Question

The value of ${lim}_{x\leftarrow 0}\frac{\frac{d}{dx}{\int }_2^{x^2}\sqrt{\cos t}dt}{1-\sqrt{\cos x}}$

Answer 1

$\underset{x\to 0}{lim}\frac{\frac{d}{dx}{\int }_2^{x^2}\sqrt{\cos t}dt}{1-\sqrt{\cos x}}\left(\frac{0}{0}form\right)$

By $L^1$- Hospital rule

$=\underset{x\to 0}{lim}\frac{\sqrt{\cos x^2}\times 2x}{-\frac{1}{2\sqrt{\cos x}}(-\sin x)}$

$=\underset{x\to 0}{lim}\frac{\sqrt{\cos x^2}\times \sqrt{\cos x}\times 4x}{\sin x}$

$=\underset{x\to 0}{lim}1\times 1\times 4\times \frac{x}{\sin x}$

$=1\times 1\times 4\times 1$

$=4$