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.

Sketch:
Assuming that $\lim_{x\rightarrow\infty}f'(x)$ exists and equals $L>0$, then there is some $M$ such that for all $x>M$, $|f'(x)-L|>\frac{L}{2}$.  Therefore, $f'(x)>\frac{L}{2}$ for $x>M$.
Observe that 
$$
f(x)=f(M)+\int_M^xf'(t)dt.
$$
For $x>M$, it follows that
$$
f(x)=f(M)+\int_M^xf'(t)dt\geq f(M)+\int_M^x\frac{L}{2}dt=f(M)+\frac{L}{2}(x-M).
$$
Since the RHS diverges to infinity, so does $f(x)$.
