Question

${lim}_{x\to \frac{\pi }{2}}\frac{a^{cotx}-a^{cosx}}{cotx-cosx}$

Answer 1

${lim}_{x\to \frac{\pi }{2}}\frac{a^{cotx}-a^{cosx}}{cotx-cosx}$

$={lim}_{x\to \frac{\pi }{2}}{a}^{cosx}\left[\frac{a^{cotx-cosx}-1}{cotx-cosx}\right]$

$=1\times loga\left[\begin{array}{cc}\because \text{as}x\to \frac{\pi }{2}& \\ cotx-cosx\to 0& \end{array}\right]$

$=loga$