Question

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

• A. $\frac{2}{3}$
• B. $\frac{2}{\sqrt{3}}$
• C. $\frac{3\sqrt{3}}{2}$
• D. $\frac{2}{3\sqrt{3}}$

Answer 1

Answer: (D)

$\underset{x\to a}{\lim }\frac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{3a+x}-2\sqrt{x}}$
$=\underset{x\to a}{\lim }\frac{\sqrt{3a+x}+2\sqrt{x}}{\sqrt{a+2x}+\sqrt{3x}}\times \frac{a+2x-3x}{3a+x-4x}$
$=\underset{x\to a}{\lim }\frac{\sqrt{3a+x}+2\sqrt{x}}{\sqrt{a+2x}+\sqrt{3x}}\times \frac{a-x}{3a-3x}$
$=\underset{x\to a}{\lim }\cdot \frac{1}{3}\left\{\frac{\sqrt{3a+x}+2\sqrt{x}}{\sqrt{a+2x}+\sqrt{3x}}\right\}\cdot \left(\frac{a-x}{a-x}\right)$
$=\frac{1}{3}\left\{\frac{\sqrt{4a}+2\sqrt{a}}{\sqrt{a+2a}+\sqrt{3a}}\right\}$
$=\frac{1}{3}\left(\frac{2\sqrt{a}+2\sqrt{a}}{\sqrt{3a}+\sqrt{3a}}\right)$
$=\frac{1}{3}\cdot \frac{4\sqrt{a}}{2\sqrt{3}\cdot \sqrt{a}}=\frac{2}{3\sqrt{3}}$