Limits PYQs Set 4 (PDF)

Set 4 of 5 · Questions 91-120 of 130

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Limits · Set 4 (Q91-120)
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91. $If lim_{x \to 0} \frac{e ^{a x} - c o s ( b x ) - \frac{c x e ^{- c x}}{2}}{1 - c o s ( 2 x )} = 17, then 5 a^{2} + b^{2} is equal to$ (1) $64$ (2) $72$ (3) $68$ (4) $76$ 92. $lim_{x \to 0} \frac{e ^{4 s i n x ∣} - 2∣ s i n x ∣ - 1}{x ^{2}}$ (1) $is equal to - 1$ (2) $does not exist$ (3) $is equal to 1$ (4) $is equal to 2$ 93. $The set of all values of a for which lim_{x \to a} ([x - 5] - [2 x + 2]) = 0, where [α] denotes the greatest integer less than or equal to α, is equal to$ (1) $(- 7.5, - 6.5)$ (2) $(- 7.5, - 6.5]$ (3) $[- 7.5, - 6.5]$ (4) $[- 7.5, - 6.5)$ 94. $Let [t] denote the greatest integer \leq t and lim_{x \to 0} x [\frac{4}{x}] = A . Then the function, f (x) = [x^{2}] sin (π x) is discontinuous, when x is equal to:$ (1) $A$ (2) $A + 1$ (3) $A + 5$ (4) $A + 21$ 95. $If lim_{x \to 0} x [\frac{4}{x}] = A then the value of x at which f (x) = [x^{2}] sin π x is discontinuous (where [.] denotes greatest integer function)$ (1) $A + 1$ (2) $A + 21$ (3) $A$ (4) $A + 5$ 96. $Let f : (- \frac{π}{2}, \frac{π}{2}) \to R be a differentiable function such that f (0) = \frac{1}{2} . If lim_{x \to 0} \frac{x \int _{0}^{x} f ( t ) d t}{e ^{x^{2}} - 1} = α, then 8 α^{2} is equal to :$ (1) $16$ (2) $2$ (3) $1$ (4) $4$ 97. $Let f, g and h be real valued functions defined on R as f (x) = {\frac{x}{∣ x ∣}, 1, x \neq = 0 x = 0, f (x) = {\frac{s i n ( x + 1 )}{( x + 1 )}, 1, x \neq = 0 x = - 1 and h (x) = 2 [x] - f (x) . Where [x] is the greatest integer \leq x . Then the value of lim_{x \to 1} g (h (x - 1)) is$ (1) $1$ (2) $sin (1)$ (3) $- 1$ (4) $0$ 98. $Let f (x) = \int_{0}^{x} (t + sin (1 - e^{t})) d t, x \in R . Then, lim_{x \to 0} \frac{f ( x )}{x ^{3}} is equal to$ (1) $- \frac{1}{6}$ (2) $\frac{2}{3}$ (3) $- \frac{2}{3}$ (4) $\frac{1}{6}$ 99. $lim_{x \to \infty} \frac{( 2 x ^{2} - 3 x + 5 ) ( 3 x - 1 ) ^{\frac{x}{2}}}{( 3 x ^{2} + 5 x + 4 ) ( 3 x + 2 ) ^{x}} is equal to:$ (1) $\frac{2}{3 e}$ (2) $\frac{2 e}{3}$ (3) $\frac{2}{3 e}$ (4) $\frac{2 e}{3}$ 100. $If lo g (1 + x) = x - \frac{x ^{2}}{2} + \frac{x ^{3}}{3} - \frac{x ^{4}}{4} + \dots \infty and lim_{x \to 0} (\frac{l o g ( 1 + x ) ^{1 + x}}{x ^{2}} - \frac{1}{x}) = k, then 12 k =$ (1) $1$ (2) $3$ (3) $6$ (4) $9$ 101. $f (x) = ⎩ ⎨ ⎧ \frac{1 - s i n ^{3} x}{3 c o s ^{2} x}, α, \frac{β ( 1 - s i n x )}{( π - 2 x ) ^{2}}, x < \frac{π}{2} x = \frac{π}{2} x > \frac{π}{2} is continuous at x = \frac{π}{2}, then α β =$ (1) $1$ (2) $-1$ (3) $2$ (4) $-2$ 102. $lim_{x \to \frac{π}{2}} (\frac{\int _{x^{3}}^{(π /2)^{3}} ( s i n ( 2 t ^{1/3} ) + c o s ( t ^{1/3} ) ) d t}{( x - \frac{π}{2} ) ^{2}}) is equal to$ (1) $\frac{5 π ^{2}}{9}$ (2) $\frac{9 π ^{2}}{8}$ (3) $\frac{11 π ^{2}}{10}$ (4) $\frac{3 π ^{2}}{2}$ 103. $lim_{x \to 0} \frac{e - ( 1 + 2 x ) ^{\frac{1}{2 x}}}{x} is equal to$ (1) $0$ (2) $\frac{- 2}{e}$ (3) $e$ (4) $e - e^{2}$ 104. $Let f (x) = {x - 1, 2 x, x is even x is odd, x \in N . If for some a \in N, f (f (f (a))) = 21, then lim_{x \to a^{-}} (\frac{∣ x ∣ ^{3}}{a} - [\frac{x}{a}]), where [t] denotes the greatest integer less than or equal to t, is equal to :$ (1) $121$ (2) $144$ (3) $169$ (4) $225$ 105. $If