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MathGroup Archive 2007

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Re: Limit question

  • To: mathgroup at smc.vnet.net
  • Subject: [mg73102] Re: Limit question
  • From: "dimitris" <dimmechan at yahoo.com>
  • Date: Fri, 2 Feb 2007 05:35:40 -0500 (EST)
  • References: <eps6fd$7he$1@smc.vnet.net>

Thanks a lot Andrzej!

Regards
Dimitris



>Although I can't find any clear documentation for this, it seems to
>me to be very clearly implicit in the definition of Limit. Since
Limit always computes directed limits and since it works in the
complex plane, it would be a very serious defect if you could not
compute directional limits in the direction of arbitrary non-zero
complex numbers. I mean, what sense could you make of

In[12]:=
Limit[x, x -> I]

Out[12]=
I

if the direction of the limit was always along the real axis? (In
fact, if you try to specify such contradictory directions, the
Direction option is simply ignored). Note also the he existence of
objects such as DirectedInfinity[I], which are documented, and which
are analogous to Infinity and -Infinity (which are always converted
to DirectedInfinity[1] and DirectedInfinity[-1]). This again would
not make much sense unless limits could be considered for all
directions in the complex plane.

Andrzej Kozlowski

On Feb 1, 9:56 am, "dimitris" <dimmec... at yahoo.com> wrote:
> Practising with the Limit command I discover the following setting
>
> In[487]:=
> (Limit[Log[z], z -> -1, Direction -> #1] & ) /@ {I, -I}
>
> Out[487]=
> {(-I)*Pi, I*Pi}
>
> So my simple question is if it is somewhere documented in the
> Mathematica Book that you can use other values than just {1,-1} for
> the Direction Option. Note that
>
> In[481]:=
> Information[Direction]
>
> >From In[481]:=
>
> "Direction is an option for Limit. Limit[expr, x -> x0, Direction ->
> 1] computes the limit as x approaches x0 from smaller \
> values. Limit[expr, x -> x0, Direction -> -1] computes the limit as x
> approaches x0 from larger values. Direction -> Automatic \
> uses Direction -> -1 except for limits at Infinity, where it is
> equivalent to Direction -> 1."
>
> >From In[481]:=
>
> Attributes[Direction] = {Protected}
>
> Note also that even in this linkhttp://support.wolfram.com/mathematica/kernel/Symbols/
> in which for some Built-In functions you can find additional
> information (for examplehttp://support.wolfram.com/mathematica/kernel/Symbols/System/
> Oscillatory.htmlhttp://support.wolfram.com/mathematica/kernel/Symbols/System/
> Integrate.html )
> there is no other information.
>
> Dimitris



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