How to prove that $\int_{-\infty}^{\infty}f'(t)\exp(-t^2)\text{d}t$ is finite when $\lim_{|t|\to \infty}f(t)\exp(-t^2) = 0$? Given a function $f$ which is continuously differentiable over the real line, and given that $\displaystyle \lim_{|t|\to \infty}f(t)\exp(-t^2) = 0$ how does one prove that $\displaystyle \int_{-\infty}^{\infty}f'(t)\exp(-t^2)\text{d}t$ is finite?

I think you can't. Just looking at the positive side of your integral, consider 
$$f:t\mapsto \frac{e^{t^2}}{\sqrt{t+1}}$$
You do have $\lim_{t\to+\infty} f(t)e^{-t^2}=0$, but 
$$f'(t)e^{-t^2} = \frac{2t\sqrt{t+1}-\frac{1}{\sqrt{t+1}}}{t+1} \sim_{t\to+\infty} 2\sqrt t$$
which is not integrable on $[0,+\infty[$.
