It is well-known that if you have an inverse system of abelian groups $(A_n)$ (this works in several other nice categories) in which all the maps are surjective (or at least satisfy the Mittag-Leffler Condition), and if you have a short exact sequence of inverse systems $0\to (A_n)\to (B_n)\to (C_n)\to 0$, then taking the limit is exact and you get another short exact sequence $0\to \lim A_n \to \lim B_n \to \lim C_n \to 0$.
Hartshorne warns that this is not the case with abelian sheaves on a space. In particular, you can have all the maps of $(\mathcal{F}_n)$ surjective, and a short exact sequence $0\to (\mathcal{F}_n)\to (\mathcal{G}_n)\to (\mathcal{J}_n)\to 0$ but you only get left exactness $0\to \lim \mathcal{F}_n\to \lim \mathcal{G}_n\to \lim \mathcal{J}_n$
I.e. you get that $\lim^1(\mathcal{F}_n)\neq 0$ despite satisfying surjectivity of maps. Is there a canonical example of this happening?
My first guess was that this had to be related to the fact that you can have a surjective map of sheaves $\mathcal{F}\to \mathcal{G}$, yet still have an open set for which $\mathcal{F}(U)\to \mathcal{G}(U)$ is not surjective. The canonical example of when this happens is to use the exponential map on the sheaf of holomorphic functions on $\mathbb{C}^\times$, but it is very non-obvious to me how to turn this into an example of the above.