1. ## TEST Preparations; Integrals.

Let $F(x)=e^{-x^2}\int_0^x e^{t^2}dt$

I have proved the following(I don't know if this will be useful):

$F'(x)+2xF(x)=1$

Now, I need to find for which values of $a$ we get:
$lim_{x\to 0}\frac{F(x)}{x^a}=1$

2. Expand the function $F(x)$ is a Taylor series at zero.

You should get something like $F(x) \approx x-\frac{2}{3}x^3+\mathcal{O}(x^5)$

This gives

$\frac{F(x)}{x^a}\approx x^{1-a}-\frac{2}{3}x^{3-a}+\mathcal{O}(x^{5-a})$

I hope this helps.