
Re: Functions asymptotic to the xaxis
Posted:
Jul 28, 2011 9:05 AM


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 welldefined 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 > k1.
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.
Rob Johnson <rob@trash.whim.org> take out the trash before replying to view any ASCII art, display article in a monospaced font

