Re: Is zero even or odd?
From: Dave Seaman (dseaman_at_no.such.host)
Date: 12/28/04
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Date: Tue, 28 Dec 2004 17:38:31 +0000 (UTC)
On Tue, 28 Dec 2004 16:35:39 +0000, John Woodgate wrote:
> I read in sci.electronics.design that Dave Seaman <dseaman@no.such.host>
> wrote (in <cqs0gk$gpp$1@mailhub227.itcs.purdue.edu>) about 'Is zero even
> or odd?', on Tue, 28 Dec 2004:
>>That limit cannot exist because 0^x is undefined for all x < 0.
> 'Undefined' is a human artefact. ANYTHING can be defined; if it's badly
> defined, the definition may cause a contradiction.
There is no way to define 0^x for negative x such that the laws of
exponents are preserved.
> In this case, the nature of negative powers of 0 does not affect the
> limit, as x tends to 0 from positive values, of x^0. Now, are there any
> grounds for supposing (or even proving) that the limit, as x tends to 0
> from positive values, of 0^x differs from the above limit value?
Then you don't know the definition of limit. In the case of real
functions, it goes like this:
Definition. Let f: D -> R, where D is a subset of the reals,
and let L be a real number. Given a in D, we say lim_{x->a} f(x)
= L if, for every epsilon > 0, there exists delta > 0 such that
0 < |x-a| < delta => |f(x) - L| < epsilon. (*)
Compare this with the definition of the right-hand limit:
Definition. Let f: D -> R, where D is a subset of the reals,
and let L be a real number. Given a in D, we say lim_{x->a+} f(x)
= L if, for every epsilon > 0, there exists delta > 0 such that
0 < x-a < delta => |f(x) - L| < epsilon. (**)
Notice the crucial difference between (*) and (**), where |x-a| is
replaced by x-a. The (**) condition requires x to be greater than a, but
(*) allows x to be on either side of a, as long as it is suitably close.
-- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. <http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
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