Can a black hole reach a limit where it can no longer attract more matter?

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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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.