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Re: Limit[f[x], x->a] vs. f[a]. When are they equal?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg118446] Re: Limit[f[x], x->a] vs. f[a]. When are they equal?
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Thu, 28 Apr 2011 06:36:22 -0400 (EDT)
  • References: <ip6834$bmt$1@smc.vnet.net> <4DB8C302.3060402@cs.berkeley.edu>

On 28 Apr 2011, at 03:29, Richard Fateman wrote:

> On 4/26/2011 3:51 AM, Andrzej Kozlowski wrote:
> .. skipped and snipped...
>
>
>> and so on. In Mathematica is no sense in taking limits as z ->ComplexInfinity
>
>> without specifying a direction as there is no natural direction.
>
> So the idea of a limit as x->x0  makes no sense if x0 is a member of some set of numbers, symbols, whatever.  Maybe the documentation for Limit should provide some information on this?
>
> I don't know what you have written previously on this topic and have no intention of looking it up.  But in a system which includes ComplexInfinity, a concept which unifies at one "place" positive real infinity and negative real infinity, it becomes tricky to also have the separate values +Infinity and -Infinity.
>
> some of my thoughts are in section 4 of
> http://www.cs.berkeley.edu/~fateman/papers/interval.pdf
>
> RJF
>
>
>
>
>


The basic problem is one that is shared by all algebra systems that do not require the user to specify the mathematical context he is working in. Thus Mathematica allows one to perform all kinds of operations for which there is no mathematical context at all, e.g. adding strings to numbers etc. I think anyone who has grasped the basic principle involved here (i.e. that you have to think about what you are doing) will have any problems and I don't see much point in expending the already enormous documentation to deal with what should be obvious to any person intelligent enough to make use of Mathematica.

The problem with various kinds of infinities (and by the way, they all have FullForm DirectedInfinity[z], with Infinity being DirectedInfinity[1], - Infinity, DirectedInfinity[-1] and ComplexInfinity - DirectedInfinity[])  arise from the fact that although the real numbers are naturally embedded in the complex numbers (e.g. as real Banach algebras), the compactifications that are normally used to  deal with infinite limits etc. are different and incompatible. The space of real numbers is usually compactified by adding two points Infinity and -Infinity, the space of complex numbers is compactified by adding just one point - ComplexInfinity. In fact, one can also compactify the real numbers by adding just one Infinity (making +Infinity and -Infinity equal) and getting topologically a circle which is embedded in the Riemann sphere, but this is rarely done as it is not very useful in analysis. In the complex case the reason for using one point compactification is that one gets th
 e!
 Riemann sphere, which  is a complex manifold, so one can extend the concept of analycity to functions defined on the Riemann sphere and taking values in the Riemann sphere. Of course, in the complex case one can also use the "real" compactification, obtained by adding DirecttedInfinity[z] for every direction z. In this case one gets a space diffeomorphic to the two dimensional disc - which is not a complex manifold but whose interior is diffeomorphic to the complex plane. Thus the concepts of limits, derivatives etc are still preserved and one can obtain a lot of useful information by using this approach.

Although these two compactifications are not compatible, there is no reason why they should be any more confusing to an informed user than the ability to add strings to numbers is. In fact I once suggested that an options should be available for the user to decide which compatification he wants to use when taking limits etc, but now I think that this additional functionality would almost never be used and thus is not worth the effort. In my opinion things work pretty well as there are now (barring bugs and user ignorance, both of which are fundamental "facts of life").

Andrzej Kozlowski




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