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MathGroup Archive 2003

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Re: Limits: Is there something I'm missing Here?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg39390] Re: Limits: Is there something I'm missing Here?
  • From: "David W. Cantrell" <DWCantrell at sigmaxi.org>
  • Date: Thu, 13 Feb 2003 04:57:06 -0500 (EST)
  • References: <b2d0n3$e5g$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

"David Park" <djmp at earthlink.net> wrote:
> Ashraf,
>
> Strictly speaking you are correct. The limit does not exist.

It doesn't exist _in the real or complex number systems_, true. However, if
one deals with the projective extension (one-point compactification) of
either the reals or complexes, then the limit does actually exist.

> But there is such a thing as a one-sided limit, which is not a true limit
> but still useful.

Correct if, by "true limit", you mean a limit which is independent of the
direction of approach.

> In this case even the one-sided limits do not really exist because the
> result is unbounded.

Ha, ha. Yes, they do not _real_ly exist; that is, they do not exist in the
real number system. Indeed, in an introductory calculus course dealing with
the reals, whenever one says that a limit "is +oo" or "is -oo", one is
not actually speaking of the limit existing at all. Rather, one is merely
indicating that the limit fails to exist because the function increases or
decreases without bound.

However, if we use the affine extension (two-point compactification) of
the reals, then the unilateral (one-sided) limits actually do exist. And
Mathematica does these correctly.

> But again, it is useful to signify this by saying that the limit is
> infinite. So it is rather loose language by both mathematicians and
> Mathematica.

Regarding "rather loose language" on the part of mathematicians:
I must agree with you _if_ the context is that of strictly the reals or
complexes. It is surely confusing to some students for us to say that a
certain limit _is_ ... while we actually mean that the limit fails to
exist. It seems duplicitous.

That's why I prefer to deal with appropriate extensions of the reals or
complexes. Then, whenever we say that a limit _is_ ..., it truly is!

It is quite regrettable that the documentation for Limit doesn't make
completely obvious the fact that, in absence of a specified Direction,
a default Direction is assumed! (Or did I just miss the bold type someplace
warning about that?)

And it's unfortunate IMO that Mathematica's Limit, in absence of a
specified Direction, doesn't give the omnidirectional limit. For the
problem at hand: If we deal with the projective extension of the complexes,
the omnidirectional limit of 1/x as x -> 0 is, truly is, ComplexInfinity.

David Cantrell


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