A) \[log\text{ }a\]
B) \[log\text{ }2\]
C) \[a\]
D) \[log\text{ }x\]
E) none of these
Correct Answer: A
Solution :
\[\underset{x\to \frac{\pi }{2}}{\mathop{\lim }}\,\frac{{{a}^{\cot x}}-{{a}^{\cos x}}}{\cot x-\cos x}\] \[=\underset{x\to \frac{\pi }{2}}{\mathop{\lim }}\,\frac{\left[ \begin{align} & 1+\cos x\log a+\frac{{{\cot }^{2}}x}{2!}{{(\log a)}^{2}}+.... \\ & -1-\cos x\log a-\frac{{{\cos }^{2}}x}{2!}{{(\log a)}^{2}}+... \\ \end{align} \right]}{\cot x-\cos x}\] \[=\underset{x\to \frac{\pi }{2}}{\mathop{\lim }}\,\frac{\left[ \begin{align} & \log a(\cot x-\cos x)+\frac{{{(\log a)}^{2}}}{2!} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,({{\cot }^{2}}x-{{\cos }^{2}}x)+... \\ \end{align} \right]}{\cot x-\cos x}\] \[=\underset{x\to \frac{\pi }{2}}{\mathop{\lim }}\,\log a+\frac{{{(\log a)}^{2}}}{2!}(\cos x+\cos x)+...\] \[=\log a\]
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