Can someone help me with this limit? $ \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!

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