I want to learn about how to compute the completions of local rings. For example, I want to be able to compute the completions of
\begin{align*} \left(\frac{\mathbb{C}[x,y]}{(y^2 - x)}\right)_{(x,y)} && \left(\frac{\mathbb{C}[x,y,z]}{(x^4 + y^4 - z)}\right)_{(x,y,z)} \end{align*}
with respect to their maximal ideals. With the first example, I expect it to be isomorphic to $\mathbb{C}[[x]]$ but I'm not sure how to show this using the inverse limit definition. When looking at multiplying the maximal ideal together, I found the following relations \begin{align*} (x,y)\cdot(x,y)&=(x^2,xy,y^2) = (x^2,xy,x) = (x) \\ (x)\cdot(x,y) &= (x^2, xy) \\ (x^2,xy)\cdot(x,y) &= (x^2, x^2y, x^2y, xy^2) = (x^2,x^2y, x^2y,x^2) = (x^2) \end{align*} giving a recurrence relation on the ideals, but I fail to see why the completion isn't $\mathbb{C}[y][[x]]$, which isn't what I'd expect since the first scheme is smooth.