Question

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

Answer 1

$x-a=(\sqrt{x}{)}^2-(\sqrt{a}{)}^2=(\sqrt{x}+\sqrt{a})(\sqrt{x}-\sqrt{a})$

$⟹\frac{\sqrt{x}-\sqrt{a}}{x-a}=\frac{1}{\sqrt{x}+\sqrt{a}}$

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