calc4 limit question

[shizzy]

Having some trouble with this problem:

lim            1 - x^2 - y^2
               ----------------
(x,y)->(0,0)       x^2 + y^2

after a little manipulation(r = x^2 + y^2):

lim       1 - r
         -------
r->0        r

curious if I am doing this right? The way it looks to me this is undefined, unless I am missing something. Help please!!

 

[Euler]

I think you did most of the steps right, though when you substituted the numerator, you substituted r right into x^2-y^2, when r=x^2+y^2.

My answer, doing the same thing, was:

 

[shizzy]

yeah I skipped some steps there. I factored out a -1 from it so:

1 - x^2 - y^2

became

1 -(x^2 + y^2) = 1 - (r) = 1 - r

unless I messed that up somehow. Sometimes I do !!

 

[Gene]

I don't see that it helps but I have the feeling that the substitution you want is
x=rcos(t)
y=rsin(t)
It leads to essentially the same thing except it is
(1-r²)/r²
The lim(x,y) -> (0,0) bothers me too. I keep seeing the circle equation and x & y can't both be zero at the same time in a circle graph. That might explain why it is undefined.
As I said, I don't see that this helps, but it is something to think about.
-------------------
Gene