Can you help with the problem below ?
$\displaystyle
\lim_{x\to{{\pi}\over{2}}^-}\left(\tan x\right)^{\cos x}
$
Thanks in advance.
$\displaystyle \lim_{x\to{{\pi}\over{2}}^-}\left(\tan x\right)^{\cos x}$
$\displaystyle = exp \{\ \lim_{x\to{{\pi}\over{2}}^-} \cos(x) \ln(\tan(x)) \}\ $
$\displaystyle = exp \{\ \lim_{x\to{{\pi}\over{2}}^-} \sin(x) \frac{\ln(\tan(x))}{\tan(x)} \}\$
We know $\displaystyle \lim_{u\to\infty} \frac{ \ln(u) }{u} = 0 $
so it is equal to
$\displaystyle exp \{\ 0 \}\ = 1$