A) \[1/2\]
B) \[0\]
C) \[1\]
D) None of these
Correct Answer: A
Solution :
\[\underset{x\to \infty }{\mathop{\lim }}\,\left( \sqrt{x+\sqrt{x+\sqrt{x}}} \right.-\sqrt{x}\] \[\times \left. \frac{\sqrt{x\sqrt{x+\sqrt{x}}}+\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x+}}}\sqrt{x}} \right)\] \[=\underset{x\to \infty }{\mathop{\lim }}\,\left( \frac{\sqrt{x+\sqrt{x}}}{\sqrt{x+\sqrt{x+\sqrt{x}}}+\sqrt{x}} \right)\] \[=\underset{x\to \infty }{\mathop{\lim }}\,\frac{(\sqrt{1+\sqrt{y}})/\sqrt{y}}{\sqrt{\frac{1}{y}+\frac{\sqrt{1+}\sqrt{y}}{\sqrt{y}}+\frac{1}{\sqrt{y}}}}\] \[\left( Put\,\,x=\frac{1}{y} \right)\] \[=\underset{y\to 0}{\mathop{\lim }}\,\frac{\sqrt{1+\sqrt{y}}}{\sqrt{1+\sqrt{y}(1+\sqrt{y})+1}}=\frac{1}{2}\]
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