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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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