Question

Evaluate ;-
$\underset{x\to 1}{lim}\left(1-x\right)\tan \left(\frac{\pi x}{2}\right)$

Answer 1

Solution-

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

Apply L Hospital rule-

$\underset{x\to 1}{lim}\frac{-1}{-cose{c}^2\left(\frac{\pi x}{2}\right)}\times \frac{2}{\pi }=\frac{-1}{-1\times \frac{\pi }{2}}$

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