Question

Evaluate the following limit:

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

Answer 1

$\underset{x\to 0}{lim}\frac{\sqrt{(1+\sin x)}-\sqrt{(1-\sin x)}}{x}$
$=\underset{x\to 0}{lim}\frac{\sqrt{(1+\sin x)}-\sqrt{(1-\sin x)}}{x}.\frac{\sqrt{(1+\sin x)}+\sqrt{(1-\sin x)}}{\sqrt{(1+\sin x)}+\sqrt{(1-\sin x)}}$
$=\underset{x\to 0}{lim}\frac{2\sin x}{x}.\frac{1}{\sqrt{(1+\sin x)}+\sqrt{(1-\sin x)}}=\frac{2}{1+1}=1$