Question

Evaluate:

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

Answer 1

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

$\underset{x\to a}{lim}\frac{1}{\sqrt{x}+\sqrt{a}}=\frac{1}{\sqrt{a}+\sqrt{a}}=\frac{1}{2\sqrt{a}}$

$\underset{x\to a}{lim}\frac{\sqrt{x}+\sqrt{a}}{x-a}=\frac{\sqrt{a}+\sqrt{a}}{2a}\Rightarrow \frac{2\sqrt{a}}{2a}=\frac{1}{\sqrt{a}}$