If nothing can travel faster than the speed of light, then what does light do as it approaches a black hole?

light travels at same speed. just that the space is bent in such a way that moving ahead in time is equivalent to moving into the black hole. there is no other choice. Bending of space ime is commonly perceived as gravitation.

well if light went faster then it's own speed which is supposeably impossiable, it is in theory suppose to completely stop, meaning that all time would freeze for that particular object moving faster than the speeed up light, good question, and i really dont know the answer, but maybe the light moves at a constant speed, but idk

One concept here is to be corrected
the speed of light is not constant.  .  .  . ( my concept only)
proof:
-light traveling in a spectrum is deflected in a different direction according to the color.
- light traveling in different medium will have different velocities according to the medium
- light, space and time are warped according to the theory of relativity
- there is a theory about the nature of light. Light is composed of a very fine particle, wave particle or energy particle

So, true to what you know.
black hole will absorb all, including the light.
not even the principle that light is the fasted velocity ever
velocity is not a force.

The light is destroyed in away. The light I think will speed up because the Black Hole would count as a vacuum perhaps? E=MC^2 Energy= Mass x Speed Of Light in a Vacuum ^ 2

The speed of light appears the same no matter how intense the local gravitational field is.  Light traveling near or into a black hole will bend and appear to bend, but it will not slow down as it does.  However, light that comes out from very near a black hole will be noticeably red-shifted.

Light never changes the speed. Not with gravity neither in electromagnetic fields, nor in event horizon of a black hole. It means photon speed remains constant.
Then, why it cannot escape from a black hole. Because black hole due to its mass, bends space-time around itself extraordinarily where that part of the space-time curls up on itself and create a kind of trap. Light which moves along the space-time is bended through the center of that mass and never able to turn back or another direction to move away from that trap.

I know your next question. Now, if photons never change their speed, how come Light travels slower when it enters to a dense opaque medium?
There they talk about two types of velocity. One is the velocity of individual photons, which never ever changes. The second is the group velocity of light, can change when it travels in denser mediums. Light is scattered, absorbed, re-emitted by the surrounding blocking atoms and it takes longer to travel certain length compare to its travel in empty space. Meaning its group velocity is calculated less, but individually photons run at same velocity, speed of light.

it goes around in circles around the black holes event horizon, at the speed of light.

As the light approaches a black hole, it get's caught by it, getting trapped by the singularity, a point of zero volume and infinite mass and density. According to Newton's laws, the light will need an infinite effort to escape, effort that isn't reached by the light's capabilities. So the light remains in that single point called singularity.

think of a car traveling past you beeping it's horn, the horn sounds high pitched when it approaches you and low pitched when it moves away, this is because the speed of sound is constant in air, if you try to throw sound you essentially squash it like a spring which shortens it's wavelength so it will sound higher pitched.
the speed of light is constant in vacuum just like sound in air, when the gravity of the black hole tries to accelerate it, instead of getting faster it's wavelength will get shorter so the light will get bluer, turn into UV then X-ray then gamma ray and it's wavelength will just keep getting shorter and shorter until it hits the center.
This is called the 'doppler effect' and it occurs whenever you try to accelerate a wave.

the end
.

It follows the curved space at the event horizon and orbits the black hole at that point forever.