In this talk, we provide an introduction to the study of the derived functors of the inverse limit functor and in particular to the more set theoretic aspects of this study. We present a pair of complementary results, due to Goblot and Mitchell, connecting the cofinalities of partial orders with the vanishing of derived limits of inverse systems indexed along them, and provide combinatorial set theoretic proofs of special cases of these results. Time permitting, we will also discuss some connections between derived limits and the projective dimension of rings.