Problem:Lef f be the function defined by 
> ;
2 x y f(x, y) =  2 4 x + y
when (x,y) is not (0,0) and define 
> ;
f(0, 0) = 0
Is f continuous at (0,0)? Explain.

If we use the path y=a x^m with x>0 we obtain, 
> Limit('x*a^2*x^(2*m)/(x^2+a^4*x^(4*m))',x=0) =
> limit(a^2*x^(2*m1)/(1+a^4*x^(4*m2)),x=0);
2 (2 m) 2 (2 m  1) x a x a x lim  = lim  x > 0 2 4 (4 m) x > 0 4 (4 m  2) x + a x 1 + a x
when m=1/2 this last limit produces 
> ;
2 a  4 1 + a
so the limit depends on a and thus it DNE. The function f is therefore not continuous at (0,0) since it doesn't have a limit at (0,0). 