I have done the following problem but I wonder whether there is a generalization of the statement for the usage of direct limit. I will denote $lim$ as direct limit with the open set containing $p$.
Let $F'\to F$ be an injective sheaf morphism. Then I have $(F/F')_p\cong (F_p/F'_p)$.
I do not care for the sheafification at the moment as there is no difference to distinguish the stalk on sheaf or pre-sheaf as they are the same through connoical isomorphism on stalk level.
Q: Is there a generalization of about direct limit usage? (i.e. $lim(\frac{A}{B})=\frac{lim A}{lim B}$ obvious? Sounds like quotient of the limit exists if each limit exists. However it does not make sense unless I can make sure the sequence of B is always a subset of any sequence of $A$. In particular, I wish to say if I have another direct limit system $lim'$ such that $lim'B\cong lim B$, then I conclude $\frac{lim A}{lim'B}\cong\frac{lim A}{lim B}\cong lim(\frac{A}{B})\cong lim'(\frac{A}{B})$. This might be the another version of freshman's dream? )
I am too lazy to draw a lot of diagram. I want to think an intuitive way of direct limit for this. I hope there is an obvious way to see this.
Let $I$ be a directed set, and let $\{A_i\}, \{B_i\}, \{C_i\}$ be directed systems of abelian groups indexed by $i$. For each $i \in I$, let
$$\{A_i\} \rightarrow \{B_i\} \rightarrow \{C_i\} $$
be an exact sequence of directed systems. By this I mean a collection of exact sequences $ A_i \rightarrow B_i \rightarrow C_i$ indexed by $I$, which are compatible with the homomorphisms $A_i \rightarrow A_j, B_i \rightarrow B_j,C_i \rightarrow C_j$ for all $i \leq j$.
The universal property of direct limits induces a sequence of abelian groups
$$\varinjlim A_i \rightarrow \varinjlim B_i \rightarrow \varinjlim C_i$$
which by a standard exercise is exact e.g. see here Why do direct limits preserve exactness? .
In particular, if you started with a short exact sequence
$$0 \rightarrow \{A_i\} \rightarrow \{B_i\} \rightarrow \{C_i\} \rightarrow 0$$
then for each $i$, you have $B_i/A_i \cong C_i$, and the above results lets you say that
$$\varinjlim B_i / \varinjlim A_i \cong \varinjlim C_i \cong \varinjlim B_i/A_i$$
In your case, $I$ is the collection of open neighborhoods of $p$ ordered by reverse inclusion ($U \leq V$ if and only if $U \supseteq V$).