  We show that almost split sequences in the category of comodules over a
coalgebra with finite-dimensional right-hand term are direct limits of almost
split sequences over finite dimensional subcoalgebras. In previous work we
showed that such almost split sequences exist if the right hand term has a
quasifinitely copresented linear dual. Conversely, taking limits of almost
split sequences over finte-dimensional comodule categories, we then show that,
for countable-dimensional coalgebras, certain exact sequences exist which
satisfy a condition weaker than being almost split, which we call ``finitely
almost split''. Under additional assumptions, these sequences are shown to be
almost split in the appropriate category.
