Answer
$0$
Work Step by Step
Given: $\lim\limits_{(x,y) \to (0,0)}f(x,y)=\frac{x^{3}-y^{3}}{ {x^{2}+xy+y^{2}}}$
Consider $f(x,y)=\frac{x^{3}-y^{3}}{ {x^{2}+xy+y^{2}}}=\frac{(x-y)(x^{2}+xy+y^{2})}{x^{2}+xy+y^{2}}$
$f(x,y)=(x-y)$
Put $x=0,y=0$ , we get
$\lim\limits_{(x,y) \to (0,0)}f(x,y)=\lim\limits_{(x,y) \to (0,0)}(x-y)=0$
Hence,$\lim\limits_{(x,y) \to (0,0)}f(x,y)=\frac{x^{3}-y^{3}}{ {x^{2}+xy+y^{2}}}=0$
