MathGroup Archive 2011

[Date Index] [Thread Index] [Author Index]

Search the Archive

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

  • To: mathgroup at smc.vnet.net
  • Subject: [mg118381] Re: Limit[f[x], x->a] vs. f[a]. When are they equal?
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Mon, 25 Apr 2011 07:28:43 -0400 (EDT)


----- Original Message -----
> From: "Richard Fateman" <fateman at eecs.berkeley.edu>
> To: mathgroup at smc.vnet.net
> Sent: Sunday, April 24, 2011 7:25:35 AM
> Subject: [mg118359] Re: Limit[f[x], x->a] vs. f[a]. When are they equal?
> On 4/23/2011 10:53 AM, Leonid Shifrin wrote:
> > Richard,
> >
> > On Sat, Apr 23, 2011 at 4:49 AM, Richard Fateman
> > <fateman at cs.berkeley.edu <mailto:fateman at cs.berkeley.edu>> wrote:
> >
> >     Let c=ComplexInfinity
> >
> >     then Mathematica (7.0) "knows" that Sin[c] is Indeterminate
> >
> >     but Limit[Sin[x],x->c] is not evaluated.
> >
> >     I would have thought that if f[a] is known, Limit[f[x],x->a] is
> >     also known.
> >
> >
> > Not necessarily. The limiting procedure has nothing to do with the
> > value of the function at a given point, it has to do with values of
> > the function in the neighborhood of that point.
> >
> > Regards,
> > Leonid
> >
> >
> >     RJF
> >
> >
> 
> I agree not necessarily, but sine() is continuous and differentiable
> everywhere.
> I think a computer algebra system should do better.
> 
> 
> I poked around some more..
> 
> Limit[1 - Exp[I x], x -> 0] yields 0
> Limit[1/x, x->0] yields Infinity
> 
> Limit[1/(1 - Exp[I x]), x -> 0] yields I*Infinity. ??
> I would think this would be a place for ComplexInfinity.

The series expansion would be 1/(1-(1+I*x+...)) = 1/(-I*x+...) --> I*Infinity. This is using the Mathematica limit direction convention (more below) so the 'x' is regarded as positive.


> All in Mathematica 7.0

If I remember correctly we try to return unevaluated in preference to Indeterminate for Limit.

Recall that limits are in general path dependent. The examples you show just above all have clearly defined paths. Per documentation, direction is from positive to negative unless specified otherwise by option, or forced otherwise in approaching an infinity. When the point approached is ComplexInfinity==DirectedInfinity[] we neither are given nor infer a direction. As sine has an essential singularity at (complex) infinity, there is no viable limit nor even a plausible Interval-type result. Ergo, it is unevaluated.

Daniel Lichtblau
Wolfram Research


  • Prev by Date: Re: Limit[f[x], x->a] vs. f[a]. When are they equal?
  • Next by Date: Re: Mathematica 8: Shortcut Ctrl+Tab for switching between notebooks?
  • Previous by thread: Re: Limit[f[x], x->a] vs. f[a]. When are they equal?
  • Next by thread: Re: Limit[f[x], x->a] vs. f[a]. When are they equal?