Question

Evaluate the limit:
$\underset{x\to \frac{\pi }{2}}{lim}\frac{e^{\cos x}-1}{\cos x}$

Answer 1

Given:
$\underset{x\to \frac{\pi }{2}}{lim}\frac{e^{\cos x}-1}{\cos x}$

$\text{Put}y=\cos x\text{As}x\to \frac{\pi }{2},\cos x\to 0\therefore y\to 0$

$=\underset{y\to 0}{lim}\left[\frac{e^y-1}{y}\right]$

$=\log e=1[\because \underset{x\to 0}{lim}\left(\frac{a^x-1}{x}\right)=\log a]$

Therefore,
$\underset{x\to \frac{\pi }{2}}{lim}\frac{e^{\cos x}-1}{\cos x}=1$