Plot of $\int^x_0 e^{-t^2} dt$

Question

I know that the function is strictly monotone, since $f'(x)$ is always positive, and it is convex in $]-\infty, 0]$ and concave in $]0, +\infty[$ because if its second derivative.

However, I am also provided with the information that

$\int^{+\infty}_{-\infty} e^{-t^2} dt = \sqrt{\pi}$

but I don't know how to use this information.

I'm sure it somehow shows that it converges to $-\infty$ and $+\infty$, but how do I prove that?

Answer 1

The additional information tells you that $y=\pm\sqrt\pi/2$ are horizontal asymptotes.