Any counterexample for inverse limit functor not to be right exact

Question

We know that inverse limit is a "left" exact functor on the category of modules in the sense that whenever $r:(A_i,α_j^i )→(B_i,β_j^i )$ and $s:(B_i,β_j^i )→(C_i,γ_j^i )$ are transformations of inverse systems over an index set $I$ between modules, and $0→A_i→B_i →C_i→0$ is exact for each $i∈I$, where the first function is $r_i$ and the second is $s_i$, then there is an induced (left) exact sequence $0→\varprojlim A_i→\varprojlim B_i→\varprojlim C_i$ . I search for a counterexample to the latter to be not "right" exact. Thanks very much in advance!