Question

$\underset{x\to 1}{lim}(1-x)\tan \left(\frac{\pi x}{2}\right)$ is $\frac{b}{\pi }.$ Find $b$

Answer 1

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