No, the speed of light does not increase beyond c in the vicinity of black holes, that is impossible, because it would violate the general theory of relativity, however, we don't know what happens after something enters a black hole because light cannot escape from one, it is infinitely red-shifted (i.e. we can't probe a black hole beyond Schwarzschild's radius because that would require passing through it, which would lead the person which passed beyond the horizon to being literally crushed out of existence).
According to Penrose's theorem every black hole with an event horizon contains a singularity, around which the laws of physics are supposed to break down (according to some physicists). Singularities are the reason why Einstein introduced the cosmological constant into his equations, something that he later called "the biggest blunder of his life", since it turns out that singularities are a part of spacetime, and not just flaws in the mathematics used to describe spacetime, which was what Einstein thought.
An answer to the original question is that the speed of light is not infinite, it is definitely finite. Additionally it's impossible to reach the speed of light since that would require an infinite amount of energy because, as you approach the speed of light, your mass would start to increase. The closer an object gets to the speed of light the more mass it has which, in turn, requires that to get that additional increase in speed you would need even more energy than before (i.e. as an object approaches c, the object's mass approaches infinity and to accelerate an object with an infinite mass you need infinite energy).
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You are wrong. According to Stephen Hawking's theories, certain paticles (I forget which) can travel faster than the speed of light at the event horizon of a black hole. As was said earlier by Elledan we don't know for sure around black holes. >>
Amish, stop pulling false info out of your a$$... quote the theorem
Maybe you're confusing this with Hawking radiation, but the effect is not due to matter traveling faster than light, it's due to quantum mechanics and relativity.