The derived functors $\lim^n$ of the inverse limit measure $n$-dimensional obstructions to the inverse limit functor preserving exact sequences. Whether or not certain derived limits vanish carries surprisingly deep set theoretic content involving higher dimensional infinitary combinatorics. The vanishing of certain derived limits $\lim^n \mathbf{A}$ has been the focus of considerable set theoretic focus and has implications in both strong homology and condensed mathematics. This talk will focus on the combinatorial principles behind the vanishing of $\lim^n \mathbf{A}$ and recent developments regarding when they do and do not vanish.