What is the limit
$${{\lim }_{x\to\infty}}x^\epsilon$$
for an infinitesimal $\epsilon$? Does it give zero or infinity?
Note that I'm considering the infinitesimals described in
http://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis
EDIT:
Since I was asked to show some of my own thoughts on the subject, I'd like to contribute the following:
$$\lim_{x\to\infty} x^\epsilon = \lim_{x\to\infty} \exp(\log (x^\epsilon))= \lim_{x\to\infty} \exp(\epsilon\log(x))\\= \lim_{x\to\infty}(1+\epsilon\log(x))= \lim_{x\to\infty}(1+\epsilon)=1+\epsilon$$
Where we used a series expansion of $\exp(y)$ and the properties of infinitesimals $\epsilon^2=0$ and $a\epsilon=\epsilon$ for any number $a$. However, I am not sure if this is some sort of cheating my way around the limit at hand or not. Help would be appreciated!