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...