Задача №1126
Условие
Найти предел \(\lim_{z\to{1}}(1-z)\tg\frac{\pi{z}}{2}\).
Решение
\[
\lim_{z\to{1}}(1-z)\tg\frac{\pi{z}}{2}
=\left[0\cdot\infty\right]
=\left[\begin{aligned}& t=z-1;\\& t\to{0}.\end{aligned}\right]=\\
=\lim_{t\to{0}}\left(-t\cdot\tg\frac{\pi(t+1)}{2}\right)
=\lim_{t\to{0}}\left(t\cdot\ctg\frac{\pi{t}}{2}\right)
=\lim_{t\to{0}}\left(\frac{1}{\frac{\sin\frac{\pi{t}}{2}}{\frac{\pi{t}}{2}}}\cdot\frac{2\cos\frac{\pi{t}}{2}}{\pi}\right)
=\frac{2}{\pi}.
\]
Ответ:
\(\frac{2}{\pi}\)