Differential Calculus PYQ Topic Analysis

Evaluate $lim_{x \to 0} (1 + sin x)^{1/ x}$ . – 2019
$lim_{x \to 0} \frac{x ( 1 + a cos x ) - b sin x}{x ^{3}}$ exists. Find also the value of the limit. – 2021
evaluate $lim_{x \to 1} (\frac{x}{x - 1} - \frac{1}{lo g x})$ . – 2024
Evaluate $lim_{x \to 0} x^{5} sin x$ . – 2023

$f (x) = {x^{2} - 16 x, x < 912 x, x \geq 9$ . Is $f (x)$ continuous at $x = 9$ ? Determine whether $f (x)$ is differentiable at $x = 9$ . – 2019
$f (x) = {3 x + 3, x \geq 0 x + 3, - \frac{3}{2} \leq x < 0 - 3 x - 3, x < - \frac{3}{2}$ . Discuss continuity and differentiability at $x = - \frac{3}{2}$ . – 2020
$f (x) = {1, x < 01 + sin x, 0 \leq x < \frac{π}{2} 2 + (x - \frac{π}{2})^{2}, x \geq \frac{π}{2}$ . Discuss continuity and differentiability at $x = 0$ . – 2021
$f (x) = ∣ x - 1∣ + ∣ x + 3∣$ . Examine continuity and differentiability at $x = - 3$ and $x = 1$ . – 2022
$f (x) = {\frac{1}{2} - x, 0 < x < \frac{1}{2} \frac{1}{2}, x = \frac{1}{2} \frac{3}{2} - x, \frac{1}{2} < x < 1 \frac{5}{2} - 2 x, x \geq 1$ . Discuss continuity at $x = \frac{1}{2}$ and the limit at $x = 1$ . – 2023
$f (x) = {3 + x^{2}, x < 01 + sin x, 0 \leq x < \frac{π}{2} 2 + (x - \frac{π}{2})^{2}, x \geq \frac{π}{2}$ . Discuss existence of $f^{'} (x)$ at $x = \frac{π}{2}$ and $x = 0$ . – 2023
Discuss differentiability at $x = \frac{π}{2}$ of $f (x) = {1 + sin x, 0 \leq x < \frac{π}{2} 2 + (x - \frac{π}{2})^{2}, x \geq \frac{π}{2}$ . – 2024

If $y = sin^{- 1} x$ , show that $(1 - x^{2}) y_{n + 2} - (2 n + 1) x y_{n + 1} - n^{2} y_{n} = 0$ . – 2019
If $y = e^{c o s^{- 1} x}$ , prove $(1 - x^{2}) y_{n + 2} - (2 n + 1) x y_{n + 1} - (n^{2} + 1) y_{n} = 0$ . – 2021
If $y = sin (m sin^{- 1} x)$ , prove $(1 - x^{2}) y_{n + 2} - (2 n + 1) x y_{n + 1} + (m^{2} - n^{2}) y_{n} = 0$ . – 2022
If $y = tan^{- 1} x$ , prove $(1 + x^{2}) y_{n + 1} + 2 n x y_{n} + n (n - 1) y_{n - 1} = 0$ . – 2023
If $y = sin (m sin^{- 1} x)$ , prove $(1 - x^{2}) y_{n + 2} - (2 n + 1) x y_{n + 1} + (m^{2} - n^{2}) y_{n} = 0$ . – 2024

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