Let $\{\mathcal{F}_i\}_{i\in \mathbb{N}}$ be an inverse system of sheaves of abelian groups on a space $X$. Then for any $x\in X$ we have a natural map $$\left(\lim_i \mathcal{F}_i\right)_x\rightarrow \lim_i \mathcal{F}_{i,x}.$$ However, as a was pointed out on https://mathoverflow.net/q/80322, this morphism is neither injective or surjective in general. Is there a way to gain any sort of information about the stalks of $\lim_i \mathcal{F}_i$ from the stalks of the $\mathcal{F}_i$? For instance, if the $\mathcal{F}_{i+1}\rightarrow \mathcal{F}_i$ are surjective, does that help?
As far as I know, the category $\mathbb{N}$ is not $L$-finite, as in that case the filtered colimit and $\lim_{\mathbb{N}}$ would commute by https://mathoverflow.net/q/140164.
Other Edit: Is the situation better if we assume that the limit is exact, by which I mean that the higher limits $R^i\lim_{j\in I} \mathcal{F}_j=0$ for $i>0$ (where by better I mean there we can get some information from $\lim_i \mathcal{F}_{i,x}$ about $\left(\lim_i \mathcal{F}_i\right)_x$)?