Question

The value of ${lim}_{x\to 0}\frac{{\int }_0^{x^2}se{c}^{2}tdt}{xsinx}$ is

• A. 3
• B. 2
• C. 1
• D. -1

Answer 1

The correct option is B 2

${lim}_{x\to 0}\frac{{\int }_0^{x^2}{\sec }^{2}tdt}{x\sin x}\left(\frac{0}{0}\right)$

$ApplyL-Hospitalrule,={lim}_{x\to 0}\frac{{\sec }^2\left(x^2\right).2x}{\sin x+\left(\cos x\right)x}={lim}_{x\to 0}\frac{2{\sec }^2\left(x^2\right)}{\frac{\sin x}{x}+\cos x}=\frac{2}{1+1}=1$

Hence, this is the answer.