Why is this limit of 2 variables undefined? it looks like both = 0

[mr_coffee]

Hello everyone...I have the following problem, it tellls me to find the limit.
I attached the image which has my work, it looks like when u let y = 0, then let x = 0, both come out to 0! but the book says undefined, what am i doing wrong?
thanks.

Here is the image: http://show.imagehosting.us/show/800815/0/nouser_800/T0_-1_800815.jpg
that link seems to be slow so try this one:
http://img427.imageshack.us/img427/1834/lastscan8bp.jpg

 

[iNCREDiBLE]

You can show that the limit is undefined by showing that the limit depends on how you approach zero.
You can show that $f(x,y) \longrightarrow \frac{1}{2}$ if you approach zero along the curve $(\sqrt{y},y)$.
So you have showed that the limit is both 0 and 1/2, i.e. it must be undefined.

 

[mr_coffee]

hm..thanks for the responce, but how did u figure out that u should approach zero along $(\sqrt{y},y)$ to show its 1/2?

 

[iNCREDiBLE]

To prove that lim f(x,y) does not exist, it suffices to show that the limit along one curve into (a,b) differs from the limit along a second curve. If lim f(x,y) does exist, however, then computing limits along individual curves will prove nothing (although, such computations will likely help to build understanding). As it turns out, proving that a limit exists requires a significant amount of mathematical rigor.

I didn't know. I just sat down and tried some "ordinary" curves. If you try y = kx², (x, kx²), you can show that the limit depends on k.