Question

Evaluate ${lim}_{x\to a}\frac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{3a+x}-2\sqrt{x}}(a\ne 0)$

Answer 1

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

$=\underset{x\to a}{lim}\frac{(\sqrt{a+2x}-\sqrt{3x})}{(\sqrt{3a+x}-2\sqrt{x})}\times \frac{(\sqrt{3a+x}+2\sqrt{x})}{(\sqrt{3a+x}+2\sqrt{x})}\times \frac{(\sqrt{a+2x}+\sqrt{3x})}{(\sqrt{a+2x}+\sqrt{3x}}$

$=\underset{x\to a}{lim}\frac{(a+2x-3x)}{(3a+x-4x)}\times \frac{\sqrt{3a+x}+2\sqrt{x}}{(\sqrt{a+2x}+\sqrt{3x})}$

$=\underset{x\to a}{lim}\frac{(a-x)}{3(a-x)}\times \frac{\sqrt{4a}+2\sqrt{a}}{\sqrt{3a}+\sqrt{3a}}$

$=\frac{1}{3}\times \frac{4\sqrt{a}}{2\sqrt{3}\sqrt{a}}$

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