- #1
juantheron
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Homework Statement
[tex]\displaystyle \lim_{x\rightarrow \frac{\pi}{4}}\left(\tan x\right)^{\tan 2x}[/tex]
Homework Equations
The Attempt at a Solution
let [tex]\frac{\pi}{4}-x = t\Leftrightarrow x=\frac{\pi}{4}-t[/tex] and [tex]t\rightarrow 0[/tex] [tex]\lim_{t\rightarrow 0}\left(\tan \left(\frac{\pi}{4}-t\right)\right)^{\tan \left(\frac{\pi}{2}-2t\right)}[/tex]
[tex]\lim_{t\rightarrow 0}\left(\frac{1-\tan t}{1+\tan t}\right)^{\frac{1}{\tan 2t}}[/tex]
as [tex]t\rightarrow 0,\tan t\approx t,\tan 2t\approx 2t[/tex]
[tex]\lim_{t\rightarrow 0}\left(\frac{1- t}{1+ t}\right)^{\frac{1}{2t}}[/tex]
after that how can i solve