Question

Evaluate:
$\underset{x\to 0}{lim}\frac{\sqrt{1+\sin x}-\sqrt{1-\sin x}}{x}$

Answer 1

$\underset{x\to 0}{lim}\frac{\sqrt{1+\sin x}-\sqrt{1-\sin x}}{x}$

it is of the form of $\frac{0}{0}$

So, we will apply L Hospital rule

${lim}_{x\to 0}\frac{\frac{d}{dx}(\sqrt{1+\sin x}-\sqrt{1-\sin x})}{\frac{d}{dx}x}={lim}_{x\to 0}\frac{\frac{\cos x}{2\sqrt{1+\sin x}}+\frac{\cos x}{2\sqrt{1-\sin x}}}{1}=\frac{\frac{1}{2}+\frac{1}{2}}{1}=1$