  We study sequences of 3-dimensional solutions to the Ricci flow with almost
nonnegative sectional curvatures and diameters tending to infinity. Such
sequences may arise from the limits of dilations about singularities of Type
IIb. In particular, we study the case when the sequence collapses, which may
occur when dilating about infinite time singularities. In this case we classify
the possible Gromov-Hausdorff limits and construct 2-dimensional virtual
limits. The virtual limits are constructed using Fukaya theory of the limits of
local covers. We then show that the virtual limit arising from appropriate
dilations of a Type IIb singularity is always Hamilton's cigar soliton
solution.
