Question

Вычислить предел lim(1-x)^2*tg(пx/2) x->1

Answer 1

$\lim\limits _{x \to 1} \, \left ( (1-x)^2\cdot tg\frac{\pi x}{2}\right )=[0\cdot \infty ]=\lim\limits _{x \to 1} \frac{(1-x)^2}{ctg\frac{\pi x}{2}} =[ \frac{0}{0} ]=\\\\=\lim\limits _{x\to 1} \frac{-2(1-x)}{-\frac{1}{sin^2\frac{\pi x}{2}}\cdot \frac{\pi}{2}} =\lim\limits _{x\to 1} \frac{-4(1-x)\cdot sin^2\frac{\pi x}{2}}{\pi } =[\frac{-4\cdot 0\cdot 1}{\pi }]=0$