Re: [Axiom-developer] wrong sign in limit?

[Waldek Hebisch]

Dr.-Ing. Ingo D. Rullhusen wrote:
> 
> 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.
> 
> 
> Z:=sqrt(a^2+h^2)-a
>  
> N:=a*sqrt(a^2+h^2) - a^2 - h^2
> 
> 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 )
>  
> dN:=differentiate( N, h, 2 )
>  
> eval(dZ/dN,h=0)
>  
> 
>                1
>    (6)  - ----------
>             +--+
>             | 2
>           2\|a   - a
> 
> (see sign)

Square root has two branches, the result depends on which branch
you choose.  Limit decided that sqrt(a^2) = -a and the result
is correct for this branch (and wrong for the other branch).
de L'Hospital rule can be used only when limit on N is 0, which
hold when sqrt(a^2) = a, but the result is wrong for the other
branch.  If you want result which is correct for both branches
you need something like:

-sqrt(a^2)/a^2

but such expressions cause trouble so limit decided just to pick
one branch...

-- 
                              Waldek Hebisch
address@hidden