What happens when two black holes get too close?
That is a very interesting question and it happens to be on the fore-front of modern Astrophysics and General Relativity.
If two bodies (stars, black holes, anything) get close enough to be gravitationally bound they will develop some sort of orbit in which the two bodies circle each other.
Einstein's theory of General Relativity predicts that when two very massive objects(like Black Holes and Neutron Stars) are caught in an orbit like this they will generate Gravitational Waves (Ripples in Space-Time). These Gravitational Waves carry energy away from the orbiting bodies which means that with each successive revolution they get closer to each other. Eventually the orbit gets so small that the two bodies merge.
The merger of two black holes entails a lot of complicated General Relativity and some non-intuitive physics, but theorists use large supercomputers to simulate such events. There are two primary results to these mergers of which we can be sure of:
1. A single, more massive black hole
2. An extremely powerful burst of gravitational waves
There are currently projects like LIGO which will attempt to directly detect the disturbances in space time caused by gravitational waves. They primarily expect to "see" the results of neutron star and black hole binary mergers.
There was also a recent discovery of a black hole merger in a distant galaxy though this was discovered by observing the jet streams caused by the black holes. You should note that those results are still preliminary and unconfirmed.
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