# lim (1-x)tan{pie×x/2}x_1

Arun Kumar IIT Delhi
9 years ago
Hi

$\\\lim _{x\to \:1}\left(\left(1-x\right)\tan \left(\frac{\pi x}{2}\right)\right) \\=\lim _{x\to \:1}\left(\frac{1-x}{\frac{1}{\tan \left(\frac{\pi x}{2}\right)}}\right) \\Apply \,L\,Hospital \\=\lim _{x\to \:1}\left(\frac{-1}{-\frac{\pi }{2\cos ^2\left(\frac{\pi x}{2}\right)\tan ^2\left(\frac{\pi x}{2}\right)}}\right) \\=\lim _{x\to \:1}\left(\frac{2\cos ^2\left(\frac{\pi x}{2}\right)\tan ^2\left(\frac{\pi x}{2}\right)}{\pi }\right) \\=\lim _{x\to \:1}\left(\frac{2\cos ^2\left(\frac{\pi x}{2}\right)\tan ^2\left(\frac{\pi x}{2}\right)}{\pi }\right) \\Concept \\\lim _{x\to a}[\frac{f\left(x\right)}{g\left(x\right)}]=\frac{\lim _{x\to a}f\left(x\right)}{\lim _{x\to a}g\left(x\right)}\mathrm{,\:where\:}\lim _{x\to a}g\left(x\right)\ne 0$

$\\=\frac{\lim _{x\to \:1}\left(2\cos ^2\left(\frac{\pi x}{2}\right)\tan ^2\left(\frac{\pi x}{2}\right)\right)}{\lim _{x\to \:1}\left(\pi \right)} \\\lim _{x\to \:1}\left(2\cos ^2\left(\frac{\pi x}{2}\right)\tan ^2\left(\frac{\pi x}{2}\right)\right)=2 \\\lim _{x\to \:1}\left(\pi \right)=\pi \\\\so=2/\pi$

Thanks & Regards, Arun Kumar, Btech,IIT Delhi, Askiitians Faculty