A) \[\sqrt{2}a\]
B) \[1/\sqrt{2a}\]
C) 2a
D) \[1/2a\]
Correct Answer: B
Solution :
\[\underset{x\to a}{\mathop{\lim }}\,\,\frac{\sqrt{3x-a}-\sqrt{x+a}}{x-a}\] \[=\underset{x\to a}{\mathop{\lim }}\,\,\frac{\sqrt{3x-a}-\sqrt{x+a}}{(x-a)}\times \frac{\sqrt{3x-a}+\sqrt{x+a}}{\sqrt{3x-a}+\sqrt{x+a}}\] \[=\frac{2}{2\sqrt{2a}}=\frac{1}{\sqrt{2a}}\] Aliter : Apply L-Hospital?s rule \[\underset{x\to a}{\mathop{\lim }}\,\,\frac{\sqrt{3x-a}-\sqrt{x+a}}{x-a}=\underset{x\to a}{\mathop{\lim }}\,\,\frac{3}{2\,\sqrt{3x-a}}-\frac{1}{2\,\sqrt{x+a}}\] \[=\frac{3}{2\sqrt{2a}}-\frac{1}{2\sqrt{2a}}=\frac{1}{\sqrt{2a}}.\]
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