Question

$\underset{x\to \frac{\pi }{2}}{lim}\frac{a^{\cot x}-a^{\cos x}}{\cot x-\cot x},a>0$

Answer 1

We have,

${lim}_{x\to \frac{\pi }{2}}\frac{a^{\cot x}-a^{\cos x}}{\cot x-\cos x}{lim}_{x\to \frac{\pi }{2}}\frac{a^{\cos x}\left(a^{\cot x-\cos x}-1\right)}{\left(\cot x-\cos x\right)}{lim}_{x\to \frac{\pi }{2}}\frac{a^{\cot x-\cos x}-1}{\cot -\cos x}=\log a$

$Hence,solved.$