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MathGroup Archive 2003

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Re: RE: Limits: Is there something I'm missing Here?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg39369] Re: [mg39349] RE: [mg39333] Limits: Is there something I'm missing Here?
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Thu, 13 Feb 2003 04:51:57 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

The answer to the last Ashraf's last question is mathematically rather 
uninteresting: in Mathematica Limit[f[x], x -> 0] *means* Limit[f[x], x 
-> 0, Direction -> -1].

As for the other points, well it is true that even the one sided limits 
of 1/x "do not exist" if a limit is required to be a real number. But 
it is often convenient to consider the "extended" real line with two 
additional points, called -Infinity and Infinity (one speaks of 
"compactifying" the real line, which then becomes topologically 
equivalent to a closed interval). This can be made perfectly rigorous, 
although of course the object thus obtained is no longer a field in the 
algebraic sense (in other words, you can't perform usual arithmetic 
with Infinity and -Infinity).


Andrzej Kozlowski
Yokohama, Japan
http://www.mimuw.edu.pl/~akoz/
http://platon.c.u-tokyo.ac.jp/andrzej/

On Wednesday, February 12, 2003, at 05:52 PM, David Park wrote:

> Ashraf,
>
> Strictly speaking you are correct. The limit does not exist. But there 
> is
> such a thing as a one-sided limit, which is not a true limit but still
> useful.
>
> In this case even the one-sided limits do not really exist because the
> result is unbounded. But again, it is useful to signify this by saying 
> that
> the limit is infinite. So it is rather loose language by both 
> mathematicians
> and Mathematica.
>
> Perhaps one of the mathematicians in the group will give you a fuller
> explanation.
>
> David Park
> djmp at earthlink.net
> http://home.earthlink.net/~djmp/
>
> From: Ashraf El Ansary [mailto:Elansary at btopenworld.com]
To: mathgroup at smc.vnet.net
> To: mathgroup at smc.vnet.net
>
> Dear all,
> One thing I've noticed that if we have a function which has two 
> different
> limits (given two different directions) at one points , mathematica 
> would be
> still give an answer though to my understanding the limit doesn't 
> exist in
> such a case.
>
> Consider the following example:
> a[x_]:=1/x
>
> Limit[a[x],x->0,Direction->+1]    +Inf
>
> Limit[a[x],x->0,Direction->+1]    -Inf
>
> Limit[a[x],x->0].            +Inf.... Maybe my calculus knowledge is a 
> bit
> rusty but does the limit exist in this case??
>
>
>
> Thank you
>
>
>
>
>
>



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