Question

lim (x,y) goes to (0,0) (6x^3( y))/(2x^4+y^4)

why the limit does not exist ?

Answer 1

check if the equation is a proper function (1 to 1)

Answer 2

Check different paths of approach towards (0,0).

Fix \(y=0\), then
\[\lim_{(x,y)\to(0,0)}\frac{6x^3y}{2x^4+y^4}=\lim_{x\to0}\frac{0}{2x^4}=0\]
Fix \(x=0\), then
\[\lim_{(x,y)\to(0,0)}\frac{6x^3y}{2x^4+y^4}=\lim_{y\to0}\frac{0}{y^4}=0\]
Let \(y=x\), then
\[\lim_{(x,y)\to(0,0)}\frac{6x^3y}{2x^4+y^4}=\lim_{x\to0}\frac{6x^4}{3x^4}=2\]
The limit is path-dependent, so it does not exist. (not sure if those are the right words)