$\underset{x\to 0}{lim} \frac{{\sin }^{2}x}{\sqrt{2}-\sqrt{1+\cos x}}$ equals.

Solution:

$\underset{x\to 0}{lim} \frac{{\sin }^{2}x}{\sqrt{2}-\sqrt{1+\cos x}} \left(\frac{0}{0}\text{ form }\right)$

So, using L' Hospital' s rule, we get

$\underset{\mid x\to 0}{lim} \frac{2\sin x\cos x}{\frac{-1}{2\sqrt{1+\cos x}}\times (-\sin x)}=\underset{x\to 0}{lim} 2\cos x\times 2\sqrt{1+\cos x}=4\sqrt{2}$