Finding $\lim_{x \to \infty}\int_0^x{e^{-x^2+t^2}}\,dt$ If we aren’t able to solve the integral $\int e^{-x^2}\,dx$, then how is it possible to find the $\lim_{x \to \infty}\int_0^x{e^{-x^2+t^2}}\,dt$? This was given to me by my prof, and I asked him multiple times if it was able to be solved. He said yes, but I’m just not getting it. Any thoughts?

Note that
$$\int_{0}^{x}{e^{-x^2+t^2}}\,dt=e^{-x^2}\int_{0}^{x}e^{t^2}\,dt=\frac{\int_{0}^{x}e^{t^2}\,dt}{e^{x^2}}$$
Now, to solve the limit, use Fundamental Theorem of Calculus and L'Hospital's Rule:
$$\lim_{x\to\infty}\frac{\int_{0}^{x}e^{t^2}\,dt}{e^{x^2}}=\lim_{x\to\infty}\frac{e^{x^2}}{2xe^{x^2}}=0$$
