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MathGroup Archive 1999

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Re: Limits of multi-var. functions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg19923] Re: Limits of multi-var. functions
  • From: Phil Mendelsohn <mend0070 at tc.umn.edu>
  • Date: Tue, 21 Sep 1999 02:22:53 -0400
  • Organization: University of Minnesota, Twin Cities Campus
  • References: <7rsh34$3gf@smc.vnet.net> <7s1o5r$9l6@smc.vnet.net> <7s4gjg$qbu$4@dragonfly.wolfram.com>
  • Sender: owner-wri-mathgroup at wolfram.com

"Kai G. Gauer" wrote:
> 
> Paul Abbott wrote:

> > The syntax you want is
> >
> >     Limit[Limit[x^2*y^2 + 3*y - 2*x*y^5, x -> 2], y -> 3]
> >
> > or
> >
> >     Limit[Limit[x^2*y^2 + 3*y - 2*x*y^5, y -> 3], x -> 2]
> >
> > both of which give you the same result.
> >

Kai Gauer replied:
 
> Ok, but any student of mathematics would obviously know that it is NOT always
> necessarily the case that:
> 
>     lim[lim[f(x,y)]] <> lim[lim[f(x,y)]] <> lim [f(x,y)]
>     x=a y=b               y=b x=a               (x,y)=(a,b)
> 
> Can anyone modify Limit for multiple variables to do the right thing and
> differentiate when to use which version of limit?
> 
> By the way, I can think of a lot of functions in which the first two equations
> are the same, but by choosing another (aritrary) "path" to (a,b) gives an
> answer of undefined/no limit.

Yeah, that troubled me too.  I have not established proof, but wonder if
the function can be written in polar coordinates, could the limit as
r->0 be taken as the limit of the function.  Do you have a
counter-example?

Best,

Phil Mendelsohn


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