A) \[\frac{1}{4}\]
B) \[\frac{1}{24}\]
C) \[\frac{1}{16}\]
D) \[\frac{1}{8}\]
Correct Answer: C
Solution :
\[\underset{x\to \frac{\pi }{2}}{\mathop{\lim }}\,\frac{\cot x(1-\sin x)}{-8{{\left( x-\frac{\pi }{2} \right)}^{3}}}\] \[=\underset{x\to \frac{\pi }{2}}{\mathop{\lim }}\,\frac{\tan \left( \frac{\pi }{2}-x \right)}{8\left( \frac{\pi }{2}-x \right)}\frac{\left( 1-\cos \left( \frac{\pi }{2}-x \right) \right)}{{{\left( \frac{\pi }{2}-x \right)}^{2}}}\] \[=\frac{1}{8}.1.\frac{1}{2}=\frac{1}{16}\]
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