# Multivariable Epsilon-Delta Proof?

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.

• For problems of this kind it helps to write $x=r\cos \phi$, $\>y=r\sin\phi$ and to see what can happen when $r\to0+$. – Christian Blatter Sep 10 '13 at 16:27
• @NicholasR.Peterson You're right. I gotta pay more attention. – Git Gud Sep 10 '13 at 16:50
• @GitGud You should see some of the comments/answers I've posted before my morning coffee has kicked in :-) – Nick Peterson Sep 10 '13 at 16:55
• @NicholasR.Peterson $\ddot \smile$ – Git Gud Sep 10 '13 at 16:59

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 $\infty$.
• That's the intuition. To be more formal, I've given you a blueprint for showing that given any $\epsilon>0$, for every $\delta>0$ there exists two points that have distance at most $\delta$ from each other and differ by more than $\epsilon$ -- showing that no limit can exist. – Nick Peterson Sep 10 '13 at 23:45