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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.

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    On the limit, have you tried taking a large number $N$ and working out whether you can identify a value of $x$ which would make $f(x)>N$?2012-05-15

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