TITLE: Multivariable Epsilon-Delta Proof?
QUESTION [7 upvotes]: I am lost on this problem:
State whether the following limit exists and prove it:
$$
\lim_{(x,y) \rightarrow (0,0)} \frac{\sqrt[2]{|x|}y}{x^2+y^2}
$$
All of the examples in class used the $\epsilon$ - $\delta$ proof technique. 
I am still getting used to this proof technique and I understand what $\epsilon$ and $\delta$ represent, but I don't know how to begin (or end) $\epsilon$ - $\delta$ proofs.
What I know:
Trying to prove $\forall\epsilon>0, \exists \delta$ such that $|\sqrt[2]{x^2+y^2}|<\delta \implies |\frac{\sqrt[2]{|x|}y}{x^2+y^2}-?|<\epsilon$
Specific questions: How do I proceed with the proof if I don't know what ? is? What does $|f(\mathbf x)-\mathbf a|<\epsilon$ mean in a multivariable case? What are general $\epsilon$ - $\delta$ proof techniques?
I think if I can figure out this problem I can figure out the rest.

REPLY [3 votes]: With multivariable limits, the thing to remember is this: the limit can only exist if the path limits along all paths exist and are equal, in which case the multivariable limit in and the path limits agree.
So, to find your "?", start by taking a path limit.  For instance, along the line $y=x$, we have
$$
\lim_{x\rightarrow0}\frac{\sqrt{\lvert x\rvert}x}{2x^2}=\lim_{x\rightarrow0}\frac{\sqrt{\lvert x\rvert}}{2x}
$$
But this limit doesn't exist!  Approaching $(0,0)$ along $y=x$ for $x<0$ you get $-\infty$, and approaching $(0,0)$ along $y=x$ for $x>0$ you get $\infty$.
So, you need to switch it up here: you need to show that the limit does not exist!