Question

Solve $\underset{x\to \frac{\pi }{2}}{lim}\left(\sec x-\tan x\right)$

Answer 1

Simplifying the limit $\underset{x\to \frac{\pi }{2}}{lim}\left(\sec x-\tan x\right)$.

$\underset{x\to \frac{\pi }{2}}{lim}\left(\sec x-\tan x\right)=\underset{x\to \frac{\pi }{2}}{lim}\left(\frac{1-\sin x}{\cos x}\right)$

$=\frac{1-\sin \frac{\pi }{2}}{\cos \frac{\pi }{2}}$

$=\frac{0}{0}$

This gives the indeterminate value, so applying L’Hospitals Rule,

$\underset{x\to \frac{\pi }{2}}{lim}\left(\frac{1-\sin x}{\cos x}\right)=\underset{x\to \frac{\pi }{2}}{lim}\left(\frac{-\cos x}{-\sin x}\right)$

$=\frac{\cos \frac{\pi }{2}}{\sin \frac{\pi }{2}}$

$=\frac{0}{1}$

$=0$