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MathGroup Archive 1999

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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                           




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