Can gravity cause an object to reach the speed of light?

[Peter Frame]

We all know that nothing can go faster than light. This is because as something approaches the speed of light, its mass will approach infinity.
What if something was falling towards a immense body very very far away. Of course the object will accelerate and gain velocity and get close to the speed of light. Its mass would increase. Gravity increases with mass so why would this object not be able to reach the speed of light?

 

[ShayanJ]

We all know that nothing can go faster than light. This is because as something approaches the speed of light, its mass will approach infinity.
What if something was falling towards a immense body very very far away. Of course the object will accelerate and gain velocity and get close to the speed of light. Its mass would increase. Gravity increases with mass so why would this object not be able to reach the speed of light?

This is based on some misconceptions about several things and is wrong.
At first, mass increase is not that much of a real effect. Its, like time dilation and length contraction, something that happens relatively. I mean, A sees B in motion and so sees his mass increasing and B sees A in motion and so sees his mass increasing. A and B notice nothing about themselves because they both think they're at rest.
Second, people don't use relativistic mass and mass increase nowadays. They're useless troublesome notions. Its just that the formula for energy and momentum depend on velocity and are frame-dependent, as they were in Newtonian mechanics. So the last paragraph should be said about energy and momentum.
Third, You're talking about gravity here and so its out of SR. SR is only a special case of GR for a space-time which is empty of large amounts of mass and energy concentrated in such a small region to cause considerable gravity.
Fourth, in GR the effect of gravity on objects, doesn't depend on their mass. Imagine a hose hanging from top of a building. We can throw little balls inside it and they go along the hose until they reach the ground. Now if I bend the hose, all the balls' paths takes the shape of the hose no matter what is their mass or size. Its the same with gravity. Gravity is the change in the shape of space-time and matter has to follow that shape when moving (in space and time) and it really doesn't matter what is its the mass!

 

[Ibix]

Current thinking is that the idea of "mass increasing as you get closer to the speed of light" is a bad one to use. It used to be quite a popular view, but has fallen out of favour, since it often leads to misconceptions like yours. We used to talk about "rest mass", which is an invariant quantity, and "relativistic mass", which grew as the object accelerated. "Mass" might mean either, depending on context. In modern terminology, "mass" or "rest mass" are used interchangeably to refer to the invariant quantity, and "relativistic mass" isn't used. I suggest you

In any case, the "strength" of gravity depends on the rest mass, which doesn't change when the object accelerates. One simple way to see that this must be true is that there isn't an absolute concept of speed in relativity. If you aren't accelerating, you can consider yourself at rest, and if I'm moving relative to you, you say I'm moving. But I can consider myself at rest and you as moving, as long as I'm not accelerating myself (this is the point of the "what time does Oxford stop at this train?" quote often attributed to Einstein). If gravity depends on relativistic mass, then I see myself as normal and you moving with a large relativistic mass, so things ought to fall towards you. You see yourself as normal and me moving with a large relativistic mass, so things ought to fall towards me. Those two views aren't compatible, and there's no way to make them compatible except to drop the idea of "relativistic mass causing gravity".

Gravity in General Relativity is a very complex beast. Energy of motion does, actually, feed into it, but not in the simple "make it stronger" way you are thinking.