Question

Evaluate:

$\underset{x\to 0}{lim}\frac{\sqrt{1+2x^2}-\sqrt{1-2x^2}}{x^2}$.

Answer 1

Now,

$\underset{x\to 0}{lim}\frac{\sqrt{1+2x^2}-\sqrt{1-2x^2}}{x^2}$

$=\underset{x\to 0}{lim}\frac{(1+2x^2)-(1-2x^2)}{x^2(\sqrt{1+2x^2}+\sqrt{1-2x^2})}$

$=\underset{x\to 0}{lim}\frac{4}{(\sqrt{1+2x^2}+\sqrt{1-2x^2})}$

$=\frac{4}{2}$

$=2$.