A) \[\log \,2\]
B) \[1\]
C) \[2\]
D) None of these
Correct Answer: A
Solution :
\[\underset{x\to \pi /2}{\mathop{\lim }}\,\frac{{{2}^{\cos x}}-{{2}^{\cos x}}}{\cot \,x-cosx}\] \[=\underset{x\to \pi /2}{\mathop{\lim }}\,\frac{{{2}^{\cos x}}({{2}^{\cot x-\cos x}}-1)}{\cot \,\,x-\cos x}\] \[={{2}^{\cos (\pi /2)}}\underset{x\to \pi /2}{\mathop{\lim }}\,\left( \frac{{{2}^{\cot \,x-\cos \,x}}-1}{\cot \,x\,-\,\cos \,x} \right)\] \[=1.\,\log \,2\] \[\left( \because \,\,\underset{x\to 0}{\mathop{\lim }}\,\frac{{{a}^{x}}-1}{x}=\log \,a \right)\] \[=\log \,\,2\]
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