Question

The value of $\underset{x\to 0}{lim}\frac{{\int }_0^{x^2}{\sec }^{2}tdt}{x\sin x}$ is

• A. 2
• B. 1
• C. 0
• D. 3

Answer 1

The correct option is B 1

Formulas required:-

1.${lim}_{x\to 0}\frac{\sin x}{x}=0$

2.${lim}_{x\to 0}\frac{\tan x}{x}=0$

Solution:-

$\underset{x\to 0}{lim}\frac{{\int }_0^{x^2}{\sec }^{2}tdt}{x\sin x}$

$=\underset{x\to 0}{lim}\frac{{\left(\tan t\right)}_0^{x^2}}{x\sin x}$

$=\underset{x\to 0}{lim}\frac{\tan x^2}{x\sin x}=\underset{x\to 0}{lim}\frac{\tan x^2}{x^2\frac{\sin x}{x}}$

$=\underset{x\to 0}{lim}\frac{\tan x^2}{x^2}\frac{1}{{lim}_{x\to 0}\frac{\sin x}{x}}$

$=1\times 1=1$