Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math

Topic: Functions asymptotic to the x-axis
Replies: 9   Last Post: Jul 29, 2011 5:21 AM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Rob Johnson

Posts: 1,771
Registered: 12/6/04
Re: Functions asymptotic to the x-axis
Posted: Jul 28, 2011 9:28 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

In article <20110728.040300@whim.org>,
Rob Johnson <rob@trash.whim.org> wrote:
>In article <20110727220251.J48523@agora.rdrop.com>,
>William Elliot <marsh@rdrop.com> wrote:

>>Assume f:R -> R and for all x, lim(n->oo) f(nx) = 0.
>>
>>If f is continuous, does lim(x->oo) f(x) = 0?
>>
>>Is continuity needed?

>
>The problem as stated is trivial by letting x = 1. Therefore, I
>assume that you intend n to be an integer.
>
>Continuity is necessary. Let f be defined as
>
> k
> f(n/?^k) = ---------
> k + n/?^k
>
> f(x) = 0 elsewhere
>
>where ? is the golden ratio. f is well-defined and for
>all x, lim_{n->oo} f(nx) = 0, where n is restricted to
>the integers.
>
>If n = [k ?^k], then f(n/?^k) > 1/2 and n/?^k > k-1.
>
>Thus, f(x) > 1/2 for arbitrarily large x = n/?^k.
>
>I haven't yet found what kind of continuity is necessary
>to insure that lim_{x->oo} f(x) = 0.


Sorry. This doesn't show that continuity is necessary, but that
there are discontinuous functions that satisfy the hypotheses but
not the conclusion.

Rob Johnson <rob@trash.whim.org>
take out the trash before replying
to view any ASCII art, display article in a monospaced font



Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.