x - 1} = 5, then a + b is equal to :$ (1) $-4$ (2) $5$ (3) $-7$ (4) $1$ 68. $l im_{x \to 0} cosec x (2 cos^{2} x + 3 cos x - cos^{2} x + sin x + 4) is:$ (1) $0$ (2) $\frac{1}{15}$ (3) $\frac{1}{2 5}$ (4) $- \frac{1}{2 5}$ 69. $lim_{x \to π /4} \frac{cot ^{3} x - tan x}{cos ( x + π /4 )} is:$ (1) $4$ (2) $42$ (3) $82$ (4) $8$ 70. $Let f be a differentiable function such that f^{’} (x) = 7 - \frac{3}{4} \frac{f ( x )}{x}, (x > 0) and f (1) \neq = 4 . Then lim_{x \to 0^{+}} xf (\frac{1}{x}) :$ (1) $exists and equals \frac{4}{7} .$ (2) $exists and equals 4$ (3) $does not exist.$ (4) $exists and equals 0$ 71. $Let f : R^{+} \to R^{+} be a function satisfying f (x) - x = λ (constant), \forall x \in R^{+} and f (xf (y)) = f (xy) + x, \forall x, y \in R^{+} . Then lim_{x \to 0} \frac{( f ( x ) ) ^{\frac{1}{3}} - 1}{( f ( x ) ) ^{\frac{1}{2}} - 1} =$ (1) $\frac{1}{3}$ (2) $0$ (3) $\frac{2}{3}$ (4) $1$ 72. $The value of the limit lim θ \to 0 \frac{tan ( π cos ^{2} θ )}{sin ( 2 π sin ^{2} θ )} is equal to :$ (1) $- \frac{1}{2}$ (2) $- \frac{1}{4}$ (3) $0$ (4) $\frac{1}{4}$ 73. $Among (^{S 1})^{: l i m_{n \to \infty}} \frac{1}{n ^{2}} (^{2 + 4 + 6 + ... + 2 n})^{= 1} (S 2) : lim_{n \to \infty} \frac{1}{n ^{16}} (1^{15} 2^{15} + 3^{15} + ... + n^{15}) = \frac{1}{16}$ (1) $Both (S1) and (S2) are true$ (2) $Only (S1) is true$ (3) $Both (S1) and (S2) are false$ (4) $Only (S2) is true$ 74. $Let f (x) = 5 - \| x - 2 \| and g (x) = \| x + 1 \|, x \in R . If f (x) attains maximum value at α and g (x) attains minimum value at β, then lim_{x \to - αβ} \frac{( x - 1 ) ( x ^{2} - 5 x + 6 )}{x ^{2} - 6 x + 8} is equal to :$ (1) $½$ (2) $- 3/2$ (3) $- 1/2$ (4) $3/2$ 75. $lim_{x \to 0} \frac{co s _{(} s in x _{)} - cos x}{x ^{4}} is equal to :$ (1) $\frac{1}{3}$ (2) $\frac{1}{4}$ (3) $\frac{1}{6}$ (4) $\frac{1}{12}$ 76. $Let a be an integer such that lim_{x \to 7} \frac{18 - [ 1 - x ]}{[ x - 3 a ]} exists, where [t] is greatest integer \leq t. Then a is equal to :$ (1) $-6$ (2) $-2$ (3) $2$ (4) $6$ 77. $If the function f (x) = ⎩ ⎨ ⎧ \frac{1}{x} lo g_{e} (1 + \frac{x}{a}), k, \frac{c o s ^{2} x - s i n ^{2} x - 1}{x ^{2} + 1 - 1}, x < 0 x = 0 x > 0 is continuous at x=0, then \frac{1}{a} + \frac{1}{b} + \frac{4}{k} is equal to :$ (1) $-5$ (2) $5$ (3) $-4$ (4) $4$ 78. $If lim_{x \to 0} \frac{α e ^{x} + β e ^{- x} + γ s in x}{x s i n ^{2} x} = \frac{2}{3}, where α, β, γ \in R, then which of the following is NOT correct ?$ (1) $α^{2} + β^{2} + γ^{2} = 6$ (2) $α β + β γ + γ α + 1 = 0$ (3) $α β^{2} + β γ^{2} + γ α^{2} + 3 = 0$ (4) $α^{2} - β^{2} + γ^{2} = 4$ 79. $Let β = lim_{x \to 0} \frac{α x - ( e ^{3 x} - 1 )}{α x ( e ^{3 x} - 1 )} for some α \in R . Then the value of α + β is :$ (1) $\frac{14}{5}$ (2) $\frac{3}{2}$ (3) $\frac{5}{2}$ (4) $\frac{7}{2}$ 80. $If α > β > 0 are the roots of the equation a x^{2} + bx + 1 = 0, and lim_{x \to \frac{1}{α}} (\frac{1 - c o s ( x ^{2} + b x + a )}{2 ( 1 - α x )})^{\frac{1}{2}} = \frac{1}{k} (\frac{1}{β} - \frac{1}{α}), then k is equal to$ (1) $2 β$ (2) $α$ (3) $2 α$ (4) $β$ 81. $If lim x \to 0 (\frac{1}{x ^{8}} (1 - cos \frac{x ^{2}}{2} - cos \frac{x ^{2}}{4} + cos \frac{x ^{2}}{2} cos \frac{x ^{2}}{4})) = 2^{- k}, then the value of k is:$ 82. $Let f be a differentiable function in the interval (0, \infty) such that f (1) = 1 and lim_{t \to x} \frac{t ^{2} f ( x ) - x ^{2} f ( t )}{t - x} = 1 for each x > 0 . Then 2 f (2) + 3 f (3) is equal to_____$ 83. $lim x \to 2 \frac{3 ^{x} + 3 ^{3 - x} - 12}{3 ^{- \frac{x}{2}} - 3 ^{1 - x}} is equal to \dots..$ 84. $Suppose a differentiable function f (x) satisfies the identify f (x + y) = f (x) + f (y) + x y^{2} + x^{2} y, for all real x and y . If lim x \to 0 \frac{f ( x )}{x} = 1, then f^{’} (3) is equal to \dots$ 85. $Let f (x) = l im_{x \to 0} \sum (\frac{tan ( x / 2 ^{r + 1} ) + tan ^{3} ( x / 2 ^{r + 1} )}{1 - tan ^{2} ( x / 2 ^{r + 1} )}) . Then l im_{x \to 0} \frac{e ^{x} - f ^{f (x)}}{( x - f ( x ) )} is equal to$ 86. $Evaluate lim_{x \to 2} \frac{3 ^{x} + 3 ^{x - 1} - 12}{3 ^{\frac{- x}{2}} - 3 ^{1 - x}}$ 87. $If lim x \to 0 \frac{a e ^{x} - b cos x + c e ^{- x}}{x sin x} = 2, then a + b + c is equal to____$ 88. $If the value of is equal to e^{n}, then a is equal to ___$ 89. $If lim_{x \to 1} \frac{s i n ( 3 x ^{2} - 4 x + 1 )}{2 x ^{3} - 7 x ^{2} + a x + b} = - 2, then the value of (a-b) is equal to$ 90. $Let f(x) = x^{6} + 2 x^{4} + x^{3} + 2x + 3, x \inR.Then the natural number for which lim_{x \to 1} \frac{x ^{n} f ( 1 ) - f ( x )}{x - 1} = 44 is .$