Traveling along the x-axis, we have
[tex]\displaystyle\lim_{(x,y)\to(0,0)}\frac{x^4-20y^2}{x^2+10y^2}=\lim_{x\to0}\frac{x^4}{x^2}=\lim_{x\to0}x^2=0[/tex]
On the other hand, along the y-axis we get
[tex]\displaystyle\lim_{(x,y)\to(0,0)}\frac{x^4-20y^2}{x^2+10y^2}=\lim_{y\to0}\frac{-20y^2}{10y^2}=\lim_{y\to0}(-2)=-2[/tex]
Therefore the limit doesn't exist.
