Problem:Show that the following limit does not exist. 
> ;
4 4 x y lim  x, y > (0, 0) 2 4 3 (x + y )

We only need to find two paths ariving to (0,0) along which the limits are different. We consider the paths y=x and x=y^2. When y=x with x>0 we have, 
> Limit(`x^4*x^4`/`(x^2+x^4)^3`,x=0) = Limit(x^4*x^4/(x^2+x^4)^3,x=0);
8 x^4*x^4 x lim  = lim  = 0 x > 0 (x^2+x^4)^3 x > 0 2 4 3 (x + x )
but along x=y^2 with y>0 we have, 
> Limit(`y^8*y^4`/`(y^4+y^4)^3`,y=0) = Limit(y^8*y^4/(y^4+y^4)^3,y=0);
y^8*y^4 lim  = lim 1/8 = 1/8 y > 0 (y^4+y^4)^3 y > 0
Since the two paths produce different limits then the limit does not exist. 