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MathGroup Archive 2001

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Re: Re: Limit and Abs


I'd hate to enter an argument on this topic, but even if, as you say,
"Interval[{0,1}]
provides more useful information than merely saying "does not exist".", the
fact remains that the definition of limit is precise.  I wonder if it is
true that "Mathematica has a more generalized notion of
limit than is often used". If such were the case, it would be necessary to
redefine established mathematical concepts, wouldn't it?

Tomas

----- Original Message -----
From: "David W. Cantrell" <DWCantrell at sigmaxi.org>
To: mathgroup at smc.vnet.net
Subject: [mg31256] [mg31230] Re: Limit and Abs


> Tomas Garza <tgarza01 at prodigy.net.mx> wrote:
> > And what, may I ask, is the "correct answer"? Your function is periodic
> > with period Pi. The only answer is
> >
> > In[1]:=
> > Limit[Cot[a]/(2 + Cot[a]), a -> Infinity]
> >
> > Out[1]=
> > Limit[Cot[a]/(2 + Cot[a]), a -> Infinity]
> >
> > And, unless there is a misprint in your second expression below, the
> > limit with *n* -> Infinity has to remain unevaluated, because a is not
> > defined. But, then, if there is a misprint and you meant a -> Infinity
> > instead of n->Infinity,
>
> As we now know, he didn't mean either! But taking a -> Infinity does
> raise an interesting point.
>
> > the function is again periodic with period Pi
> > (look at the graphs). The limit doesn't exist, and Mathematica has no
> > alternative but to leave the expression unevaluated.
>
> False! As others have pointed out, Mathematica does have other
> alternatives, and the one it chooses, returning a limit of 1, is
> incorrect. In my opinion, it should have returned Interval[{0,1}]
> as the answer instead. Mathematica has a more generalized notion of
> limit than is often used. For example, although most of us would
> normally say simply that Limit[Sin[a], a -> Infinity] does not exist,
> Mathematica gives Interval[{-1,1}]. This is correct in a sense. I have
> no objection to it; indeed, it provides more useful information than
> merely saying "does not exist".
>
>   David Cantrell
>
> > From: "Oliver Friedrich" <oliver.friedrich at tz-mikroelektronik.de>
To: mathgroup at smc.vnet.net
> To: mathgroup at smc.vnet.net
> > Subject: [mg31256] [mg31230]  Limit and Abs
> >
> > > Hallo,
> > >
> > > if I evaluate
> > > Limit[Cot[a]/(Cot[a]+2),a->Infinity]
> > > i get the correct answer.
> > >
> > > But I want to evaluate
> > >
> > > Limit[Abs[Cot[a]]/(Abs[Cot[a]]+2),n->Infinity]
> > >
> > > and that's being returned unevaluated.
>
> --
> -------------------- http://NewsReader.Com/ --------------------
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  • References:
    • Limit and Abs
      • From: "Oliver Friedrich" <oliver.friedrich@tz-mikroelektronik.de>
    • Re: Limit and Abs
      • From: "David W. Cantrell" <DWCantrell@sigmaxi.org>
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