I need to show that if f:(0,infty)->R is a differentiable function, and lim f'(x)=0, then lim f(x)/x=0?

Question

The proof must be rigorous and only use the definitions of limits and derivative and theorems on differentiable functions (excluding l'Hopital).

Thanks for your help

Answer 1

You mean lim as x -> infinity? Intuitively, that does make sense. If lim f'(x) = 0. it means that f(x) is tending to a constant value. Whatever that constant value, f(x)/x must tend to zero as x gets larger and larger.

Essentially, I think you just need to put that argument in terms of the definitions to make it rigorous. (I'm not doing your homework for you, though!)