Limits PYQs Set 1 (PDF)

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Limits · Set 1 (Q1-30)
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1. $Let f (x) = sin x, g (x) = cos x, h (x) = x^{2} then lim_{x \to 1} \frac{f ( g ( h ( x ) ) ) - f ( g ( h ( 1 ) ) )}{x - 1} =$ (1) $0$ (2) $- 2 sin 1 cos (cos 1)$ (3) $\infty$ (4) $- 2 sin 1 cos 1$ 2. $lim x \to 0 \frac{x ( e ^{\frac{( 1 + x ^{2} + x ^{4} - 1 )}{x}} - 1 )}{1 + x ^{2} + x ^{4} - 1}$ (1) $is equal to 0.$ (2) $is equal to e$ (3) $is equal to 1.$ (4) $does not exist.$ 3. $lim_{x \to 1} \frac{tan ( x ^{2} - 1 )}{x - 1} is equal to$ (1) $2$ (2) $\frac{1}{2}$ (3) $-2$ (4) $\frac{- 1}{2}$ 4. $Let f be a differentiable function on R such that f (2) = 1, f^{’} (2) = 4 . Let lim x \to 0 (f (2 + x))^{3/ x} = e^{α} . Then the number of times the curve y = 4 x^{3} - 4 x^{2} - 4 (α - 7) x - α meets x axis is:$ (1) $3$ (2) $1$ (3) $2$ (4) $0$ 5. $Let f : R \to R be a differentiable function satisfying f^{’} (3) + f^{’} (2) = 0 . Then lim_{x \to 0} (\frac{1 + f ( 3 + x ) - f ( 3 )}{1 + f ( 2 - x ) - f ( 2 )})^{\frac{1}{x}} is equal to:$ (1) $1$ (2) $e^{- 1}$ (3) $e$ (4) $e^{2}$ 6. $lim x \to 0 \frac{2 ^{2 x} - 2 ^{x + 1} + 2 - cos 2 x}{x ^{2}} =$ (1) $2 + log 2$ (2) $2 + (log 2)^{2}$ (3) $2 + (log 4)^{2}$ (4) $2 + log 4$ 7. $lim x \to 0 (\frac{3 x ^{2} + 2}{7 x ^{2} + 2})^{\frac{1}{x ^{2}}} is equal to:$ (1) $e^{- 2}$ (2) $e^{2}$ (3) $e^{2} 7$ (4) $e^{\frac{3}{7}}$ 8. $For each t \in R, let [t] be the greatest integer less than or equal to t . Then, lim x \to 1_{+} \frac{( 1 - \| x \| + sin \| 1 - x \| ) sin ( \frac{π}{2} [ 1 - x ] )}{\| 1 - x \| [ 1 - x ]}$ (1) $equals 1$ (2) $equals 0$ (3) $equals - 1$ (4) $does not exist$ 9. $Given below are two statements: Statement I: lim x \to 0 (\frac{tan ^{- 1} x + log _{e} \frac{1 + x}{1 - x} - 2 x}{x ^{5}}) = \frac{2}{5} Statement II: lim x \to 1 (x^{\frac{2}{1 - x}}) = \frac{1}{e ^{2}} In the light of the above statements, choose the correct answer from the options given below:$ (1) $Statement I is false but Statement II is true$ (2) $Both Statement I and Statement II are false$ (3) $Both Statement I and Statement II are true$ (4) $Statement I is true but Statement II is false$ 10. $lim_{x \to 0} \frac{2 sin x - sin 2 x}{x ^{3}} is equal to$ (1) $1$ (2) $0$ (3) $-1$ (4) $does not exist$ 11. $lim_{x \to 0} (\frac{3 x ^{2} + 2}{7 x ^{2} + 2})^{\frac{1}{x ^{2}}} is equal to$ (1) $e^{- 2}$ (2) $e^{2}$ (3) $e^{2/7}$ (4) $e^{3/7}$ 12. $lim_{x \to \frac{π}{2}} (\frac{1}{( x - \frac{π}{2} ) ^{2}} \int_{x^{3}}^{(\frac{π}{2})^{3}} cos (\frac{1}{t ^{3}}) dt) is equal to$ (1) $\frac{3 π}{8}$ (2) $\frac{3 π ^{2}}{4}$ (3) $\frac{3 π ^{2}}{8}$ (4) $\frac{3 π}{4}$ 13. $lim_{x \to \frac{π}{2}} (\frac{2 x - π}{cos x}) is equal to$ (1) $0$ (2) $\frac{1}{2}$ (3) $-2$ (4) $5$ 14. $Let f (x) = ⎩ ⎨ ⎧ (1 + a x)^{1/ x}, 1 + b, \frac{( x + 4 ) ^{1/2} - 2}{( x + c ) ^{1/3} - 2}, x < 0 x = 0 x > 0 be continuous at x = 0 . Then e^{a} b c is equal to:$ (1) $64$ (2) $48$ (3) $36$ (4) $72$ 15. $Let f : (0, \infty) \to (0, \infty) be a differentiable function such that f (1) = e