How to explain that black holes mergers take finite time but black hole forming infinite? Through detection of gravitational waves we observe coalescences of black holes, black holes and neutron stars, or even neutron stars into black hole, although forming of a black hole should take infinitely long time for an outside observer.
 A: Technically, the final phase of a compact binary coalescence, the ringdown, takes infinitely long. The ringdown is an exponentially decaying process, and a well known property of a decay exponential is that it never truly vanishes. More practically, the exponential decay is so quick that within a fraction of second it has become zero for all intents and purposes.
On a more general level I point out that the statement that "forming of a black hole should take infinitely long time for an outside observer" betrays a fundamental misunderstanding of one of the key features of general relativity. In general relativity does not feature a unique  notion of global time, and unlike special relativity it isn't even possible to assign a unique global time to the local frame of an individual observer. The more correct physical statement is that the formation of an event horizon does not happen in the causal past of any outside observer. (The astute reader will note that this statement is actually tautology.)
The past light cone of an observer, however, is not what we would usually call "now". E.g. A supernova observed millions of light years away, will typically  also be referred to as having happened millions of years ago. The possible choices of a "now" in general relativity are so-called spacelike hypersurfaces. For any observer outside of a black hole there will be many choice of hypersurface that will in fact intersect the event horizon. For these notions of "now" the black hole will have formed in a finite amount of time.
