Задача №2012
Условие
Найти предел \(\lim_{x\to\frac{\pi}{4}}(\tg{x})^{\tg{2x}}\).
Решение
\[
\lim_{x\to\frac{\pi}{4}}(\tg{x})^{\tg{2x}}
=\lim_{x\to\frac{\pi}{4}}(1+\tg{x}-1)^{\frac{2\tg{x}}{1-\tg^2{x}}}
=\lim_{x\to\frac{\pi}{4}}\left((1+\tg{x}-1)^{\frac{1}{\tg{x}-1}}\right)^{\frac{-2\tg{x}}{\tg{x}+1}}
=e^{-1}
=\frac{1}{e}.
\]
Ответ:
\(\frac{1}{e}\)