A) \[\sqrt{2a}\]
B) \[\frac{1}{\sqrt{2a}}\]
C) 2a
D) \[\frac{1}{2a}\]
Correct Answer: B
Solution :
\[\underset{x\to a}{\mathop{\lim }}\,\frac{\sqrt{3x-a}-\sqrt{x+a}}{x-a}\] \[=\underset{x\to a}{\mathop{\lim }}\,\frac{\sqrt{3x-a}-\sqrt{x+a}}{x-a}\] \[\times \,\frac{(\sqrt{3x-a}+\sqrt{x+a})}{(\sqrt{3x-a}+\sqrt{x+a})}\] \[=\underset{x\to a}{\mathop{\lim }}\,\frac{3x-a-(x+a)}{(x-a)(\sqrt{3x-a}+\sqrt{x+a})}\] \[=\underset{x\to a}{\mathop{\lim }}\,\frac{2x-2a}{(x-a)(\sqrt{3x-a}+\sqrt{x+a})}\] \[=\frac{2}{(\sqrt{2a}+\sqrt{2a})}=\frac{1}{\sqrt{2a}}\] Alternate Solution: \[\underset{x\to a}{\mathop{\lim }}\,\frac{\sqrt{3x-a}-\sqrt{x+a}}{x-a}\] Using L Hospitals rule \[=\underset{x\to a}{\mathop{\lim }}\,\frac{\frac{1(3)}{2\sqrt{(3x-a)}}-\frac{1(1)}{2\sqrt{(x+a)}}}{1}\] \[=\frac{3}{2\sqrt{2}a}-\frac{1}{2\sqrt{2a}}\] \[=\frac{1}{\sqrt{2a}}\]
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