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

Let e = epsilon and d = delta

We need to find a d st if x^2 + y^2 < d^2

then| xy - 0| = |xy| < e

in terms of polar coordinates xy = r^2cos(t)sin(t)

It follows |xy| < r^2 = x^2 + y^2

take d = sqrt(e)

then if x^2 +y^2 < d^2 |xy| < e

It should be fairy obvious http://www.mathhelpforum.com/math-he...d51ea326-1.gif is false

since if you approach (0,0) along the line y = x the limit is infinity

oh, hey, no offense but that second part was useless. I'm pretty sure I wont get away with writing "it's obviously false" on a test question.

And I have no idea what happened on the first part... Why is everything squared!?! o.o

For the first part

we're saying that if the distance from a pt (x.y) is less than a distance d from (0,0) etc. --- what is the distance the formula from a pt in the plane to the origin?

As to the second part read the entire answer if you appraoch (0,0)


along the line y = x what is lim 1/(xy)
(x,y)->(0,0) ?

Yes, I get it, it's false, obviously. I even said up in my question that it's false. That's why I made it up, because it is false.

I need to see a limit proof of a FALSE limit. Like, disprove it.