Inverse Limit of Sheaves 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.
 A: Let $X = \text{Spec}\;ℂ[x]$, and $\mathcal{G}⊂\mathcal{O}_X$ be the subsheaf of regular (rational) functions with no pole at $0$ or $∞$, and at most simple poles elsewhere.  Thus if $U = ℂ \backslash \lbrace a_1,\dotsc,a_m \rbrace$ or $ℂ \backslash \lbrace a_1,\dotsc,a_m,0 \rbrace$, with $a_1,\dotsc,a_m$ distinct and nonzero, then $\mathcal{G}(U) = \lbrace\dfrac{f}{∏_{i=1}^m(x-a_i)} \vert \deg(f)≤m \rbrace$.  Then let $\mathcal{G}_n = \bigoplus_{d=1}^n\mathcal{G}$, with $\mathcal{G}_{n+1}↠\mathcal{G}_n$ by projection onto the first $n$ components.  Let $\mathcal{F}_n = \bigoplus_{d=1}^n\mathcal{F}^d$, where $\mathcal{F}^d⊂\mathcal{G}$ is the subsheaf of functions with a zero of order at least $d$ at $0$.  Then the sheaf (not presheaf!) $\mathcal{G} / \mathcal{F}^d$ is isomorphic to the constant sheaf $ℂ[x]/(x^d)$.  Therefore for nonempty $U$, $(\varprojlim\;\mathcal{G}_n/\mathcal{F}_n)(U) = ∏_{d=1}^∞ ℂ[x]/(x^d)$, whereas $\varprojlim\;\mathcal{G}_n(U) = ∏_{d=1}^∞ \mathcal{G}(U)$.  Passing to stalks, we see  $\varprojlim\;\mathcal{G}_n → \varprojlim\;\mathcal{G}_n/\mathcal{F}_n$ is not surjective, since $ℂ(x) → ℂ[[x]]$ is not.
Remark:  Obviously this example is not as simple or enlightening as yours.  However, one can argue that the example needs to be at least this complex, I believe.  Without going into details, let me at least say that surjectivity of a sheaf map is a local property, so it takes more than a global topological obstruction to block it.
A: A natural example of this is the inverse system $\left \{ \mathbb Z/ \ell^n \mathbb Z \right \}_{n\ge 1}$ of $\ell-$adic sheaves in the étale topology. In this case $\lim^1 \left \{ \mathbb Z/ \ell^n \mathbb Z \right \}_{n\ge 1} \ne 0 $. 
However, the same example does not work in the Zarisky topology (by flasqueness of the sheaves involved) or in the pro-étale topology (by the existence of weakly contractible objects), i.e., it holds for two different reasons: sheaves are not interesting enough (Zarisky) and the topology is very good (pro-\'etale).
