Re: Re: Limits of multi-var. functions
- To: mathgroup at smc.vnet.net
- Subject: [mg19948] Re: [mg19923] Re: Limits of multi-var. functions
- From: me at talmanl1.mscd.edu
- Date: Wed, 22 Sep 1999 04:11:24 -0400
- Sender: owner-wri-mathgroup at wolfram.com
Phil Mendelsohn wrote:
> 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?
The polar coordinates approach sometimes works. But not for
2
x y
f[x. y] = ----------------,
4 2
x + y
which has no limit at the origin. This is in spite of the fact that
the limit as x -> 0 along lines of the form y = m x always gives zero.
To see that the limit doesn't exist, try letting x -> 0 along curves
of the form y = m x^2.
A QuickTime animation of this surface can be found from
http://clem.mscd.edu/~talmanl/MathAnim.html
--Lou Talman