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MathGroup Archive 2007

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Re: Bug with Limit?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg73931] Re: [mg73866] Bug with Limit?
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Sat, 3 Mar 2007 01:08:04 -0500 (EST)
  • References: <200703021134.GAA03672@smc.vnet.net>

As has been pointed out on this list very many times before (inluding  
one of last week's threads)   Mathematica's Limit computes only  
directional limits in the complex plane, and Limit[f[x],x->a] is  
simply a shorthand for Limit[f[x],x->a,Direction->-1]. This is not a  
bug but a deliberate part of the design.  If a limit in the complex  
plane exists, it can be found by using Series:

Normal[ Series[Abs[x]/x, {x, 0, 1}]] /. x -> 0

Indeterminate

Normal[Series[1/(x - 3), {x, 3, 1}]] /. x -> 3

ComplexInfinity

Both of these show you the nature of the problem. In fact, Limit  
first calls Series to check if a limit in the complex palne exists  
and returns the answer given by Series when it does and only when it  
does not it returns a one sided Limit. This seems to me entirely  
reasonable and it in my opinion doing it in a different way would  
involve unnecessary duplication of code.

I also would like to add that having taught mathematics at  
universities (in various countries) for many years I do not feel much  
sympathy with complaints that this could be confusing for students.  
Mathematica is not actually "educational software". It is a  
sophisticated multi-purpose programming and computational tool. It  
is, of course,  an excellent tool with which one can make educational  
tools, but it is up to teachers who wish to use it for education to  
implement such tools themselves in the form of notebooks and  
packages. There is no reason why any compromises should be made in  
design of Mathematica to deal with problems of students who do not  
have yet acquired sufficient understanding of the mathematical  
concepts involved. In my opinion, this is a problem for educators and  
not for the designers of Mathematica.

Andrzej Kozlowski






On 2 Mar 2007, at 12:34, Sergio Miguel Terrazas Porras wrote:

> Hi guys,
>
> I was teaching a class and was discussing discontinuous functions.
> We came across f(x) = Abs(x)/x, and g(x) = 1/(x-3).
> The first does not have a limit as x -> 0 and the second does not  
> have a
> limit as x -> 3.
> The unilateral limits of both are different.
> When I specified the direction, Mathematica 5.1 gave the correct  
> answer.
> However, when no direction was specified, Mathematica 5.1 gave  
> (seemingly)
> by default the value of de right handside limit.
>
> This is plain wrong, and could lead to problems, specially for a  
> student.
> Any comments?
>
> Thank you.
> Sergio Terrazas
>



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