Limit of monotonic functions at infinity I understand that if a function is monotonic then the limit at infinity is either $\infty$,a finite number or $-\infty$.
If I know the derivative is bigger than $0$ for every $x$ in $[0, \infty)$ then I know that $f$ is monotonically increasing but I don't know whether the limit is finite or infinite.
If $f'(x) \geq c$ and $c \gt 0$ then I know the limit at infinity is infinity and not finite, but why? How do I say that if the limit of the derivative at infinity is greater than zero, then the limit is infinite?

You can also prove it directly by the Mean Value Theorem:
$$f(x)-f(0)=f'(\alpha)(x-0) \geq cx \,.$$
Thus $f(x) \geq cx + f(0)$.
