Don't know how to solve limits that have tan(x) as a power. I have these two problems.
The first one is:
$$\lim_{x \to \pi/2} \left(\frac{2x} \pi\right)^{\tan(x)}$$
and the second one is:
$$\lim_{x \to \pi/4} (\tan(x))^{\tan(2x)}$$
The way I'm trying to solve them is by taking the natural logarithm of both sides and then trying to apply l'Hôpital's rule on the limit's side. However, in both cases, I found myself stuck trying to figure out how to find the limit of the denominator, which happens to be $$\frac{1} {\tan x}$$ And $$\frac{1}{\tan(2x)}$$.

Note that $\lim_{x \to \pi/2} \tan(x) $ does not exist, since $\lim_{x \to \pi/2^-} \tan(x) = +\infty $ and $\lim_{x \to \pi/2^+} \tan(x) = -\infty $. But this is not a serious problem, since $\lim_{x \to \pi/2} \dfrac{1}{\tan(x)} = 0$. So you have an indet $\dfrac{0}{0}$ and you can apply l'hopital's rule.
