When should we take direct limit and when should we take inverse limit? We know that we can take direct limit for a direct system and inverse limit for an inverse system. For example, when can defined the stalk of a presheaf $\mathcal{F}$ on a topological space $X$ at a point $P\in X$ by $$ \lim_{\rightarrow_{p \in U \text{ open}}} \mathcal{F}(U). $$
It seems that we get something small. We can take inverse limit of the inverse system $\{k[x]/(x)^n, n=0, 1, 2, \ldots\}$. Then we obtain $k[[x]]$ which is larger than $k[x]$. It seems that by taking inverse limit we add something to $k[x]$. It seems that taking inverse limit is to add something and taking direct limit is to get something small. Is this correct? If a system is both direct system and inverse system, how do we know that we need to take direct limit or inverse limit? Thank you very much.
 A: This is not a full answer, but some thoughts about direct and inverse limits.
I always think of the direct limit of a system $(A_i,\varphi_i)_{i \in I}$ as a union $A = \bigcup A_i$ but with compatible transition maps, i.e. an element is in this union if and only if it is compatible by those maps and the canonical injections $A_i \longrightarrow A$.
In contrast to inverse limit, where I think of them not as the union but as the cartesion product (or to be exact: a subset of the cartesian product). There you don't have canonical injections, but canonical projections $A \longrightarrow A_i$ and the transition maps of the inverse system have to be compatible with this canonical projections.
In conclusion, for me a direct limit is a really tiny subset of the union of all spaces and the inverse limit is a really big subset of the cartesian product, roughly speaking. 
I hope that intuition helped you!
A: It is a bit strange to compare direct and inverse limits. In spite of the (very) misleading terminology, they are not variations of the same concept.
Inverse limit is a special case of the notion of limit, which searches for a universal object factorizing morphisms to a diagram. Another way to put it : limits solve problems about the representability of some contravariant functors.
Direct limit is a special case of the notion of colimit, which searches for a universal object factorizing morphisms from a diagram.  Another way to put it : colimits solve problems about the representability of some covariant functors.
So, in my understanding, trying to compare direct and inverse limits in a category $\mathcal C$ is about comparing objects (and morphisms) of $\mathcal C$ and $\mathcal C^{\mathrm{op}}$, which are (a priori) very different categories...
