Question

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

Answer 1

$\underset{x\to a}{lim}\frac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{3a+x}-2\sqrt{x}}\left(\frac{0}{0}\right)$

L Hospital

$=\underset{x\to a}{lim}\frac{\frac{2}{2\sqrt{a+2x}}-\frac{2}{2\sqrt{3x}}}{\frac{1}{2\sqrt{3a+x}}-\frac{1}{-2\sqrt{x}}}$

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

$=\frac{\frac{2-3}{2\sqrt{3}\sqrt{9}}}{\frac{1-4}{4\sqrt{9}}}$

$=\frac{6}{2\sqrt{3}}\times \frac{4}{3}=\frac{2}{3\sqrt{3}}$