I am beginner to algebraic geometry and I was reading Vakil's notes https://math.stanford.edu/~vakil/245/245class2.pdf
I am stuck on the part where he said to take the example $K[x,y]$ and localize it at a divisor $x=0$. I trust that we get a DVR; however, I am confused as to how to find the order of zeros and poles of a rational function. The function he gave seems to not makes sense as we localize at $x=0$ so the denominator should not be divisible by $x$. So let's consider a new example like $\frac{x^{2}-3y}{x^{2}+y}$.
I looked at
How to compute the order $\text{ord}_P (f)$ for $f \in K(C)$
and it seems that I write my rational function as a unit times its power of the generator (I believe it is called the uniformizer). But none of the above is divisble by $x$ so we have the order at $x=0$ of $\frac{x^{2}-3y}{x^{2}+y}$ is 0 as we can write it as $\frac{x^{2}-3y}{x^{2}+y}x^{0}$? This seems very awkward to me as no value of $y$ is specified so I am sure this is not the right way to go.
Edit: It seems this part has nothing to do with the localizing at $x=0$.
Also, I am not sure what to do in the next question (to find order of pole/zeros of $\frac{y}{x}$ at $y^{2}=x^{3}$). Edit: I looked into Order of the pole of projective curve using uniformizer and it seems that to do the second part, we have if $P=(0,0)$, then $0=ord_{P}(y^{2}/x^{3})=ord_{P}(1)=2ord_{P}(y)-3ord_{P}(x)$. What exactly should I do next? I was thinking maybe this can help me find $ord_{P}(y)$ and $ord_{P}(x)$ so I can find $ord_{P}(y/x)$.