Let f(x,y)=((x^(2)y^(2))/(x^(2)+y^(2))), classify the behavior of f near the critical point (0,0).
I don't know how you are supposed to approach this on your course,
but the following works:
Along the $\displaystyle x \,$ and $\displaystyle y\,$ axes the function is a constant equal to zero. Along
any other ray through the origin we may put $\displaystyle y=\lambda x\,$, when:
$\displaystyle f(x,\lambda x)=\frac{\lambda^2 x^2}{1+\lambda^2}\,$
which has a quadratic like mininum at $\displaystyle x=0\,$.
A surface plot shows what this looks like (see attachment)
(this assumes that we give the function a value of $\displaystyle 0\,$ at $\displaystyle x=y=0\,$)
RonL