# Homework Help: Integral estimate.

1. May 10, 2010

### Malmstrom

1. The problem statement, all variables and given/known data
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?

2. Relevant equations
Actually prove that the limit is zero.

3. The attempt at a solution
Should I use the dominated convergence theorem? Can't find the right function to dominate this one...

Last edited: May 11, 2010
2. May 11, 2010

### 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...

Last edited: May 11, 2010