suppose f is a positive function. if lim_{x->a} f(x) = 0 and lim_{x->a }g(x) = infinity, show that
lim_{x->a} [f(x)]^{g(x) }= 0
this shows that 0 to the infinite power is not an indeterminate form.
If I were you, I would say it is a known fact that:
if $\displaystyle |r|<1$ then $\displaystyle \lim _{x \to \infty } r^x = 0$.
Because $\displaystyle \lim _{x \to a } f(x) = 0$ then $\displaystyle 0\le |f(x)|<0.5$ for all $\displaystyle x$ near $\displaystyle a$.
But we are given $\displaystyle \lim _{x \to a } g(x) = \infty$ so $\displaystyle (0.5)^{g(x)}\to 0$.
So $\displaystyle \lim _{x \to a} f(x)^{g(x)} = 0$ because $\displaystyle 0\le|f(x)|^{g(x)}\le (0.5)^{g(x)}\to 0$
Note that $\displaystyle |f(x)|^{g(x)}=|f(x)^{g(x)}|$