I am trying to understand inverse limits and so I decided to try and compute one of such limits. We are going to work in the category of rings. Let $R$ be a ring. First, I set up an inverse system. Let $A_i:=R[x_1,\dots,x_i]$ denote a ring of polynomials over $R$ in $i$ indeterminates and let $f_{ij}:A_j \rightarrow A_i$ be morphism sending $x_j,\dots,x_{i+1}$ to $0$ and $x_1,\dots,x_i$ to themselves. Then $((A_i)_{i \in \mathbb{N}},(f_{ij})_{i \leq j \in \mathbb{N}}$) is my inverse system. Then, according to wiki, the inverse limit $A$ of our inverse system is
\begin{equation*}A=\underleftarrow{\text{lim}}A_i=\Bigg\{\overrightarrow{a}\in\prod_\limits{i \in \mathbb{N}}A_i|a_i=f_{ij}(a_j) \; \forall \; i\leq j \in \mathbb{N} \Bigg\}. \end{equation*}
So if we look at some elements of $A$ (take $R=\mathbb{Z})$ we have, for example:
\begin{equation*} (\dots,3x_{3}^4-5x_3x_2x_{1}^2+5,5,5) \; \text{or} \; (\dots,x_{3}+x_2+x_1+4,x_2+x_1+4,x_1+4) \end{equation*}.
My suspicion is that $A \cong R[x_1,x_2,\dots]$ and so the inverse limit would be just that with nutaral projections. Is that correct? Is this the right way of looking at these sort of constructions?