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Topic: Does this imply that lim x --> oo f'(x) = 0?
Replies: 18   Last Post: May 26, 2013 1:28 AM

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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
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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.
>
>
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> 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

_______________________

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





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