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RE: 0^0 = ?
From: |
Randy Gober |
Subject: |
RE: 0^0 = ? |
Date: |
Fri, 14 Nov 2003 21:09:06 -0600 |
Yes, but lim(x->0) log(x) does not exist, however The lim(x->0+)=-inf
-----Original Message-----
From: Mike Miller [mailto:address@hidden
Sent: Friday, November 14, 2003 9:06 PM
To: Randy Gober
Cc: address@hidden
Subject: RE: 0^0 = ?
On Fri, 14 Nov 2003, Randy Gober wrote:
> Goldberg's proof is flawed. He writes that f(x)^g(x)=
> e^[g(x)log{f(x)}], but
> f(x) = 0, => log(f(x)) = -inf
> So g(x)log(f(x)) is a 0*inf form, which itself is indeterment.
>
> (similar problem with the limit at the end: lim(x->0) x*log(a_1*x) )
Randy-- Tell me if I'm wrong: x approaches zero quickly enough to offset
the rate at which log(a*x) approaches -Inf. Let x=exp(y), then
x*log(a_1*x) = exp(y)*(a_1 + y)
What happens to that expression as y approaches -Inf? It seems to me that
it approaches zero. Same for lim(x->0) x*log(a_1*x).
What is lim(x->0) (x^2)*(1/x)? Is it 0*Inf, and therefore indeterminate? Of
course not. It converges to zero.
Mike
Goldberg's proof:
> http://docs.sun.com/source/806-3568/ncg_goldberg.html
>
> Here is the part we wanted to see:
>
> "In the case of 0^0, f(x)^g(x) = e^[g(x)log{f(x)}]. Since f and g are
> analytic and take on the value 0 at 0, f(x) = a_1*x^1 + a_2*x^2 + ...
> and
> g(x) = b_1*x^1 + b_2*x^2 + .... Thus lim(x->0) g(x)*log[f(x)] =
> lim(x->0) x*log[x(a_1 + a_2*x + ...)] = lim(x->0) x*log(a_1*x) = 0. So
> f(x)^g(x) -> e^0 = 1 for all f and g, which means that 0^0 = 1."
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-------------------------------------------------------------
- 0^0 = ?, Cong, 2003/11/12
- RE: 0^0 = ?, Randy Gober, 2003/11/13
- RE: 0^0 = ?, Boud Roukema, 2003/11/14
- RE: 0^0 = ?, John W. Eaton, 2003/11/14
- Re: 0^0 = ?, Geraint Paul Bevan, 2003/11/14
- RE: 0^0 = ?, Ted Harding, 2003/11/14
- RE: 0^0 = ?, John W. Eaton, 2003/11/14
- RE: 0^0 = ?, Mike Miller, 2003/11/14