Limit of a Function

Question

Show that the limit of the below function does not exist

lim _{(x,y)→(0,0)} { [ 4*(x^2)*(y^2) ] }/{ [ 2*(x^3) + 3*(y^3) ] }

Answer 1

lim _{(x,y)→(0,0)} { [ 4*(x^2)*(y^2) ] }/{ [ 2*(x^3) + 3*(y^3)] Let we tend to (0,0) along line y = x. Then

lim _{(x,y)→(0,0)} { [ 4*(x^2)*(y^2) ] }/{ [ 2*(x^3) + 3*(y^3) ]} = lim _x→04x⁴/5x³) = lim_x→04x/5 = 0;

but we tend to (0,0) along line y = x^1/3 then lim _{(x,y)→(0,0)} { [ 4*(x^2)*(y^2) ] }/{ [ 2*(x^3) + 3*(y^3) ] } =

lim_x→0[4x^8/3/(2x³ + 3x)] DNE because degree at the bottom > degree at the top

And because given limit DNE