A) \[\frac{1}{\sqrt{2}}\]
B) \[\frac{1}{2}\]
C) \[\frac{1}{2\sqrt{2}}\]
D) 1
Correct Answer: B
Solution :
\[\underset{x\to \pi /4}{\mathop{\lim }}\,\,\frac{(\sqrt{2}-\sec x)\,\cos x\,(1+\cot x)}{\cot x\,[2-{{\sec }^{2}}x]}\] \[=\underset{x\to \pi /4}{\mathop{\lim }}\,\frac{\sin x\,(1+\cot x)}{(\sqrt{2}+\sec x)}=\frac{\frac{1}{\sqrt{2}}(2)}{\sqrt{2}+\sqrt{2}}=\frac{1}{2}.\] Aliter : Apply L-Hospital?s rule.
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