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MathGroup Archive 2001

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Re: Re: Limit question

  • To: mathgroup at smc.vnet.net
  • Subject: [mg31530] Re: Re: Limit question
  • From: Otto Linsuain <linsuain at andrew.cmu.edu>
  • Date: Fri, 9 Nov 2001 06:13:36 -0500 (EST)
  • Approved: Steven M. Christensen <steve@smc.vnet.net>, Moderator
  • Sender: owner-wri-mathgroup at wolfram.com

Hi Erk, well your question is tricky. I can't think of an answer that is
reasonable and not misleading. I wonder what answer one gets with the
improved Limit function from that other package. More or less reasonable
would be

Limit[1/x,x->0] = If[Direction -> 1, -Infinity, Infinity]

But this would be OK for the real line, where one has two possible
directions. This is wrong on the complex plane, where Mathematica thinks
by default. Perhaps ComplexInfinity is the most reasonable answer here.
Mathematica outputs ComplexInfinity when it encounters a divergence with
unclear phase. But what about the cases where the limits depend on the
direction  but are not divergent, like it is 0 from the left, 1 from the
right, 1/2 from above, etc? What should the output be then?

One could trace the root of the problem to this point: the concept of
limit can be introduced fundamentally (and in a manner suitable for
coding) to real-valued functions of one real variable. There the only
problem one encounters is the presence of two possible directions. The
complications with complex variables arise from the fact that one now has
two real-valued functions of TWO real variables Taking a limit in a
complex variable amounts to taking two limits in TWO real variables (the
second "TWO" is crucial), and taking multiple limits depends on the order
in which they are taken, or on which variable moves faster! But this
complication arose when building up from the one-variable case, which can
be handled easily, to the two-variable case, which is harder. By taking
the one-complex-variable case (which is really a two-real-variables case)
as the fundamental case, Mathematica gives up control of the cases that
could be controlled. It is like introducing R^n (and n-dimensional vector
space) to kids, before they know R (the reals), or R^Infinity (an
infinite-dimensional vector space) before they understand R^n.

In two words, Mathematica did not observe the time-honored principle of
building up from simple to complex.

Cheers,

Otto.

On Thu, 8 Nov 2001, Erk Jensen wrote:
>
> Thanks a lot for your many comments, I got the point.
>
> I understand that Direction->Automatic actually takes Direction->-1. This
> explains Mathematica's behaviour.
>
> So in your opinion, should then Limit[1/x,x->0] remain unevaluated by
> default? Or what doyou think would be a reasonable answer for
> Direction->Automatic?
>
> Cheers
>    -erk-
>



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