
Re: Does this imply that lim x > oo f'(x) = 0?
Posted:
May 24, 2013 3:42 AM


On Thu, 23 May 2013, steinerartur@gmail.com wrote:
> Suppose f:[0, oo) > R is increasing, differentiable and has a finite > limit as x > oo. Then, must we have lim x > oo f'(x) = 0? I guess > not, but couldn't find a counter example.
No, it doesn't.
For x in [0,1], let f(x) = 0. For x in [n, n+1/2], let f(x) = 1  1/n.
For x in [n+1/2, n+1], . . let f(x) = 1  1/n + (x  n + 1/2)(1/n  1/(n+1)).
To assure f' exists, round the corners at n, n + 1/2, for all n in N.
To make f strictly increasing, slope slightly upwards the horizontal portion. gtest

