In theory they can eventually die (they do emit a special kind of radiation called Hawking radiation, and eventually this would reduce them to nothing), but our current calculations say that it would take longer than the lifetime of the universe for this to happen!
Because of all radiation travelling at the speed of light, when we look at things that are really far away in the universe, we’re looking at them as they were very far back in time. If we look far enough, we can see galaxies from very near the beginning of the universe, and we can see they had black holes in them. So some have existed as long as galaxies have!
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