A) \[\infty \]
B) 0
C) 1
D) \[\frac{1}{2}\]
Correct Answer: D
Solution :
[d]:\[\underset{x\to \infty }{\mathop{\lim }}\,\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}\] \[=\underset{x\to \infty }{\mathop{\lim }}\,\frac{\sqrt{x+\sqrt{x}}}{\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}}\] (Using Reationalisation) \[=\underset{x\to \infty }{\mathop{\lim }}\,\frac{\sqrt{1+\frac{1}{\sqrt{x}}}}{\sqrt{x+\sqrt{\frac{1}{x}+\frac{1}{x\sqrt{x}}}}+1}=\frac{1}{1+1}=\frac{1}{2}\]
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