Задача №1127

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