First slide

Introduction to limits

Question

$The value of \lim_{x \to 0} \frac{\int_{0}^{x^{2}} s e c^{2} t d t}{x \sin x} is$

Moderate

A

0
B

3
C

2
D

1

Solution

$\begin{array}{l} \lim_{x \to 0} \frac{\int_{0}^{x^{2}} \sec^{2} t d t}{x \sin x} \\ = \lim_{x \to 0} \frac{\frac{d}{d x} \int_{0}^{x^{2}} \sec^{2} t d t}{\frac{d}{d x} x \sin x} (By L' Hospital rule) \\ = \lim_{x \to 0} \frac{\sec^{2} x^{2} \cdot 2 x}{\sin x + x \cos x} \\ = \lim_{x \to 0} \frac{2 x \sec^{2} x^{2}}{x (\frac{\sin x}{x} + \cos x)} \\ = \lim_{x \to 0} \frac{2 \sec^{2} x^{2}}{\frac{\sin x}{x} + \cos x} \\ = \frac{2 \sec^{2} 0}{1 + 1} \end{array}$
$\begin{array}{l} = \frac{2}{2} \\ = 1 \end{array}$

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