Does infinity exist?

If it does, we'll never know. It would take infinitely long for light to reach us from the far edge of infinity, so we will never see it in the lifetime of the universe. And a black hole isnt infinitely dense. Whilst its so dense that not even light can escape from it, the mass is still finite, equivalent to a stellar mass many times that of our sun, compressed into an extremely small space. 

Infinity doesn't exist : black holes have enormous density , but less than infinity (horizon of events is > 0)

what difference does it make?  

Infinity is a concept and concepts exist.  To understand the answer to your question you will need to understand the mathematics of cosmology.  It will take you years to learn the language of physics.

They are learning more about black holes all the time.  There was an expulsion of light out of a black hole.  Now they have to explain that.  Check out Cosmos on YT

Infinity is the idea, not a number. Infinity cannot be reached. Everything in the universe has some limit, some finite number even if it has a very high order of magnitude. Black holes are not infinitely dense; they are dense enough that they surpass a threshold for light to escape its effects on space-time. In order to be infinitely dense, it would need to be either infinitely massive or infinitely small. They are made up of a finite amount of mass, and they take up a finite amount of space, so they cannot be infinitely dense.

Do you know what infinity means, pal. Yes something continues with out end that is infinity does space end? NO! It goes on and on and continues to grow so YES INFINITY DOES EXIST!

Infinity is a way to say "we have no idea what number would fit the situation, and whatever number we can guess is likely too small (a bigger value could exist and we can't know)"  So, clearly, infinity as an idea exists.  We cannot put a limit in quantity on the number value for a thing.

In practical terms, it is not obvious whether or not "infinity" truly exists.  We cannot know because the act of knowing would make a limit and thus make it not infinite.

Infinite density means that it is so dense that we cannot say how dense it is because whatever number we come up with, it will be smaller than what actually is, somewhere in there.

However, the problem might not actually be a matter if unquantifiable mass per unit volume, so much as we cannot quantify it.  That is not the same thing.  The density clearly is going to be something if you could only look and see.  But we have no idea what it might be, and we cannot put a limit to it based on what we know.

Our inability to put a number on it does not mean that a number cannot be put on it.  Our limitations do not govern reality.

Why do we call the value of Pi as infinity.

I know a girl named Infinity... I'll leave this answer right there... the rest of the question complicates matters.