Finding limit of a function

Question

$$\lim_{x\to0}\left(\ln(\cot x)\right)^{\tan x}$$

The answer is obviously 1 but how do I reach that conclusion without L'hospital rule

Answer 1

write it in the form $$e^{\lim_{x \to 0}\frac{\ln(\ln(\cot(x)))}{\cot(x)}}$$ and use the rules of L'Hospital.

Answer 2

Put $n=\cot x$ and rewrite as $\lim_{n\to\infty}\ln(n)^{1/n}$. Then from the inequalities $$1\le\ln(n)^{1/n}\le n^{1/n}$$ by using the squeeze theorem you get the result.
EDIT: This $n$ substitution works for the right-sided limit. To the left of the origin the function is complex, so it doesn't work there.