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Path: utzoo!utcsri!greg
From: greg@utcsri.UUCP (Gregory Smith)
Newsgroups: net.lang.c
Subject: Re:  LPow correction
Message-ID: <3055@utcsri.UUCP>
Date: Mon, 30-Jun-86 16:19:25 EDT
Article-I.D.: utcsri.3055
Posted: Mon Jun 30 16:19:25 1986
Date-Received: Mon, 30-Jun-86 16:21:22 EDT
References: <1809@brl-smoke.ARPA>
Reply-To: greg@utcsri.UUCP (Gregory Smith)
Organization: CSRI, University of Toronto
Lines: 82
Summary: Why 0^0 is undefined


Question: What should 0^0 come out as?
Well, what we have here is a two-dimensional limit.
I.e. what is
			Lim x^y
			(x,y)-> (0,0)

The point (0,0) in the x,y plane can be approached in many ways:
E.g. the following is the approach along the x-axis:

	let x=t, y=0
			Lim t^0		=	1
			t->0+
No problem there.
This is the approach along the y-axis:

	let y=t, x=0
			Lim 0^t		=	0
			t->0+
No problem there.
What about the general solution, along the line y=mx?
( y-axis not included in this one )

	let y=mt, x=t
		L   =	Lim t^mt
			t->0+

	ln(L) = Lim m * t * ln(t)

			m * ln(t)
		= Lim ----------
			1/t

Apply L'Hopital's rule:
			 m*  1/t
		= Lim ----------
		 t->0+	- 1/(t^2)

		= Lim - m * t	=  0
 		  t->0+

Thus the value of L in this class of approaches is exp(0) = 1.

Since the value of the limit depends on the path of approach, the
limit of x^y as (x,y)->(0,0) is deemed to be undefined. As least
that's what they learned me in Advanced Calculus.
So if you don't want to trap 0^0 as an error you could pick 0 or 1,
whichever you feel is better. But the limit is mathematically undefined.

>>>>	limit of x^x as x->0+ is precisely 1

Yes it is. This is the approach along the x=y line, and it is
not a complete characterization of 0^0.

>> This isn't a truth, it's a consequence of the definition of the system
>> as self consistant.
>
The system still isn't consistent. 0^X is 0 in general. X^0 is 1 in general.
( what this all means is that if you built a 3-d model of the plane
z=x^y it would have a vertical cliff or other such weirdness at x=y=0.)

>Fortunately I don't subscribe to the school of thought that says all
>mathematics is simply rearrangement of symbols according to formal rules.
>As a physicist/engineer, the above limit has real meaning for me.
>So there.

I hope it has a little more meaning now.

I have only included straight-line approaches. You can use the general
parametric approach along the curve ( f(t), g(t) ) where f(0)=g(0)=0.
I believe that the case y=mx is the same as this general case, provided

			       g(t)
			Lim    ----
			t->0+  f(t)

is defined and equal to m.

-- 
"Shades of scorpions! Daedalus has vanished ..... Great Zeus, my ring!"
----------------------------------------------------------------------
Greg Smith     University of Toronto      UUCP: ..utzoo!utcsri!greg