An inverse limit

Let $k$ be a field.

Consider the inverse limit

$\varprojlim k[x,y]/(y\cdot x^n)$.

I wonder if there is a nice description of this ring?

Geometrically, we look at the union of the line $y=0$ along with an infinitesimal neighborhood of the line $x=0$. But what happens in the limit? I think we get $k[y][[x]]$? What is the geometric interpretation of this?

Any help will be appreciated.

Solution 1:

Let $R=\varprojlim k[x]/(x^n) = k[[x]]$, the ring of power series in $x$. The natural map $k[x]\rightarrow R$ is injective.

Then your limit is the subring of $R[y]$ with constant terms in $k[x]$.