       Re: Limits of multi-var. functions

• To: mathgroup at smc.vnet.net
• Subject: [mg19860] Re: [mg19795] Limits of multi-var. functions
• From: "David Park" <djmp at earthlink.net>
• Date: Sun, 19 Sep 1999 01:20:43 -0400
• Sender: owner-wri-mathgroup at wolfram.com

Phil,

You have to take two limits in succession, or by following some path in the xy-plane.
If you get different answers depending upon the order or path, then the limit does
not really exist. Keeping this in mind, here is one method:

Fold[Limit[#1, #2] &, x^2 y^2 - 2x y^5 + 3y, {{x -> 2}, {y -> 3}}]
{-927}

Fold[Limit[#1, #2] &, x^2 y^2 - 2x y^5 + 3y, {{y -> 3}, {x -> 2}}]
{-927}

Here is a second method which approaches along a specific line in the xy-plane.

f[x_, y_] := x^2 y^2 - 2x y^5 + 3y
Limit[f[x, 3/2x], {x -> 2}]
{-927}

David Park

>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}]
>
>for example, but this gave me two results; one case if x approaches 2
>(leaving y unevaluated) and the other case if y approaches 3 (leaving x
>unevaluated.)  I'd like to evaluate for both simultaneously.
>
>Did I miss something -- I do know that convergence of the limit is
>proportionally more complex in several vars, but expected that Mathematica would
>do it.  (Running Mathematica v3.0.1, Linux)
>
>Thanks -- Phil Mendelsohn
>
>

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