A) \[\sqrt{2}\]
B) 2
C) \[4\]
D) \[4\sqrt{2}\]
Correct Answer: D
Solution :
| \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\sin }^{2}}x}{\sqrt{2}-\sqrt{1+\cos x}}\] |
| \[=\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\sin }^{2}}x\left( \sqrt{2}+\sqrt{1+\cos x} \right)}{1-\cos x}\] |
| \[=\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\sin }^{2}}x}{{{x}^{2}}}\left( \frac{1}{\frac{1-\cos x}{{{x}^{2}}}} \right)\left( \sqrt{2}+\sqrt{1+\cos x} \right)\] |
| \[=\frac{2\sqrt{2}}{1/2}=4\sqrt{2}.\] |
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