Question

Consider the function $\frac{\sqrt{(1+x)}-\sqrt{(1-x)}}{x}$ Find the limit as $x\to 0.$

Answer 1

Required limit is, $=\underset{x\to 0}{lim}\frac{\sqrt{(1+x)}-\sqrt{(1-x)}}{x}$
$=\underset{x\to 0}{lim}\frac{\sqrt{(1+x)}-\sqrt{(1-x)}}{x}.\frac{\sqrt{(1+x)}+\sqrt{(1-x)}}{\sqrt{(1+x)}+\sqrt{(1-x)}}$
$=\underset{x\to 0}{lim}\frac{(1+x)-(1-x)}{x[\sqrt{(1+x)}+\sqrt{(1-x)}]}$
$=\underset{x\to 0}{lim}\frac{2}{\sqrt{(1+x)}+\sqrt{(1-x)}}=\frac{2}{1+1}=1$.