# find the limit: $\lim_{ x \to0^+ }(\ln x)^{\cot x}$

find the limit:

$$\lim_{ x \to0^+ }(\ln x)^{\cot x}$$

my try :

$$\lim_{x\rightarrow x_0}{f(x)^{g(x)}}=\left( \lim_{x\rightarrow x_0}{f(x)}\right)^{\left( \lim_{x\rightarrow x_0}{g(x)}\right)}$$

$$\lim_{ x \to0^+ }(\ln x)^{\cot x}=(\lim_{ x \to0^+ }(\ln x))^{\lim_{ x \to0^+ }(\cot x)}$$

now ?

• The identity you are trying to use is not always valid! – b00n heT Feb 16 '17 at 13:50
• The function is not well defined in the neighborhood of $0$. – Zhanxiong Feb 16 '17 at 14:18

The limit doesn't exist. Notice that when $x\in(0,1)$, then $\ln(x)$ is negative.
Likewise, when $x\in(0,1)$, $\cot(x)$ is not always an integer.
When $\cot(x)$ is an integer, depending on if it is even or odd, the limit goes to positive or negative infinity, so the limit doesn't exist for $\cot(x)\in\mathbb N$ either.