
Re: Does this imply that lim x > oo f'(x) = 0?
Posted:
May 24, 2013 3:28 AM


On Thu, 23 May 2013, baclesback@gmail.com wrote: > On Thursday, May 23, 2013 11:11:12 AM UTC4, 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.

