Question

Using L'Hospital's Rule, evaluate:
$\underset{x\to \frac{\pi }{2}}{lim}x\tan x-\frac{\pi }{4}\sec x$

Answer 1

$\underset{x\to \frac{\pi }{2}}{lim}x\tan x-\frac{\pi }{4}\sec x$

$\underset{x\to \frac{\pi }{2}}{lim}x\frac{\sin x}{\cos x}-\frac{\pi }{4}\frac{1}{\cos x}$

$\underset{x\to \frac{\pi }{2}}{lim}\frac{4x\sin x-\pi }{4\cos x}$

$\underset{x\to \frac{\pi }{2}}{lim}\frac{4(x\cos x+\sin x)}{-4\sin x}$

$=\frac{\frac{\pi }{2}\times 0+1}{-1}$

$=\frac{1}{-1}$

$=-1$