Question

Solve $\underset{x\to a}{lim}\frac{\sqrt{1+ax}-\sqrt{1-ax}}{x}$

Answer 1

$\underset{x\to a}{lim}\frac{\sqrt{1+ax}-\sqrt{1-ax}}{x}$
$\underset{x\to a}{lim}\frac{(\sqrt{1+ax}-\sqrt{1-ax})}{x}\times \frac{(\sqrt{1+ax}+\sqrt{1-ax})}{(\sqrt{1+ax}+\sqrt{1-ax})}$
$=\underset{x\to a}{lim}\frac{1+ax-1+ax}{x(\sqrt{1+ax}+\sqrt{1-ax})}$
$=\underset{x\to a}{lim}\frac{2ax}{x(\sqrt{1+ax}+\sqrt{1-ax})}$
$=\left(\frac{2a}{\sqrt{1+a^2}+\sqrt{1-a^2}}\right)$