Задача №1996

\[ \lim_{x\to{1}}(1-x)\tg\frac{\pi{x}}{2}=\left|\begin{aligned}&t=x-1;\\&x=t+1;\\&t\to{0}.\end{aligned}\right| =\lim_{t\to{0}}(-t)\cdot\tg\left(\frac{\pi{t}}{2}+\frac{\pi}{2}\right) =\lim_{t\to{0}}t\cdot\ctg\frac{\pi{t}}{2} =\lim_{t\to{0}}\frac{t}{\tg\frac{\pi{t}}{2}} =\frac{2}{\pi}\lim_{t\to{0}}\frac{1}{\frac{\tg\frac{\pi{t}}{2}}{\frac{\pi{t}}{2}}}=\frac{2}{\pi}. \]