Recall the Theorem on Formal Functions [Hartshorne, III.11.1] Let $f:X \to Y$ be a projective morphism of noetherian schemes, let $\mathcal{F}$ be a coherent sheaf on $X$ and let $y\in Y$. Then the natural map $$ R^i f_\ast (\mathcal{F})_y^\hat{} \to \varprojlim H^i(X_n, \mathcal{F}_n) $$ is an isomorphism for all $i \geq 0$.
My question is: what's the best way to think about this result? Hartshorne mentions that this is just a way to compare the cohomology of infinitesimal neighborhoods of the fibers to the stalks of the higher direct image sheaves. Does taking inverse limits not muddy this idea? Also, the important corollaries (Zariski's main theorem, Stein factorization) only use the case $i=0$. Is this the case I should remember or are there applications for $i > 0$?