A) \[-1/2\]
B) \[1/2\]
C) \[1\]
D) \[0\]
Correct Answer: B
Solution :
\[\underset{x\to \infty }{\mathop{\lim }}\,\left[ \frac{\sqrt{x+\sqrt{x}}-\sqrt{x}}{1} \right]\] \[=\underset{x\to \infty }{\mathop{\lim }}\,\left[ \frac{\sqrt{x+\sqrt{x}}-\sqrt{x}}{1}\times \frac{\sqrt{x+\sqrt{x}}+\sqrt{x}}{\sqrt{x+\sqrt{x}}+\sqrt{x}} \right]\] \[=\underset{x\to \infty }{\mathop{\lim }}\,\frac{x+\sqrt{x}-x}{\sqrt{x+\sqrt{x}}+\sqrt{x}}\] \[=\underset{x\to \infty }{\mathop{\lim }}\,\frac{\sqrt{x}}{\sqrt{x}\left[ \sqrt{1+\frac{1}{\sqrt{x}}}+1 \right]}\] \[=\underset{x\to \infty }{\mathop{\lim }}\,\frac{1}{\sqrt{1+\frac{1}{\sqrt{x}}}+1}=\frac{1}{2}\]
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