Bacle H
Posts:
283
Registered:
4/8/12


Re: Does this imply that lim x > oo f'(x) = 0?
Posted:
May 24, 2013 1:21 AM


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. > > > > Thank you
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
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Now, if f approaches a finite limit at oo , then , as n>oo f(n+1)f(n) =f'(cn) >0 .

