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