On May 24, 5:42 pm, William Elliot <ma...@panix.com> wrote: > On Thu, 23 May 2013, steinerar...@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 >
OK, so you approach the limit in smaller and smaller bumps!
Another self-similar fractal coastline but increasing!