Mass does NOT become infinite, and it does not increase.

I know, this answer is long, but I think it's everything you need to know.

Relativistic Mass is Born

In 1920, Max Born sends his manuscript for a new textbook to his publisher, and in it is the first textbook use of the term "relativistic mass." (Incidentally, he calls "rest mass" the "statical mass"). The mathematical derivation had been done earlier by Lewis (1908) and developed extensively by Tolman (1912). No one knows who actually said "relativistic mass" first so its origin is often left uncredited.

Anyway, we can see where this comes from if we start with the expression for relativistic momentum:

The momentum is expressed in 3 variables and using the rule of associativity we are free to group the variables how we choose:

This new mass, m', is the relativistic mass. Its most common mathematical expression is written as:

This expression is not without its merits. It shortens the equation for momentum, cleaning up the equations a bit, and makes for a simple reminder of how objects behave at relativistic speeds.

Spacetime Becomes Space-propertime

It is not inevitable that we would or should lump together the lorentz factor with the mass, and we are certainly within our mathematical rights to associate the lorentz factor with the velocity.

We can in fact build a whole space-propertime version of relativity and this has been done. This program was primarily led by Brehme and Loedel. There is an excellent overview of both their ideas in Special Relativity by Shadowitz (1968).

Side Note: Space-propertime diagrams are incredibly intuitive and powerful tools that can make quick work of lorentz transforms. There is an interesting, though quirky, treatment of this by Lew Epstein called "Relativity Visualized" that is excellent for the novice. With that I recommend a mathematical follow up by David Eckstein (you could even start here): Epstein explains Einstein: Read the book online

More space-propertime diagrams: UM-StL Motion-at-Any-Speed

Takeaway Point: There is absolutely no a priori reason to group the lorentz factor with the mass.

The War of the Masses

Starting in the late 1980's and raging through the 90's came the debate on what should be done with word "mass." There were those who followed Born and Pauli and decided relativistic mass should be kept, while others followed Einstein, who didn't like the idea. The decade of the 90's spawned numerous articles in both journals and online with heavy hitters coming down on both sides of the fence.

Here's a link with 30 of such arguments
http://hep.fcfm.buap.mx/cursos/2010/IFC/EnergyMass-RelativityTheory-Okun.pdf

NOTE: It was NEVER an issue if relativistic mass is right or wrong, only if it is useful, that is, if the idea did more harm than good.

It's Over - Mass is back to simplicity

Nearly a century after Einstein's 1905 papers on relativity, we are back to the beginning and mass is just mass: the sum of all the energies that make up an object. I won't elaborate on the meaning of mass, having done so elsewhere: Spacetime Traveler's answer to What is mass?

Mass is, as it should be, the single conceptually and mathematically concise relativistic invariant mass (where c = 1) and by definition:

[math]m:= \left \| \mathbf{p} \right \|=\sqrt{-\eta _{\mu \nu}p^\mu p^\nu}=\sqrt{E^2-\left |\textbf{p} \right |^2}[/math]

The relativistic mass, while handy for some, really screwed up a lot of students and lay people trying to understand relativity. If for nothing else, we will one day soon no longer have to endure these typical types of questions:

"If the mass increases as the object speeds up, where do the extra atoms come from?"

"As an object speeds up, is the extra mass coming from the zero-point energy of the universe?"

"How fast would a block of lead have to move until its mass turned into a black hole?"

etc, etc, etc, etc, etc, etc, etc, etc, etc, etc, etc, etc,...

Takeaway Point: Nothing happens to the object - forget relativistic mass.

Conclusion: Some Underlying Physics

With the expulsion of relativistic mass it might seem as though relativistic momentum is a concept in want of an explanation.

Here's how to think about it: Two Methods:
The most obvious is that energy possesses inertia, and the inertia over and above the mass is coming from the kinetic energy of the object.

Another way is to think of it in terms of time dilation. The explicit derivation requires the full lorentz transformations and is algebraically very messy, but to a first approximation this gives a nice intuitive explanation for what is happening, and this is all there is to it...

As an object moves relative to us, its clocks run slow compared to ours. If a force is then applied to the moving object then time dilation requires that the observed acceleration takes longer to happen in our frame. This reduced acceleration is what creates the illusion of increasing mass.

Yes, it's that simple.