Question

$\underset{x\to \mathrm{\pi }/2}{\lim }\frac{a^{\cot x}-a^{\cos x}}{\cot x-\cos x}$

Answer 1

$$\begin{gathered} \underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\left[\frac{a^{\cot x}-a^{\cos x}}{\cot x-\cos x}\right] \\ =\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\left[\frac{a^{\cos x}\left(a^{\cot x-\cos x}-1\right)}{\cot x-\cos x}\right] \\ x\to \frac{\mathrm{\pi }}{2} \\ \therefore \cot x-\cos x\to 0 \\ \Rightarrow a^{\cos \frac{\mathrm{\pi }}{2}}\times \log a \\ =a^0\times \log a \\ =\log a \end{gathered}$$