Does the limit exist for the following function?

Question

How can we justify that the limit does/does not exist for this function? http://uploadpie.com/7VL6o

My attempt is that:

If y=0 , we have lim = 0

if x=0 , we have lim = 0

but does that mean the limit actualy exist at 0?

My book uses another method which is set y=x so we get: lim(x,x)->(0,0) which also equals 0...

Is this enough of a proof that the limit exists at zero?

Answer 1

As you have stated, along $y=0$ the limit is $0$.

Along $x = y^2$, $\frac{xy^2}{x^2+y^4} = \frac{y^4}{y^4 + y^4} \to \frac{1}{2}$, so the limit does not exist.