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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
djmp at earthlink.net
http://home.earthlink.net/~djmp/
>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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