Prove Limit of Integral Estimate: Zero

[Malmstrom]

Homework Statement

I have to prove that the solution of an ODE can be continued to a function \in \mathcal{C}^1(\mathbb{R}). The solution is:
e^{-\frac{1}{x^2}} \int_{x_0}^x -\frac{2e^{\frac{1}{t^2}}}{t^2} dt
It is clear that this function is not defined in x=0. Its limit for x \rightarrow 0 though, seems to be zero. How do I prove it?

Homework Equations

Actually prove that the limit is zero.

The Attempt at a Solution

Should I use the dominated convergence theorem? Can't find the right function to dominate this one...

 

[lanedance]

have you tried power expanding the exponentials in powers of \frac{1}{x^2}?

may need some extra thought regarding convergence, so not sure whether it will work, but has nice from for it, so may be worth a crack...