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
> 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.