Question

Evaluate:

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

Answer 1

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

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

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

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

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

$\left[x\to 0,x^2\ne 0\right]$

$=\frac{\sqrt{a+0^0}+\sqrt{a-0^2}}{2}$

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

$=\sqrt{a}$

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