From: Heike Koch-Beuttenmueller <Heike.Koch-Beuttenmueller@rz.uni-ulm.de>
Subject: Re: Bug in Maple's normal function?
Date: Fri, 28 May 1999 10:37:14 +0200
Newsgroups: sci.math.symbolic
Keywords: Symbolic algebra problems cannot all compute this limit

Helmut Kahovec wrote:
> 
> Serguei Miridonov wrote:
> 
> > I've traced the execution of the statement "limit(expr,limpoint);" and
> > have not found that it called 'series'. It tries the function 'normal' for
> > the original expression at the limit point and since 'normal' returned
> > zero without error (see my first posting in this thread), the calling
> > function 'limit' considers that the initial expression can be calculated
> > directly at this point.
> 
> Well, I have to apologize. Inspection of the source code shows that in
> Release 4 limit() calls `limit/limit`(), `limit/series`(), and series(),
> respectively. In Release 5 the source code that is executed in this case
> has been changed completely: apparently, limit() does not call either of
> those procedures. By the way, with Release 4 you get the correct result,
> anyway:
> 
> > restart;
> > expr:=xi*R*(exp(xi*R+xi-1)-1)/(xi*R+xi-1);
> 
>                         xi R (exp(xi R + xi - 1) - 1)
>                 expr := -----------------------------
>                                 xi R + xi - 1
> 
> > limpoint:=xi=1/(R+1);
> 
>                                            1
>                         limpoint := xi = -----
>                                          R + 1
> 
> > limit(expr,limpoint);
> 
>                                   R
>                                 -----
>                                 R + 1
> 
> > limit(eval(subs(O=0,series(expr,limpoint))),limpoint);
> 
>                                   R
>                                 -----
>                                 R + 1
> 
> > limit(simplify(eval(subs(O=0,series(expr,limpoint)))),limpoint);
> 
>                                   R
>                                 -----
>                                 R + 1
> 
> Helmut
Because there are such a lot of discussions I got interested how other
CAS can handle this limit:
Axiom:
(3) -> a1

               (R + 1)xi - 1
        R xi %e              - R xi
   (3)  ---------------------------
               (R + 1)xi - 1
                                                     Type: Expression
Integer
(4) -> limit(a1,xi=1/(R+1))

          R
   (4)  -----
        R + 1

Reduce:
REDUCE 3.6, 15-Jul-95, patched to 9 Apr 97 ...

1: xi*R*(exp(xi*R+xi-1)-1)/(xi*R+xi-1);

        r*xi + xi
 r*xi*(e          - e)
-----------------------
   e*(r*xi + xi - 1)

2: limit(ws(1),xi,1/(R+1));

   r
-------

 r + 1

Mathematica:

In[1]:= expr=xi*R*(Exp[xi*R+xi-1]-1)/(xi*R+xi-1)

               -1 + xi + R xi
        (-1 + E              ) R xi
Out[1]= ---------------------------
              -1 + xi + R xi

In[2]:= Limit[expr,xi->1/(R+1)]

                                 1
Power::infy: Infinite expression - encountered.
                                 0

Out[2]= ComplexInfinity

Macsyma:

(c1) xi*R*(exp(xi*R+xi-1)-1)/(xi*R+xi-1);
                                   r xi + xi - 1
                           r xi (%e              - 1)
(d1)                       --------------------------
                                 r xi + xi - 1
(c2) limit(d1,xi,1/(R+1));
Is   r   positive, negative, or zero?
positive;
                                   r       1
                     r xi + xi + ----- + ----- - 1
                                 r + 1   r + 1
              r xi %e
(d2) limit   (------------------------------------
     xi -> 0                   r       1
                 r xi + xi + ----- + ----- - 1
                             r + 1   r + 1

                        r       1
          r xi + xi + ----- + ----- - 1
                      r + 1   r + 1
      r %e                                               r xi
 + --------------------------------------- -
-----------------------------
                          r       1                        r       1
   (r + 1) (r xi + xi + ----- + ----- - 1)   r xi + xi + ----- + ----- -
1
                        r + 1   r + 1                    r + 1   r + 1

                      r
 - ---------------------------------------)
                          r       1
   (r + 1) (r xi + xi + ----- + ----- - 1)
                        r + 1   r + 1

and this was Maple V r 5.0:

expr:=xi*R*(exp(xi*R+xi-1)-1)/(xi*R+xi-1);
                        xi R (exp(xi R + xi - 1) - 1)
                expr := -----------------------------
                                xi R + xi - 1

> limit(expr,xi=1/(R+1));

                                  0

It seems to be a difficult limit.

Mit freundlichen Grüßen

Heike Koch-Beuttenmüller

=======================================================
 Dr. Heike Koch-Beuttenmüller
 Universität Ulm
 Universitätsrechenzentrum
 Albert-Einstein-Allee 11
 D-89081 Ulm
 Germany
 e-mail: Heike.Koch-Beuttenmueller@rz.uni-ulm.de
 Telefon: 0731 5022465
=======================================================