Question

${lim}_{x\to a}\frac{\sqrt{a+2x}-\sqrt{3a}}{\sqrt{x}-\sqrt{a}}$

Answer 1

${lim}_{x\to a}\frac{\sqrt{a+2x}-\sqrt{3a}}{\sqrt{x}-\sqrt{a}}=\frac{0}{0}$

Applying L-Hospital Rule

${lim}_{x\to a}\frac{\frac{1}{2\sqrt{a+2x}}\left(2\right)}{\frac{1}{2\sqrt{x}}}={lim}_{x\to a}\frac{2\sqrt{x}}{\sqrt{a+2x}}$

$=\frac{2\sqrt{a}}{\sqrt{a+2a}}=\frac{2\sqrt{a}}{\sqrt{3}\sqrt{a}}=\frac{2}{\sqrt{3}}$