Can someone help me with this limit?

Question

$ \lim_{x \to 0}x^{|\tan x |} \overset{??}{=} 1 ??$

If you consult the graph it is intuitive to think that when $ x \to 0 $ the function tends to $1$. I have tried to lower the exponent with logarithms and apply L'Hospital's rule, but nothing. The only solution I see is to apply infinitesimal equivalents, which I have not tried yet.

Thank you all!

Answer 1

$$L=\lim_{x \to 0}x^{|\tan x |} \overset{??}{=} 1 ??$$ Take log $$\ln L =\lim_{x \to 0} {\ln x} \,{\tan x}$$ $$\ln L =\lim_{x \to 0} {\ln (1-(x+1))}\, {\tan x}$$ $$\ln L =\lim_{x \to 0} {-(x+1)} \,{ x}$$ $$\ln L= 0 \implies L=e^0=1$$

Answer 2

Hint:

Determine the limit of the logarithm, using that $\tan x\sim_0 x$.