The limit does not exist, because along the path $x = y^3$ the limit is $1/2$, whereas along the path $x = 0$ the limit is $0$.

The limit exists and is equal to $0$ because the degree of the numerator is strictly greater than the degree of the denominator.

The limit does not exist because along any straight line path $y = mx$ the limit is non-zero.

The limit exists and is equal to $1/2$.