You should notice that for some , whenever for some . Now look at the definition of derivative: . This number exists by differentiability of f.
Now, for another given , you need to prove that using above relations.
Hope this helps
Problem
Let f:R-->R be a differentiable function. Prove that if lim x-->infinity f(x) and lim x-->infinity f'(x) both exist and are finite, then lim x--> infinity f'(x) = 0.
Please help and assist in which def or theorem i should look at.
Forget what I said.
I have another idea in mind.
Use MVT on the interval :
we have: where c belongs to . Hence, .
When x goes to infinity, x+1 goes to infinity and so does c. So, both f(x) and f(x+1) go to the limit L. Therefore, f'(c) goes to L-L = 0.
Hope this helps