Explanation

(c)
$\lim_{x \rightarrow \frac{\pi}{2}}\frac{a^{\cot x} - a^{\cos x}}{\cot{x - \cos x}}$
$= \lim_{x \rightarrow \frac{\pi}{2}}\frac{\left\lbrack \begin{array}{r}
1 + \cot{x\log_{e}{a + \frac{\cot^{2}x}{2!}\left( \log_{e}a \right)^{2} + \ldots}} \\
{- 1 - cos}{x\log_{e}{a - \frac{\cos^{2}x}{2!}\left( \log_{e}a \right)^{2} - ...}}
\end{array} \right\rbrack}{\cot{x - \cos x}}$
$= \lim_{x \rightarrow \frac{\pi}{2}}\left\{ \log_{e}{a + \frac{\cot{x + \cos x}}{2!}\left( \log_{e}a \right)^{2} + ...} \right\}$
$= \log_{e}a$