It can eat matter indefinitely.
Large masses, like a star coming in at once will likely swirl around the black hole creating a very high energy disk and some of that plasma may be launched away from the black hole in a jet instead of falling in. I don’t know exactly how it works but there is good evidence of super massive black holes spewing out giant jets of gas so in that sense there may be a sort of limit to how much gas a black hole can absorb at once without throwing a bunch of it out into space in the process.
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