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
[email protected]
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