Multivariable Limits

Date: 12/01/1999 at 13:28:55
From: John McFerrin
Subject: Multivariable limits problem

I have to find a function f(x,y) such that

     lim(y->0)lim(x->0) f(x,y) = lim(x->0)lim(y->0) f(x,y)

And I can't figure out what kind of function would yield such a 
bizarre result. If you could help, I would really appreciate it.

Date: 12/02/1999 at 15:56:53
From: Doctor Schwa
Subject: Re: Multivariable limits problem

Intuitively speaking, we have

     lim(y->0) f(0,y) = lim(x->0) f(x,0)

That's easy enough to imagine: just make f(0,y) = 1 and f(x,0) = 2, 
for example. Of course these definitions contradict each other at 
(0,0), but that's no problem; f(0,0) can be anything and it won't 
affect the limits.

Now the trick is to make the function continuous enough in the 
appropriate directions that these two limits work out this way. 
Can you imagine a curved surface that would connect f(0,y) = 1 to 
f(x,0) = 2 in a continuous way (except at the origin)? If so, then 
that picture is your function.

Maybe something like f(x,y) = (2x + y)/(x + y) will do the trick. How 
did I come up with that? I just thought "hmm, how about a fraction 
that will equal 1 when x is 0 and will equal 2 when y is 0." This 
function gives problems along the whole line y = -x, though... so 
maybe a little more patching up would be necessary.

I hope this gives you some direction. If you'd like to continue this 
discussion, please do write back.

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/