If time stops inside a black hole, due to gravitational time dilation, how can it's life end after a very long time? If time doesn't pass inside a black hole, then an event to occur inside a black hole needs infinite time relative to the outside. Thus, it will never age.
Please keep it simple..
Answer
I assume you're asking how a black hole can evaporate due to Hawking radiation. The answer is that the Hawking radiation does not come from the event horizon, but instead comes from a region just outside the event horizon so time has not stopped at its position.
If you were to watch a black hole form then evaporate, you would never see an event horizon form. That's because in your co-ordinates the event horizon would take an infinite time to form. You would see the infalling matter slow and red shift, then be re-emitted as Hawking radiation without the event horizon ever having formed.
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