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

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Rob Johnson

Posts: 1,771
Registered: 12/6/04
Re: Functions asymptotic to the x-axis
Posted: Jul 28, 2011 9:28 AM
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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
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