Verify if the following limit exists (by the formal definition): $\lim_{(x,y)\to(0,0)}\frac{x^3-y^3}{x^2+y^2}$ First, I tried approaching the point through a few curves: ($x=0,\,y=0,\, y=m\cdot x,\, y=x^2,\, x=y^2)$ and to all of those I got $0$ for my limit. Since this isn't enough to prove that the limit actually exists, I need to move to the formal definition.
The definition:
If $0<\sqrt{(x-x_0)^2+(y-y_0)^2}<\delta$ then $\left|f(x,y)-L\right|<\epsilon$
But I'm having a lot of difficulty in using this definition of limits to prove if it exists or not.

let $\delta = \varepsilon$ then if $\sqrt{x^{2}+y^{2}}< \delta$ we have $\left | \frac{x^{3}-y^{3}}{x^{2}+y^{2}} \right |=\left |\frac{x^{3}}{x^{2}+y^{2}}- \frac{y^{2}}{x^{2}+y^{2}} \right |= \left | x\frac{x^{2}}{x^{2}+y^{2}}-y\frac{y^{2}}{x^{2}+y^{2}} \right |\leq \left | x-y \right |\leq \sqrt{x^{2}+y^{2}}= \delta $ and we have that $\left | f(x,y) \right |\leq \delta = \varepsilon $
