Question

The value of $\underset{x\to 0}{\lim }\frac{\underset{0}{\overset{x^2}{\int }}{\sec }^{2}tdt}{x\sin x}$

• A. 0
• B. 3
• C. 2
• D. 1

Answer 1

Answer: (D)

$\underset{x\to 0}{\lim }\frac{\frac{d}{dx}\underset{0}{\overset{x^2}{\int }}{\sec }^{2}tdt}{\frac{d}{dx}\left(x\sin x\right)}=\underset{x\to 0}{\lim }\frac{{\sec }^2{x}^2.2x}{\sin x+x\cos x}$
(by L’ Hospital rule)
$\underset{x\to 0}{\lim }\frac{2{\sec }^2{x}^2}{\left(\frac{\sin x}{x}+\cos x\right)}=\frac{2\times 1}{1+1}=1$