A) \[\frac{2}{3}\]
B) \[\frac{2}{\sqrt{3}}\]
C) \[\frac{3\sqrt{3}}{2}\]
D) \[\frac{2}{3\sqrt{3}}\]
Correct Answer: D
Solution :
\[\underset{x\to a}{\mathop{\lim }}\,\frac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{3a+x}-2\sqrt{x}}\] \[=\underset{x\to a}{\mathop{\lim }}\,\frac{\sqrt{3a+x}+2\sqrt{x}}{\sqrt{a+2x}+\sqrt{3x}}\times \frac{a+2x-3x}{3a+x-4x}\] \[=\underset{x\to a}{\mathop{\lim }}\,\frac{\sqrt{3a+x}+2\sqrt{x}}{\sqrt{a+2x}+\sqrt{3x}}\times \frac{a+2x-3x}{3a+x-4x}\] \[=\underset{x\to a}{\mathop{\lim }}\,\frac{\sqrt{3a+x}+2\sqrt{x}}{\sqrt{a+2x}+\sqrt{3x}}\times \frac{a-x}{3a-3x}\] \[=\underset{x\to a}{\mathop{\lim }}\,.\frac{1}{3}\left\{ \frac{\sqrt{3a+x}+2\sqrt{x}}{\sqrt{a+2x}+\sqrt{3x}} \right\}.\left( \frac{a-x}{a-x} \right)\] \[=\frac{1}{3}\left\{ \frac{\sqrt{4a}+2\sqrt{a}}{\sqrt{a+2a}+\sqrt{3a}} \right\}=\frac{1}{3}\left( \frac{2\sqrt{a}+2\sqrt{a}}{\sqrt{3a}+\sqrt{3a}} \right)\] \[=\frac{1}{3}.\frac{4\sqrt{a}}{2\sqrt{3}.\sqrt{a}}=\frac{2}{3\sqrt{3}}\]
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