Suppose that $X$ is a topological space with a sheaf of rings $\mathcal{O}_X$. In general, the stalk at a point $p \in X$ is the direct limit of the rings $\mathcal{O}_X(U)$ for all open sets $U$ containing $p$.
Here are two questions on computing stalks - I think both should be true, since a direct limit should be some sort of "limiting process", but that's far from convincing for me.
Can I compute the stalk of $\mathcal{O}_X$ at a point $p \in X$ by only limiting over basic open sets of $X$ containing $p$?
Can I compute the stalk of $\mathcal{O}_X$ at a point $p \in X$ by excluding some finite number of "large" open sets around $p$, and then limiting over the remaining open sets around $p$?