TITLE: An inverse limit
QUESTION [12 upvotes]: 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.

REPLY [6 votes]: 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 $k[x]$.