Does $\varprojlim\ ^1$ vanish whenever it doesn't have to account for non-right exactness of $\varprojlim$?

Question

The projective limit functor is not right-exact: if $G_\bullet\rightarrowtail H_\bullet\twoheadrightarrow K_\bullet$ is a projective system of extensions, then there is a long exact sequence $$ 0\to\varprojlim G_\bullet\to\varprojlim H_\bullet\to\varprojlim K_\bullet\to\varprojlim\ ^1 G_\bullet\to\varprojlim\ ^1 H_\bullet\to\varprojlim\ ^1 K_\bullet\to\cdots. $$ Is there an example of a projective system of extensions $G_\bullet\rightarrowtail H_\bullet\twoheadrightarrow K_\bullet$ such that the map $\varprojlim H_\bullet\to\varprojlim K_\bullet$ is surjective and such that the term $\varprojlim\ ^1 G_\bullet$ is non-zero?

Answer 1

Yes. For example, take any $G_\bullet$ for which $\varprojlim\ ^1 G_\bullet$ is non-zero, and consider the projective system $G_\bullet\rightarrowtail G_\bullet\twoheadrightarrow 0$, where the first map is the identity.