Question

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

• A. $\log a$
• B. $\log 2$
• C. $a$
• D. $\log x$

Answer 1

The correct option is A $\log a$

First we take $a^{cosx}$ common from the numerator, the limit becomes

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

Let $y=cotx-cosx$

As $x\to \frac{\pi }{2},y\to 0$

This is of the form,

$\underset{x\to \frac{\pi }{2}}{lim}{a}^{cosx}\times \underset{y\to 0}{lim}\left[\frac{a^y-1}{y}\right]$

$=1\times \log a$

$=\log a$