How to prove a limit exists using delta and epsilon?

Question

I am having issues writing a proof using deltas and epsilons for this limit: I basically do not how to construct a proof after finding a limit. My main problem with problems like that arise from the proof. I took the the path of y=x and y =-x to come up with L= 0. $$\lim_{(x,y) \to (0,0)} \frac{xy}{\sqrt{x^2+y^2}}$$

Thanks, Mo.

Answer 1

Related techniques: (I). Here is a how.

$$ \Bigg| \frac{xy}{\sqrt{x^2+y^2}}-0 \Bigg| = \frac{|x|\,|y|}{\sqrt{x^2+y^2}} \leq \frac{\sqrt{x^2+y^2}\,\sqrt{x^2+y^2}}{\sqrt{x^2+y^2}} = \sqrt{x^2+y^2} < \epsilon = \delta. $$

Note:

$$ |x| = \sqrt{x^2} \leq \sqrt{x^2+y^2}. $$

Answer 2

What do you think the limit is? If I give you an $\epsilon$ that is positive, you need to find a $\delta$ so that over the whole square $-\delta \lt x,y \lt \delta$ the expression is within $\epsilon$ of the limit. It seems that $x=\pm y$ will yield the extreme values. What are your issues?