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A star will ultimately become a black hole when it collapses due to end of fuel.
When a star burns through the last of its fuel, the object may collapse, or fall into itself. For smaller stars which are about three times the sun's mass, the new core will become a white dwarf but when a larger star collapses, it turns into a black hole.
So we can can conclude that a star will ultimately become a black hole when it collapses due to end of fuel.
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