Question

${lim}_{x\to 0}\frac{2x}{\sqrt{a+x}-\sqrt{a-x}}$

Answer 1

${lim}_{x\to 0}\frac{2x}{\sqrt{a+x}-\sqrt{a-x}}$

Rationalising the denominator

$={lim}_{x\to 0}\frac{2x}{(\sqrt{a+x}-\sqrt{a-x})}\times \frac{\sqrt{a+x}+\sqrt{a-x}}{(\sqrt{a+x}-\sqrt{a-x})}$

$={lim}_{x\to 0}\frac{2x(\sqrt{a+x}+\sqrt{a-x})}{\left(\right(a+x)-(a-x\left)\right)}$

$={lim}_{x\to 0}\frac{2x\sqrt{a+x}+\sqrt{a-x}}{2x}$

$={lim}_{x\to 0}(\sqrt{a+x}-\sqrt{a-x})=\sqrt{a}+\sqrt{a}$

$=2\sqrt{a}$