Question

Решить предел без использования правила Лопиталя Если можно, с пояснениями lim (tan(pi/(2*x))^tan(pi/x) x->2

Answer 1

$\lim\limits _{x \to 2}\, (tg\frac{\pi}{2x})^{tg\frac{\pi}{x}}=\lim\limits _{x \to 2}\Big (1+(tg\frac{\pi}{2x}-1)\Big )^{tg\frac{\pi}{x}}=\\\\\\=\lim\limits _{x \to 2}\Big (1+(tg\frac{\pi}{2x}-1)\Big )^{ \frac{1}{tg\frac{\pi}{2x}-1}\cdot (tg\frac{\pi}{2x}-1)\cdot tg\frac{\pi }{x}}=\\\\\\=\Big [\; \lim\limits _{x \to 2}(tg\frac{\pi }{2x}-1)\cdot tg\frac{\pi}{x}=\lim\limits _{x \to 2}(tg\frac{\pi}{2x}-1)\cdot \frac{2tg\frac{\pi}{2x}}{1-tg^2\frac{\pi}{2x}}=$

$=\lim\limits _{x \to 2}(tg\frac{\pi}{2x}-1)\cdot \frac{2tg\frac{\pi}{2x}}{(1-tg^2\frac{\pi}{y2x})(1+tg\frac{\pi }{2x})}=\lim\limits _{x \to 2}\frac{2tg\frac{\pi}{2x}}{-(1+tg^2\frac{\pi}{2x})} =\frac{2\cdot 1}{-(1+1)}=-1\; \Big ]=$

$=e^{-1}=\frac{1}{e}$