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Re: Limits of multi-var. functions

"Kai G. Gauer" wrote:
> Paul Abbott wrote:
> > Phil Mendelsohn wrote:
> >
> > > I suspect this is an easy question, but I'm not finding it in Help or a
> > > couple of other Mathematica books I have around.
> > >
> > > If I want to find the limit of a function of several variables, how do I
> > > do it?  In the case of a polynomial function, I tried
> > >
> > > Limit[x^2 y^2 - 2x y^5 + 3y, {x->2, y->3}]
> >
> > 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.
> >
> 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.
> ...

To state a tautology, multivariate limits are path dependent or they are
not. Those that are independent of path can be computed just fine by an
iterated limit. Those that are path dependent must of course be handled
by specifying a path. This involves parametrizing the function in terms
of a single variable, hence reduces to finding a univariate limit.

I should point out that the path independent class is in some sense the
less interesting one. For functions that are holomorphic (have power
series definitions) in some region, if a unique limit exists at a
boundary point, independent of all paths (both from inside and outside
the region), then the function extends past that point. In other words,
the point of interest is in the interior of the region for which the
function is defined.

Below is a URL to a 1997 post I made to mathgroup on this topic.

Daniel Lichtblau
Wolfram Research

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