Question

limit (tan(x))^x as x approaches 0+

Answer 1

exponentiate your property, writing \[ \large \tan (x)^x = e^{x \log ( \tan x)}\]
then use taylor approximations or the laws of Bernoulli De L'Hopital for the expression in the exponential
\[x \log ( \tan x) \tag{*} \]
You can do that because \(e \) is continuous, meaning that:
\[\large \lim_{x \to x_0} f(x) = f(x_0) = f( \lim_{x \to x_0} x) \]