Consider a path traced by an arbitrary power function [tex]y=x^n[/tex], where [tex]n\in\mathbb Z[/tex]. Then
[tex]\displaystyle\lim_{(x,y)\to(0,0)}\frac{x^2y}{x^4+y^2}=\lim_{x\to0}\frac{x^{n+2}}{x^4+x^{2n}}=\lim_{x\to0}\frac{x^{n-2}}{1+x^{2(n-2)}}[/tex]
When [tex]n=2[/tex], we have
[tex]\displaystyle\lim_{x\to0}\frac1{1+1}=\frac12[/tex]
but for any larger [tex]n[/tex], say [tex]n=3[/tex], we have
[tex]\displaystyle\lim_{x\to0}\frac x{1+x^2}=0[/tex]
Therefore the limit does not exist/is path-dependent.
