Re: Limit[f[x], x->a] vs. f[a]. When are they equal?
- To: mathgroup at smc.vnet.net
- Subject: [mg118458] Re: Limit[f[x], x->a] vs. f[a]. When are they equal?
- From: Murray Eisenberg <murray at math.umass.edu>
- Date: Fri, 29 Apr 2011 07:29:07 -0400 (EDT)
To the contrary, I would not in the least like a system that did not have +Infinity and -Infinity along with ComplexInfinity. The first two are the "points at infinity" adjoined to the real line to form its two-point compactification. And then limits as x->+Infinity or as x->-Infinity, meaningful for a function of a real variable, or that becomes in that context restricted to the real line, are just particular examples of the more general notion of limit of a net in a topological space. Here a "neighborhood" of the adjoined point +Infinity, for example, is just the complement of a compact set; and then a basic neighborhood of +Infinity is a rightward-opening ray). ComplexInfinity, on the other hand, can represent the single "point at infinity" that is adjoined to the complex plane to form its one-point compactification. And then limits as x-> ComplexInfinity is again just a special case of the more general concept of limit of a net. Here a "neighborhood" of ComplexInfinity is the complement of a compact subset of the complex plane; and then a basic neighborhood is the complement of a closed disk (say, a closed disk centered at the origin). The "More Information" section on the Mathematica ref page for Limit makes rather clear that a directional limit is at issue. On 4/28/2011 6:35 AM, 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 > > > > > > -- Murray Eisenberg murray at math.umass.edu Mathematics & Statistics Dept. Lederle Graduate Research Tower phone 413 549-1020 (H) University of Massachusetts 413 545-2859 (W) 710 North Pleasant Street fax 413 545-1801 Amherst, MA 01003-9305