mass - How is light affected by gravity?

Light is clearly affected by gravity, just think about a black hole, but light supposedly has no mass and gravity only affects objects with mass.

On the other hand, if light does have mass then doesn't mass become infinitely larger the closer to the speed of light an object travels. So this would result in light have an infinite mass which is impossible.

Any explanations?

Answer

In general relativity, gravity affects anything with energy. While light doesn't have rest-mass, it still has energy --- and is thus affected by gravity.

If you think of gravity as a distortion in space-time (a la general relativity), it doesn't matter what the secondary object is. As long as it exists, gravity affects it.