Question

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

Answer 1

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

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

$\Rightarrow \left(1\right){lim}_{x\to 1}\frac{-1}{\frac{-\pi }{2}sin\frac{\pi x}{2}}=\frac{2}{\pi }=f\left(1\right)$