Evaluate the following limits:


Evaluate the following limits:

$\lim _{x \rightarrow 0} \frac{\sqrt{2}-\sqrt{1+\cos x}}{\sin ^{2} x}$



$=\lim _{x \rightarrow 0} \frac{\sqrt{2}-\sqrt{1+\cos x}}{\sin x \times \sin x}$

$=\lim _{x \rightarrow 0} \frac{\sqrt{2}-\sqrt{1+\cos x}}{\sin x \times \sin x} \times \frac{\sqrt{2}+\sqrt{1+\cos x}}{\sqrt{2}+\sqrt{1+\cos x}}$

$=\lim _{x \rightarrow 0} \frac{2-(1-\cos x)}{\sin x \times \sin x \times \sqrt{2}+\sqrt{1+\cos x}}$

$=\lim _{x \rightarrow 0} \frac{1-\cos x}{2 \times \sin \frac{x}{2} \cos \frac{x}{2} \sin x(\sqrt{2}+\sqrt{1+\cos x})}$

$=\lim _{x \rightarrow 0} \frac{2 \times \sin \frac{x}{2} \times \sin \frac{x}{2}}{2 \times \sin \frac{x}{2} \cos \frac{x}{2} \sin x(\sqrt{2}+\sqrt{1+\cos x})}$

$=\lim _{x \rightarrow 0} \frac{\frac{\sin \frac{x}{2}}{\frac{x}{2}} \times \frac{1}{2}}{\frac{\sin x}{x}} \times \lim _{x \rightarrow 0} \frac{1}{\cos \frac{x}{2} \times(\sqrt{2}+\sqrt{1+\cos x})}$

$=\frac{1}{2} \times \frac{1}{(\sqrt{2}+\sqrt{2})}$

$=\frac{1}{4 \sqrt{2}}$

$\therefore \lim _{x \rightarrow 0} \frac{\sqrt{2}-\sqrt{1+\cos x}}{\sin x \times \sin x}=\frac{1}{4 \sqrt{2}}$


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