Given $a>1$ and $f:\mathbb{R}\backslash{\{0}\} \rightarrow\mathbb{R}$ defined $f(x)=a^\frac{1}{x}$
how do I show that $\lim_{x \to 0^+}f(x)=\infty$?
Also, is the following claim on sequences correct and can it be used somehow on the question above(by using Heine and the relationship between sequences and fucntions)? Given two sequences {$a_n$},{$b_n$} that converge to $a$ and $b$ respectively, then $a_n^{b_n}$ converges to $a^b$ as $n$ approces to $\infty$.
I think about this claim because in respect to the original question I know that $a^\frac{1}{x}$ is composed of a constant function $a$ which at any point converges to a and $\frac{1}{x}$ which converges to infinity as x approaches to zero, however I'm not sure how to make the transition between sequences and fucntions in this particular case.
I hope the question is clear.
(By the way, this isn't an homework assignment).
Thanks.