Question

Evaluate

$\underset{x\to a}{lim}\frac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{3a+x}-2\sqrt{x}},a\ne 0$

Answer 1

$\underset{x\to a}{lim}\frac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{3a+x}-2\sqrt{x}}=\underset{x\to a}{lim}\frac{\frac{2}{2\sqrt{a+2x}}-\frac{3}{2\sqrt{3x}}}{\frac{1}{2\sqrt{3a+x}}-\frac{2}{2\sqrt{x}}}=\frac{\frac{1}{\sqrt{3a}}-\frac{3}{2\sqrt{3a}}}{\frac{1}{4\sqrt{a}}-\frac{1}{\sqrt{a}}}=\frac{2}{3\sqrt{3}}$