On Thu, 23 May 2013, firstname.lastname@example.org wrote: > On Thursday, May 23, 2013 11:11:12 AM UTC-4, steine...@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. > > I think so; use the MVThm repeatedly. Starting in [0,1]: > > f(1)-f(0)=f'(c1)*1 , for c in (0,1) > > f(2)-f(1)=f'(c2)*1 ; c in (1,2) > ........... > > f(n+1)-f(n)=f'(cn)*1
> Now, if f approaches a finite limit at oo , then , as n-->oo f(n+1)-f(n) > =f'(cn) -->0 .
This can't be right because there's counter examles when f isn't monotone and nowhere do you use monotonicity.
The problem is proving c_n -> 0 while all that you have is c_n in [0,1]. Even with that, the continuity of f' is needed to complete the proof.