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Do you agree: limit(ABS(x)^n,n,∞,1); => undef (One-sided limit: n≥0)
05-22-2018, 01:13 PM (This post was last modified: 05-22-2018 03:35 PM by DrD.)
Post: #1
Do you agree: limit(ABS(x)^n,n,∞,1); => undef (One-sided limit: n≥0)
[CAS]
limit(ABS(x)^n,n,∞,1); => undef (One-sided limit: n≥0)

Notice also:
limit(0^n,n,∞,1); => undef

Wolfram, and wxmaxima disagree.
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05-22-2018, 03:30 PM
Post: #2
RE: Do you agree?
Request: better thread title. The actual phrasing doesn't really help to classify the topic. It seems either a generic discussion or one of the late heated discussion.

Wikis are great, Contribute :)
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05-22-2018, 03:38 PM
Post: #3
RE: Do you agree: limit(ABS(x)^n,n,∞,1); => undef (One-sided limit: n≥0)
Is that any better?
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05-22-2018, 05:42 PM
Post: #4
RE: Do you agree: limit(ABS(x)^n,n,∞,1); => undef (One-sided limit: n≥0)
For x>1 the limit is infinity, for x=1 it is 1 and for 0<=x<1 it is zero. With assume(x>1) your limit returns inf, as expected, and, when I do assume (x<1) and additional(x>0), it returns 0, so undef is reasonable.
Arno
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05-22-2018, 06:36 PM
Post: #5
RE: Do you agree: limit(ABS(x)^n,n,∞,1); => undef (One-sided limit: n≥0)
Limits at infinity should use the syntax limit(expression,variable,inf) for +infinity or -inf for -infinity, there is no support for sided-limit at infinity because there is no need to.
As Arno K explained, the limit of abs(x)^n at n=+infinity depends on x, you must make assumptions otherwise you get undef.
0^n is undefined, and 0^n is evaluated before the limit in limit(0^n,n,inf). You can quote to prevent evaluation. Of course that's not really the kind of computation that I run with a CAS :-)
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05-23-2018, 12:06 PM
Post: #6
RE: Do you agree: limit(ABS(x)^n,n,∞,1); => undef (One-sided limit: n≥0)
(05-22-2018 05:42 PM)Arno K Wrote:  For x>1 the limit is infinity, for x=1 it is 1 and for 0<=x<1 it is zero. With assume(x>1) your limit returns inf, as expected, and, when I do assume (x<1) and additional(x>0), it returns 0, so undef is reasonable.
Arno

My point:

Accepting limit(ABS(x)^n,n,∞,1); => undef; as a reasonable answer, (and Parisse has explained that it IS the result he wants to report here), that,
after citing three valid definitions, a result "undefined" could still be termed "reasonable:" (defined ≠ undefined).

A result something like one of these would seem to have less weakness than "undef," because limit(ABS(x)^n,n,∞,1) has definite solutions:

[[0≤x<1,0], [x=1,1], [x>1,∞]];
or: indeterminate ( could be something like: ind assume(x..?) )
or: parametric
... etc.
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05-23-2018, 04:56 PM
Post: #7
RE: Do you agree: limit(ABS(x)^n,n,∞,1); => undef (One-sided limit: n≥0)
This is the difference between an interactive or non-interactive system. The CAS is non-interactive, it means that you can not ask "Is x>1?" during a computation (like in maxima) and the result must be usable in further computations. In this example, limit would return a piecewise function because the condition can be solved symbolically, but it's not always the case and it would be much more complicated to handle (sooner or later you are faced with combinatoric complexity explosion, especially if you don't have good algorithms to simplify piecewise functions).
Another point to remember is that Giac was primarily designed to do computations that are hard or impossible to do by hand, not these kinds of limits because in my opinion, they should not be solved with a computer, I mean the computer will not give any additional value to what a STEM student should know. Of course, they could happen in an intermediate computation, but then the user should have made appropriate assumptions on the parameter.
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05-24-2018, 09:59 AM
Post: #8
RE: Do you agree: limit(ABS(x)^n,n,∞,1); => undef (One-sided limit: n≥0)
I understand, and my thanks to you, Parisse.

Its very comforting to 'think' that a technology could provide a correct answer for any STEM related problem, but it is equally disheartening to confront the reality that such a technology doesn't exist, at least not in a practical personal singular form.

Still, the hp prime, (and your CAS!) is pretty darn cool! I have enjoyed using this device, nearly every day, for several years now. It's the best dollars I have ever spent on anything similar, and no end in sight. So ... thanks for that, too!

-Dale-
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