Question

найти предел функции (фото во вложении)

Answer 1

$limlimits _{x to frac{pi}{4}}, (tgx)^{tg2x}= limlimits _{x to frac{pi }{4} }, (tgx)^{frac{2tgx}{1-tg^2x}}=[1+t=tgx; ,; tto 0; ,; tgfrac{pi}{4}=1, ]=\\= limlimits _{t to 0}; (1+t)^{ frac{2(1+t)}{1-(1+t)^2} }= limlimits _{t to 0}, (1+t)^{-frac{2+2t}{-t^2-2t}}= limlimits _{t to 0}(1+t)^{-frac{2t+2}{t(t+2)}}=\\=limlimits _{t to 0}, Big ((1+t)^{frac{1}{t}}Big )^{-frac{2t+2}{t+2}}=e^{limlimits _{t to 0}, frac{-2t-2}{t+2}}=[ frac{-2cdot 0-2}{0+2}=-1 ]=e^{-1}= frac{1}{e}$

или

$limlimits _{x to frac{pi}{4}}, (tgx)^{tg2x}=limlimits _{x to frac{pi}{4}}Big (1+(tgx-1)Big )^{frac{2tgx}{1-tg^2x}}=\\=limlimits_{x to frac{pi }{4}}, Big (Big (1+(tgx-1)Big )^{frac{1}{tgx-1}}Big )^{(tgx-1)cdot frac{2tgx}{1-tg^2x}}=\\=e^{limlimits _{x to frac{pi}{4}},frac{-(1-tgx)cdot 2tgx}{(1-tgx)(1+tgx)}}=e^{limlimits _{x to frac{pi}{4}}, frac{-2tgx}{tgx+1}}=e^{ frac{-2cdot 1}{1+1}}=e^{-1}= frac{1}{e}$