Question

Найти предел. Без Лопиталя

Answer 1

$\displaystyle \lim_{x \to 1}(x-1){\rm tg}\frac{\pi x}{2}=\left\{\begin{array}{ccc}x-1=t\\x=t+1\\t\to 0\end{array}\right\}=\lim_{t \to 0}\bigg[t\cdot{\rm tg}\bigg(\frac{\pi t}{2}+\frac{\pi}{2}\bigg)\bigg]=\\ \\ \\ =-\lim_{t \to 0}\bigg(t\cdot {\rm ctg}\frac{\pi t}{2}\bigg)=-\lim_{t \to 0}\frac{t\cos\frac{\pi t}{2}}{\sin \frac{\pi t}{2}}=-\lim_{t \to 0}\frac{t}{\sin \frac{\pi t}{2}}=-\lim_{t \to 0}\frac{t\cdot \frac{\pi}{2}}{\frac{\pi}{2}\cdot \sin \frac{\pi t}{2}}=-\frac{2}{\pi}$