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Can a black hole reach a limit where it can no longer attract more matter? If so, what happens with that black hole? Does the black hole die? Does the black hole decrease its size?

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There is no known upper limit for the mass of a black hole. A black hole stops attracting matter, when there is no matter within reach. In this case the (macroscopic) black hole is assumed to evaporate by Hawking radiation over very long time scales.

The estimates for the currently (as of early 2014) known most massive black holes (ultramassive black holes) are below about 40 billion ($4\cdot 10^{10}$) solar masses. A black hole of that mass would have a Schwarzschild radius of about 780 times the distance Earth-Sun, or about 4.5 light-days.

(Hypothetical) micro black holes are assumed to be able to evaporate faster than they can accrete matter.

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Just to add something to your answer: there is no theoretical upper limit, but they claim for an observative one arxiv.org/abs/0808.2813 It seems SuperMAssive Black Holes self-regulate to avoid to swallow/disrupt their host galaxy. According to the paper the limit is 50billion solar masses. –  Py-ser Apr 1 at 1:20

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