If $\lim_{x\to+\infty}f'(x)>0$, does that mean that $\lim_{x\to+\infty}f(x)=+\infty$ Let's assume some function $f: \mathbb{R}\to\mathbb{R}$, which is differentiable in some region $(a, +\infty)$, where $a\in(0,+\infty)$. It makes sense (to me) that the following holds true:
$$\lim_{x\to+\infty}f'(x)\in(0, +\infty) \Rightarrow \lim_{x\to+\infty}f(x)=+\infty$$
However, I am not quite sure how I would go about proving it, if it even is true.

It is true.
First of all, call $L= \lim_{x \to \infty} f'(x)$. Since $L >0$, $f'$ is eventually positive, thus $f$ is eventually increasing.
Monotone functions always have a limit, thus there exists $a= \lim_{x \to \infty} f(x)$ ($a \in (0, \infty]$).
If $a \neq \infty$, you would have an horizontal asymptote (i.e. $L=0$), so necessarily $a= \infty$.
