Question

$\underset{x\to 2}{\lim }\frac{x-2}{\sqrt{x}-\sqrt{2}}$

Answer 1

$\underset{x\to 2}{\lim }\left[\frac{x-2}{\sqrt{x}-\sqrt{2}}\right]$

It is of the form $\frac{0}{0}$.

⇒$\underset{x\to 2}{\lim }\left[\frac{{\left(\sqrt{x}\right)}^2-{\left(\sqrt{2}\right)}^2}{x-\sqrt{2}}\right]$

= $\underset{x\to 2}{\lim }\left[\frac{\left(\sqrt{x}-\sqrt{2}\right)\left(\sqrt{x}+\sqrt{2}\right)}{\left(x-\sqrt{2}\right)}\right]$

= $\sqrt{2}+\sqrt{2}$

= $2\sqrt{2}$