# Another limit problem

#### azarue

Can you help with the problem below ?

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

#### simplependulum

MHF Hall of Honor
Can you help with the problem below ?

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

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

$$\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
$$\displaystyle exp \{\ 0 \}\ = 1$$

Last edited:
AllanCuz