  We present a theorem which allows one to recognize and classify the
asymptotic behavior and causal structure of McVittie metrics for different
choices of scale factor, establishing whether a black hole or a pair
black-white hole appears in the appropriate limit. Incidentally, the theorem
also solves an apparent contradiction present in the literature over the causal
structure analysis of the McVittie solution. Although the classification we
present is not fully complete, we argue that this result covers most if not all
physically relevant scenarios.
