Definition of a limit: Multi-Variable

I know someone posted a pdf of an absolutely beautiful explanation of the delta-epsilon limit proof in R2, but this multivariate thing is staring me in the face and I'm absolutely lost all over again.

So if someone could use the definition of the limit to prove the following then It'd be great:

and, because I have a tendency to follow blindly and end up proving something that should be disproved, could someone do this one too:

ok so I have no idea how to make an obviously false statement >.>

On another note, limits are hard to Latex :D