Question

Evaluate the following
$\underset{x\to \frac{\pi }{2}}{lim}\left(\frac{\pi }{2}-x\right)\tan x$

Answer 1

$\underset{x\to \frac{\pi }{2}}{lim}(\frac{\pi }{2}-x)\tan x$

$=\underset{x\to \frac{\pi }{2}}{lim}\frac{(\frac{\pi }{2}-x)}{\cot x}$

Apply $L$ - hospital Rule

$=\underset{x\to \frac{\pi }{2}}{lim}\frac{-1}{-cose{c}^{2}x}$

$=\underset{x\to \frac{\pi }{2}}{lim}{\sin }^{2}x$

$=1$