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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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David C. Ullrich

Posts: 3,019
Registered: 12/13/04
Re: Does this imply that lim x --> oo f'(x) = 0?
Posted: May 23, 2013 12:30 PM
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On Thu, 23 May 2013 16:24:57 +0100, Robin Chapman
<R.J.Chapman@ex.ac.uk> wrote:

>On 23/05/2013 16:11, steinerartur@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.

>
>Think of a function that's flat, then rapidly jerks upward then
>flat again then rapidly jerks up and does this infinitely
>often ....



In particular, jerks upward very fast, but on a very short
interval ("short" interval relative to how fast is fast).
Then flat on a very long interval. Then jerks upward
even faster, on an even shorter interval...

Showing that f' need not even be bounded.







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