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