try this: can you prove that if f'(x) > c then
The RHS clearly does not converge to an upper limit
Hello
I'm trying to prove the following: if is differentiable on all and there exists a real positive constant such that > c for all , then the limit of f at infinity is infinity.
I've managed to prove that if the derivative is positive, then by the Mean Value Theorem the function must be strictly increasing. But I can't seem to manage to prove that the limit isn't some real number...
yesThanks to both of you...but I'm not a hundred percent sure why the RHS doesn't converge. Is it because k is a variable and can be as big as we want?
Your notation is causing the confusion. F(x) + c converges to L+c, but f(x+c) converges to the same limit as f(x) (i think!).If L was a limit of f then f + c would just converge to L + c, right?
i dont know how rigorous you're expected to be. It can be shown robustly using the mean value theorum (as demonstrated by Fernando for the case where k=0). of course, it would be easier to use fernandos result directly as he suggested in his post.Also...might it be enough to say that if f'(x) > c, then f(x) > c*x + k ?
Fernondos solution is better, but i thought this was an interesting alternative approach to the problem:
Spoiler: