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

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  • Subject: [mg118456] Re: Limit[f[x], x->a] vs. f[a]. When are they equal?
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Fri, 29 Apr 2011 07:28:45 -0400 (EDT)

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?

What the documentation says about direction of approach:

"Limit[expr,x->Subscript[x, 0]] uses the setting Direction->Automatic, 
which determines the direction from assumptions that have been given, 
using Direction->-1 as the default. For limit points at infinity, the 
direction is determined from the direction of the infinity."


> 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.

ComplexInfinity, is not a unification of positive and negative infinity. 
It is a "directionless" infinity, that is to say, it has no notion of 
how one approaches. In Mathematica it can, and often does, serve as a 
surrogate for the usual complex infinity of complex analysis.

This in no way means that the directional infinities, in particular the 
two on the real line, cannot also take on their customary meanings from, 
say, real analysis.


> some of my thoughts are in section 4 of
> http://www.cs.berkeley.edu/~fateman/papers/interval.pdf
> 
> RJF


Daniel Lichtblau
Wolfram Research


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