and lim t \to x \frac{t ^{2} f ^{2} ( x ) - x ^{2} f ^{2} ( t )}{t - x} = 0 . If f (x) = 1, then x is equal to:$ (1) $2 e$ (2) $e$ (3) $1 2 e$ (4) $\frac{1}{e}$ 16. $If lim_{x \to 0} \frac{3 + α sin x + β cos x + log _{e} ( 1 - x )}{3 tan ^{2} x} = \frac{1}{3}, then 2 α - β is equal to :$ (1) $2$ (2) $7$ (3) $5$ (4) $1$ 17. $lim n \to \infty (1 + \frac{1 + \frac{1}{2} + \dots \dots + \frac{1}{n}}{n ^{2}})^{n} is equal to$ (1) $\frac{1}{2}$ (2) $\frac{1}{e}$ (3) $1$ (4) $0$ 18. $The value of lim_{x \to 0} \frac{log _{e} ( sec ( ex ) \cdot sec ( e ^{2} x ) \dots \cdot sec ( e ^{10} x ) )}{e ^{2} - e ^{2 cosx}} is equal to$ (1) $\frac{( e ^{20} - 1 )}{2 ( e ^{2} - 1 )}$ (2) $\frac{( e ^{10} - 1 )}{2 ( e ^{2} - 1 )}$ (3) $\frac{( e ^{10} - 1 )}{2 e ^{2} ( e ^{2} - 1 )}$ (4) $\frac{( e ^{20} - 1 )}{2 e ^{2} ( e ^{2} - 1 )}$ 19. $The value of lim x \to 0^{+} \frac{cos ^{- 1} ( x - [ x ] ^{2} ) \cdot sin ^{- 1} ( x - [ x ] ^{2} )}{x - x ^{3}}, where [x] denotes the greatest integer \leq x is :$ (1) $π$ (2) $0$ (3) $\frac{π}{4}$ (4) $\frac{π}{2}$ 20. $If f (a) = 2, f^{’} (a) = 1, g (a) = - 1, g^{’} (a) = 2, then as x approaches a, \frac{g ( x ) f ( a ) - g ( a ) f ( x )}{( x - a )} approaches$ (1) $3$ (2) $5$ (3) $0$ (4) $2$ 21. $If lim_{x \to 1} \frac{x ^{4} - 1}{x - 1} = lim_{x \to k} \frac{x ^{3} - k ^{3}}{x ^{2} - k ^{2}}, then k is:$ (1) $\frac{8}{3}$ (2) $\frac{3}{8}$ (3) $\frac{3}{2}$ (4) $\frac{4}{3}$ 22. $Let lim_{n \to \infty} (\frac{n}{n ^{4} + 1} - \frac{2 n}{( n ^{2} + 1 ) n ^{4} + 1} + \frac{n}{n ^{4} + 16} - \frac{8 n}{( n ^{2} + 4 ) n ^{4} + 16} + \dots + \frac{n}{n ^{4} + n ^{4}} - \frac{2 n \cdot n ^{2}}{( n ^{2} + n ^{2} ) n ^{4} + n ^{4}}) be \frac{π}{k}, using only the principal values of the inverse trigonometric functions. Then k^{2} is equal to_______$ 23. $Let denote the fractional part of and . If and respectively denotes the left hand limit and the right hand limit of at , then is equal to$ 24. $For t > - 1, let α_{t} and β_{t} be the roots of the equation ((t + 2)^{1/7} - 1) x^{2} + ((t + 2)^{1/6} - 1) x + ((t + 2)^{1/21} - 1) = 0. if lim t \to - 1^{+} α_{t} = a and lim t \to - 1^{+} β_{t} = b then 72 (a + b)^{2} is equal to ______.$ 25. $Let [\cdot] denote the greatest integer function and f (x) = lim_{n \to \infty} \frac{1}{n ^{3}} \sum_{k = 1}^{n} (\frac{k ^{2}}{3 ^{x}}) . Then 12 \sum_{j = 1}^{\infty} f (j) is equal to____.$ 26. $lim_{x \to 0} \frac{a x ^{2} e ^{x} - b log _{e} ( 1 + x ) + cx e ^{- x}}{x ^{2} sin x} = 1 then 16 (a^{2} + b^{2} + c^{2}) is equal to$ 27. $If lim_{x \to 1} \frac{( 5 x + 1 ) ^{1/3} - ( x + 5 ) ^{1/3}}{( 2 x + 3 ) ^{1/2} - ( x + 4 ) ^{1/2}} = \frac{m 5}{n ( 2 n ) ^{2/3}}, where gcd (m, n) = 1, then 8 m + 12 n is equal to______$ 28. $If lim x \to 0 \frac{ax - ( e ^{4 x} - 1 )}{ax ( e ^{4 x} - 1 )} exists and is equal to b, then the value of a - 2 b is$ 29. $The value of lim_{x \to 0} 2 (\frac{1 - cos x cos 2 x 3 cos 3 x \dots \dots \cdot 10 cos 10 x}{x ^{2}}) is$ 30. $If lim x \to 1 \frac{x + x ^{2} + x ^{3} + \dots + x ^{n} - n}{x - 1} = 820, (n \in N) then the value of n is equal to \dots$