Fundamental Theorem of Calculus and limit I've been reading through a paper and my question has essentially came down to this:
Let 
$f(\beta) \to M$ as $ \beta \to 0$ and $f(\alpha) \to 0$ as $\alpha \to \infty.$ Prove that $M=-\int_{0}^\infty f'(x)dx$. I wanted to check this before going any further. It is probably trivial but here is my attempt.
My attempt: 
$$M=\lim_{\beta \to 0}f(\beta)$$
$$M=\lim_{\beta \to 0 , \alpha \to \infty}\left[f(\beta)- f(\alpha)\right]$$
$$M=-\lim_{\beta \to 0,\alpha \to \infty}\left[\int_\beta^\alpha f'(s)dx\right]$$
$$M=-\displaystyle \int_0^{\infty} f'(x)dx$$
as required. 

Right.
Just be aware of the fact that this last notation is a limit, maybe not an integral unlike the notation could suggest. Hence, usual results on generalized integrals can be unsuited for it.
