If $f'(x)>c, forall xin(a,+infty)$ where $c>0$. Prove that $lim_{xto+infty} f(x) = +infty$. I would say that this is trivial, how could we prove this explicitly?

Intuition suggests that a function with a positive derivative is strictly increasing. You can prove this using the mean value theorem. Next, you can use the mean value theorem to prove the function is also unbounded from above.

You now have a strictly increasing function with no upper bound, so you know its limit.

This is a direct application of Newton-Leibniz formula

$$

f(x) = f(a) + int_a^x f'(t) , dt ge f(a) + c(x-a)

$$