limit of $cos x ^{tan x}$

As far as I know, $0^\infty$ is not an indefinite form and it goes to zero. Then the limit of $(\cos x)^{\tan x}$ when $x$ goes to $\pi/2 - $ should equal $0$.

But after log transformation, its limit is $\infty$. I am not sure which one is correct, zero or infinity?

Thanks for your help.

I slightly changed the question. I realized that $x$ goes to $\pi/2 - $ (left hand limit) in the original question.

Answer

Suppose that $f$ and $g$ are functions such that as $x \to a$, $f(x) \to 0$ and $g(x) \to \infty$, and consider $h(x) = f(x)^{g(x)}$. We know that
$$
\lim_{x \to a} \ln h(x) = \lim_{x \to a} g(x) \ln f(x) = -\infty,

$$
Thus,
$$
\lim_{x \to a} h(x) = 0.
$$