I maybe should have split this up into 2 questions, but I felt like they were similar enough to be put into one.
The rings $R[x]/(x^n)$ (with $R$ an arbitrary commutative unital ring) have two sets of natural maps between them. One indexed by the poset $(\mathbb{N}, \leq)$ where for $n\geq m$, we have $R[x]/(x^n)\to R[x]/(x^m)$ which maps $x$ to $x$.
I want to try and compute the limit of this diagram, my main candidates are $R$ and $R[x]$, but am not sure either way, and wondering if it might be something more complicated.
The other natural maps are indexed by $(\mathbb{N}, |)$ and we have when $n=dm$ the maps $R[x]/(x^m)\to R[x]/(x^n)$ mapping $x$ to $x^d$. Here I would like to compute the colimit, and this feels like it should be $R[x]$ in the same way that it feels like $colim(\mathbb{Z}/n\mathbb{Z})=\mathbb{Z}$, but this being false I am wondering whether it is some "completion" of $R[x]$.
Any help would be greatly appreciated, to either one of the questions. If you think I should split this up into two questions don't hesitate to say so.
The limit of the first diagram is the ring $R[[x]]$ of formal power series; this is the one that is a completion of $R[x]$, in fact it is exactly the $x$-adic completion. This is a nice exercise (both the limit and the completion) and it is very closely analogous to the way we get the $p$-adic integers $\mathbb{Z}_p$ as either the limit of the quotients $\mathbb{Z}/p^n\mathbb{Z}$ or as the $p$-adic completion of $\mathbb{Z}$.
As for the second diagram, first of all, it is of course not true that $\text{colim}_n \mathbb{Z}/n\mathbb{Z} \cong \mathbb{Z}$. In fact $\mathbb{Z}/n\mathbb{Z}$, being torsion, does not admit any nonzero maps to $\mathbb{Z}$. The real colimit is $\mu_{\infty} \cong \mathbb{Q}/\mathbb{Z}$, the group of all roots of unity; to see this think of $\mathbb{Z}/n\mathbb{Z}$ as $\mu_n \cong \frac{1}{n} \mathbb{Z}/\mathbb{Z}$, the group of $n^{th}$ roots of unity.
Your second diagram is similar. The rings $R[x]/x^n$ don't admit any maps to $R[x]$ sending $x$ to $x$ so that can't be the colimit. Instead we can think of each ring in the diagram as $R[x^{\frac{1}{n}}]/x$ and the maps between them are inclusions consistent with this notation. Then the colimit is the increasing union of all of these, which is a ring one might call $R[x^{\frac{1}{\infty}}]/x$; formally, it is the quotient of the algebra $R[x^{\mathbb{Q}_{\ge 0}}]$ of polynomials with non-negative rational exponents $x^{\frac{p}{q}}$ by the ideal $(x)$.