[Maxima-bugs] [ maxima-Bugs-873244 ] limit(x=0,x,0) wrong

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Bugs item #873244, was opened at 2004-01-08 13:15
Message generated for change (Comment added) made by willisbl
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Category: None
Group: None
Status: Open
Resolution: None
Priority: 5
Submitted By: Stavros Macrakis (macrakis)
Assigned to: Nobody/Anonymous (nobody)
Summary: limit(x=0,x,0) wrong

Initial Comment:
limit(x=0,x,0) => 0=0.
limit(equal(x,0),x,0) => equal(x,0)

Let's leave aside the fact that 0=0 could simplify to 
True -- that is not a limit issue, but a broader issue in 
Maxima.

The real problem is that limit(x=0,x,0) is in fact FALSE, 
not TRUE.  Since x=0 is FALSE for all x#0, it is FALSE in 
particular for all 0<x<epsilon.

Presumably the problem here is that Limit is assuming 
that "=" is a continuous function of its arguments.

On the other hand, limit is *too* careful with other 
relationals:

  limit(x>0,x,0,plus) => noun
  limit(x>=0,x,0,plus) => noun

These could perfectly well return TRUE.

But that is a feature request, not a bug.


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>Comment By: Barton Willis (willisbl)
Date: 2004-01-12 15:30

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The idom "Take the <limit, derivative, cube, ...> of both 
sides" is uttered a few  thousand times per hour; for that  
reason, I think mapping limit, diff, and cube over an equality 
is sensible (not always correct, but GIGO)  If, as you suggest

   (x = x)  == true

we'd also have to live with
 
  diff(x = x,x) ---> diff(true,x) ---> huh?

I doubt this is what most users expect or want.

Polemics aside, Maxima is lackadiasical about mapping 
functions over mbags.  This needs to be cleaned up and (as 
you and I have discusssed) too much (possibly all) Maxima 
code assumes that the result of mapping a function over an 
mbag doesn't need simplification. That's another story.

Barton

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Comment By: Stavros Macrakis (macrakis)
Date: 2004-01-12 13:42

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Whether 0=0 should simplify to true is a side issue.

I don't follow your argument about 0=0 representing the set 
of all numbers.  ANY true statement represents the universal 
set (if you believe in that), whether it is {x | 0=0} or {x | 
true} or {x | Socrates is mortal}.  Note by the way that 
there is nothing telling you the domain.  0=0 could equally 
well represent the set of all determinant-1 matrices or all the 
5th degree polynomials over Z7, if that's the domain of x's 
you are manipulating.

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Comment By: Wolfgang Jenkner (wjenkner)
Date: 2004-01-12 13:42

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> The real problem is that limit(x=0,x,0) is in fact FALSE, 
> not TRUE.  Since x=0 is FALSE for all x#0, it is FALSE in 
> particular for all 0<x<epsilon.

Do you mean that in some punctured neighbourhood of 0,

is(limit(x=0,x,0)) == is(0=0) == TRUE,
limit(is(x=0),x,0) == limit(FALSE,x,0) == FALSE

But this just shows that IS and LIMIT don't commute and, in
particular, that IS(? = 0) is not continuous at 0.

But... where is Maxima wrong?


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Comment By: Stavros Macrakis (macrakis)
Date: 2004-01-12 13:37

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But mapping limit over an *equation* (not a generic mbag) is 
in general wrong if you consider an equation to be a boolean-
valued object.

Remember, limit is defined for epsilon > 0, not epsilon >= 0.

Limit(x=0,x,0) really is false, since x=0 is false for all abs(x)>0.

On the other hand, limit(sin(x)=sin(x),x,inf) is true, even 
though limit(sin(x),x,inf) is ind, and in general ind does not 
equal ind (like NaN).  And limit(x=x^2,x,inf) is false, even 
though limit(x,x,inf)=limit(x^2,x,inf)=inf.


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Comment By: Barton Willis (willisbl)
Date: 2004-01-11 15:12

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I think mapping limit over an mbag is sensible and  consistent 
with much of Maxima.  

0 = 0 could reasonally simplify to true, but it could
just as well represent the set of all real (or complex) numbers.
[ If 0 = 0, then 0 x = 0; the solution set of 0 x = 0 is 
everything.] I think Maxima shouldn't simplify  0 = 0 to 
true or anything else.

Barton

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Comment By: Barton Willis (willisbl)
Date: 2004-01-11 15:11

Message:
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I think mapping limit over an mbag is sensible and  consistent 
with much of Maxima.  

0 = 0 could reasonally simplify to true, but it could
just as well represent the set of all real (or complex) numbers.
[ If 0 = 0, then 0 x = 0; the solution set of 0 x = 0 is 
everything.] I think Maxima shouldn't simplify  0 = 0 to 
true or anything else.

Barton

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