Задача №1995
Условие
Найти предел \(\lim_{x\to\frac{\pi}{4}}\tg{2x}\cdot\tg\left(\frac{\pi}{4}-4\right)\).
Решение
\[
\lim_{x\to\frac{\pi}{4}}\tg{2x}\cdot\tg\left(\frac{\pi}{4}-4\right)=\left|\begin{aligned}&t=x-\frac{\pi}{4};\\&x=t+\frac{\pi}{4};\\&t\to{0}.\end{aligned}\right|
=\lim_{t\to{0}}\tg\left(2t+\frac{\pi}{2}\right)\cdot\tg(-t)=\\
=\lim_{t\to{0}}\ctg(2t)\cdot\tg{t}
=\lim_{t\to{0}}\frac{\tg{t}}{\tg{2t}}
=\frac{1}{2}\cdot\lim_{t\to{0}}\frac{\frac{\tg{t}}{t}}{\frac{\tg{2t}}{2t}}
=\frac{1}{2}\cdot\frac{\displaystyle\lim_{t\to{0}}\frac{\tg{t}}{t}}{\displaystyle\lim_{t\to{0}}\frac{\tg{2t}}{2t}}=\frac{1}{2}.
\]
Ответ:
\(\frac{1}{2}\)