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

\(\displaystyle exp \{\ 0 \}\ = 1\)