[Axiom-developer] wrong sign in limit?

[Dr.-Ing. Ingo D. Rullhusen]

Hello,

perhaps i have found a bug in the limit operation. I am using

                        AXIOM Computer Algebra System
                         Version: Axiom (July 2010)
              Timestamp: Wednesday August 18, 2010 at 11:16:49

on a x86_64 with linux. There seems to be an error in the sign, if you
compare the limit result with the de L'Hospital one.

>>>>Example

-- problem with limit

)clear

Z:=sqrt(a^2+h^2)-a
N:=a*sqrt(a^2+h^2) - a^2 - h^2

limit( Z/N, h=0 )

-- this seems to be wrong!
-- because de L'Hospital says

dZ:=differentiate( Z, h, 2 )
dN:=differentiate( N, h, 2 )

eval(dZ/dN,h=0)

-- which seems to be right (see graph): opposite sign!

>>>>EndExample

which generates the output:

                        AXIOM Computer Algebra System
                         Version: Axiom (July 2010)
              Timestamp: Wednesday August 18, 2010 at 11:16:49
-----------------------------------------------------------------------------
   Issue )copyright to view copyright notices.
   Issue )summary for a summary of useful system commands.
   Issue )quit to leave AXIOM and return to shell.
   Visit http://axiom-developer.org for more information
-----------------------------------------------------------------------------
 
   Re-reading compress.daase   Re-reading interp.daase
   Re-reading operation.daase
   Re-reading category.daase
   Re-reading browse.daase
(1) ->
(1) -> )read hos.input

-- problem with limit

)clear
 
   Use )clear all to clear everything in the workspace. Use )clear
      completely to clear everything in the workspace and internal
      tables. Other )clear keyword arguments are
             modes
             operations
             properties
             types
             values
      or abbreviations thereof. Issue )clear? for more information.

Z:=sqrt(a^2+h^2)-a
 

         +-------+
         | 2    2
   (1)  \|h  + a   - a
                                                     Type: Expression
Integer
N:=a*sqrt(a^2+h^2) - a^2 - h^2
 

          +-------+
          | 2    2     2    2
   (2)  a\|h  + a   - h  - a
                                                     Type: Expression
Integer

limit( Z/N, h=0 )
 

        1
   (3)  -
        a
                        Type: Union(OrderedCompletion Expression
Integer,...)

-- this seems to be wrong!
-- because de L'Hospital says

dZ:=differentiate( Z, h, 2 )
 

                  2
                 a
   (4)  -------------------
                  +-------+
          2    2  | 2    2
        (h  + a )\|h  + a
                                                     Type: Expression
Integer
dN:=differentiate( N, h, 2 )
 

                      +-------+
             2     2  | 2    2     3
        (- 2h  - 2a )\|h  + a   + a
   (5)  ----------------------------
                       +-------+
               2    2  | 2    2
             (h  + a )\|h  + a
                                                     Type: Expression
Integer

eval(dZ/dN,h=0)
 

               1
   (6)  - ----------
            +--+
            | 2
          2\|a   - a
                                                     Type: Expression
Integer
(7) ->
(7) ->

(see sign)

Thanks
   Ingo