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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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Bart Goddard

Posts: 1,706
Registered: 12/6/04
Re: Does this imply that lim x --> oo f'(x) = 0?
Posted: May 23, 2013 2:55 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply wrote in

> 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

Something like f(x) = sum_1^infinity arctan(2^n ( x-2^n) )/2^n should

work. f'(x) is a sum of terms like 1/(1 + (2^n x -2^(2n))^2.
f'(2^n)=1 plus some small positive terms. But f'(2^n+2^(n-1)) should
be pretty close to zero.

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