Question

Calculate the following limits.
$\underset{x\to 0}{lim}\frac{\sqrt{6+x}-\sqrt{6-x}}{x}.$

Answer 1

$\underset{x\to 0}{lim}\frac{\sqrt{6+x}-\sqrt{6-x}}{x}$

$=\underset{x\to 0}{lim}\frac{\sqrt{6+x}-\sqrt{6-x}}{x}\frac{\sqrt{6+x}+\sqrt{6-x}}{\sqrt{6+x}+\sqrt{6-x}}$

$=\underset{x\to 0}{lim}\frac{6+x-6+x}{x(\sqrt{6+x}+\sqrt{6-x})}$

$=\underset{x\to 0}{lim}\frac{2}{(\sqrt{6+x}+\sqrt{6-x})}$

$=\frac{2}{2\sqrt{6}}$

$=\frac{1}{\sqrt{6}}$