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