lim_{x \to \infty} ((\frac{e}{1 - e}) (\frac{1}{e} - \frac{x}{1 + x}))^{x} = α, then the value of \frac{l o g _{e} α}{1 + l o g _{e} α} equals:$ (1) $e^{- 1}$ (2) $e^{2}$ (3) $e^{- 2}$ (4) $e$ 106. $lim_{x \to \infty} \frac{( 3 x + 1 + 3 x - 1 ) ^{6} + ( 3 x + 1 - 3 x - 1 ) ^{6}}{( x + x ^{2} - 1 ) ^{6} + ( x - x ^{2} - 1 ) ^{6}} x^{3}$ (1) $is equal to 9$ (2) $is equal to 27$ (3) $does not exist$ (4) $is equal to \frac{27}{2}$ 107. $Let a be the sum of all coefficients in the expansion of (1 - 2 x + 2 x^{2})^{2023} (3 - 4 x^{2} + 2 x^{3})^{2024} and b = lim_{x \to 0} (\frac{\int _{0}^{x} \frac{log ( 1 + t )}{t ^{2024} + 1} dt}{x ^{2}}) . If the equations cx^{2} + dx + e = 0 and 2 bx^{2} + ax + 4 = 0 have a common root, where c, d, e \in R, then d : c : e equals$ (1) $2 : 1 : 4$ (2) $4 : 1 : 4$ (3) $1 : 2 : 4$ (4) $1 : 1 : 4$ 108. $The value of lim_{n \to \infty} 6 tan {\sum_{r = 1}^{n} tan^{- 1} (\frac{1}{r ^{2} + 3 r + 3})} is equal to$ (1) $1$ (2) $2$ (3) $3$ (4) $6$ 109. $lim_{x \to 0} ((\frac{1 - c o s ^{2} ( 3 x )}{c o s ^{3} ( 4 x )}) (\frac{s i n ^{3} ( 4 x )}{( l o g _{e} ( 2 x + 1 ) ) ^{5}}))$ (1) $15$ (2) $9$ (3) $18$ (4) $24$ 110. $lim_{x \to 0} (\frac{t a n x}{2 x + 4 - 2}) is equal to$ (1) $3$ (2) $4$ (3) $6$ (4) $2$ 111. $If lim_{n \to \infty} (n^{2} - n - 1 + n α + β) = 0 then 8_{(} α + β_{)} is equal to :$ (1) $4$ (2) $-8$ (3) $-4$ (4) $8$ 112. $If lim_{x \to \infty} (x^{2} - x + 1 - a x) = b, then the ordered pair (a,b) is:$ (1) $(1, \frac{1}{2})$ (2) $(1, - \frac{1}{2})$ (3) $(- 1, \frac{1}{2})$ (4) $(- 1, - \frac{1}{2})$ 113. $If f : R \to R is given by f(x) = x + 1, then the value of lim_{n \to \infty} \frac{1}{n} [f (0) + f (\frac{5}{n}) + f (\frac{10}{n}) + .... + f (\frac{5 ( n - 1 )}{n})] is:$ (1) $\frac{3}{2}$ (2) $\frac{5}{2}$ (3) $\frac{1}{2}$ (4) $\frac{7}{2}$ 114. $Let f : R \to R be a function such that f(2) = 4 and f’(2) = 1 Then, the value of l i m_{x \to 2} \frac{x ^{2} f _{(} 2 _{)} - 4 f _{(} x _{)}}{x - 2} is equal to :$ (1) $4$ (2) $8$ (3) $16$ (4) $12$ 115. $If lim_{t \to 0} (\int_{0}^{1} (3 x + 5)^{t} d x)^{\frac{1}{t}} = \frac{α}{5 e} (\frac{8}{5})^{\frac{2}{3}}, then α is equal to .$ 116. $Let [t] be the greatest integer less than or equal to t . Then the least value of p \in N for which lim_{x \to 0^{+}} (x ([\frac{1}{x}] + [\frac{2}{x}] + \dots + [\frac{p}{x}]) - x^{2} ([\frac{1}{x ^{2}}] + [\frac{2 ^{2}}{x ^{2}}] + \dots + [\frac{9 ^{2}}{x ^{2}}])) \geq 1 is equal to ____.$ 117. $Let the slope of the line 45 x + 5 y + 3 = 0 be 27 r_{1} + \frac{9 r _{2}}{2} for some r_{1}, r_{2} \in R . Then lim_{x \to 3} \frac{\int _{3}^{x} 8 t ^{2} d t}{\frac{3 r _{2} x}{2} - r _{2} x ^{2} - r _{1} x ^{3} - 3 x} is equal to_______$ 118. $If α = lim_{x \to 0^{+}} (\frac{e ^{t a n x} - e ^{x}}{t a n x - x}) and β = lim_{x \to 0} (1 + sin x)^{\frac{1}{2} c o t x} are the roots of the quadratic equation a x^{2} + b x - e = 0, then 12 lo g_{e} (a + b) is equal to______$ 119. $Let a > 0 be a root of the equation 2 x^{2} + x - 2 = 0 . If lim_{x \to \frac{1}{a}} \frac{16 ( 1 - c o s ( 2 + x - 2 x ^{2} ))}{( 1 - a x ) ^{2}} = α + β 17, where α, β \in Z, then α + β is equal to______$ 120. $If lim_{x \to 0} \frac{α x e ^{x} - β l o g _{e} ( 1 + x ) + γ x ^{2} e ^{- x}}{x s i n ^{2} x} = 10, α, β, γ \in R, then the value of α + β + γ is$