<s>[INST] How do I prove using the definition of limit that $\lim_{(x,y)\rightarrow (0,0)}\frac {x^2+y^3}{y^2 - x + xy}$? I want to use the definition of limit in order to prove $\lim_{(x,y)\rightarrow (0,0)}\frac {x^2+y^3}{y^2 - x + xy}$, but I almost crack my head open trying to do this, it seems to be easy way but y cant find a $\delta$ such that $|\frac {x^2+y^3}{y^2 - x + xy}| \leq \epsilon$, any idea of how I might be able to find an inequality to transform the denominator in a nicer thing like $\frac{1}{\left\lVert (x,y)\right\rVert}$?
 [/INST] Note that
$$
x=y^2\implies\frac{x^2+y^3}{y^2-x+xy}=\frac{y^4+y^3}{y^2-y^2+y^3}=y+1\overset{y\to0}\to1
$$
and that
$$
x=0\implies\frac{x^2+y^3}{y^2-x+xy}=\frac{y^3}{y^2}=y\overset{y\to0}{\to}0
$$
These show that the limit does not exist.
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