Re: 0.999... = 1? (I know, a beaten dead horse)

Jim Spriggs <jim.sprigs@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
> shepherdmoon@xxxxxxxxx wrote:
>
> > This is hard for me to explain because I don't know whether I am using
> > the wrong terminology or the wrong concept or both. But what I'm
> > getting at is that I can see, for example, that the limit of 1/x as x
> > approaches 0 is infinity (although I also recall reading that 1/0 is
> > undefined),
>
> It is true that
>
> the limit of 1/x as x approaches 0 is infinity . . . . (*)
>
> so long as x is greater than 0.

Perhaps unfortunately, you didn't mention compactifications of the reals
until later, in the last paragraph of yours which I quoted. So it seems
that your comments here were intended to be taken in the context of the
reals, rather than in some extension of them. In that context, your
comments here are correct. But since you did later mention the extensions,
it might be worthwhile to re-examine your earlier comments in terms of the
extensions.

The one-point extension of the reals is R U {oo}, where oo is unsigned
infinity. The two-point extension of the reals is [-oo, +oo], where -oo and
+oo are signed infinities. (BTW, the infinities in these extensions are not
merely "façons de parler". They are actual elements of these systems.)

In [-oo, +oo]:
The limit of 1/x as x approaches 0 from the right is +oo, and from the left
is -oo. Thus the bilateral limit does not exist. Also note that 1/0 is
undefined.

In R U {oo}:
The limit of 1/x as x approaches 0 (bilaterally) is oo. Also note that 1/0
is defined: 1/0 = oo. The function 1/x is continuous at x = 0.

David W. Cantrell

> (If x tends to 0 through negative
> values the limit is negative infinity. It might be better to write
> "positive infinity" in (*)) But when (*) is written out in full using
> the official definitions, the word "infinity" (or the symbol one is
> using for it) disappears. That is to say, there is no _number_ infinity
> that (*) is referring to--it is a façon de parler. (My apologies if the
> wotsit under the "c" doesn't come out right.) Not only isn't there a
> _number_ infinity that (*) is referring to, there isn't any other kind
> of entity infinity that it is referring to either.
>
> It is also true that 1/0 is undefined. There is no contradiction here
> because (*) doesn't involve x being 0, so it has nothing to do with
> 1/0. Generally "limit of thing as x tends to A..." doesn't involve x
> being equal to A in "thing". For example
>
> limit of sin(x)/x as x tends to 0 is 1
>
> is true and useful but sin(0)/0 is meaningless.
>
> > but -- unless I am wrong -- infinity cannot be used in a
> > direct calculation because it is not a number.
>
> Just to confuse you, the real numbers can be "compactified" in two ways;
> one involving a number infinity, and one involving two numbers: positive
> infinity and negative infinity. Calculating with them can be handy, for
> example in the theory of integration and measure. Probably best to
> forget about that until the need arises. Those (three) infinities are
> not real numbers, they are additions which can enter into some limited
> arithmetic.
.

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