Another limit problem

Mar 2010
26
1
Can you help with the problem below ?

\(\displaystyle
\lim_{x\to{{\pi}\over{2}}^-}\left(\tan x\right)^{\cos x}
\)

Thanks in advance.
 

simplependulum

MHF Hall of Honor
Jan 2009
715
427
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
\(\displaystyle exp \{\ 0 \}\ = 1\)
 
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