Why is the Speed of Light Unattainable for Massive Objects?

A question that is directly related to this is, "why do objects increase in mass as they get faster?" People often talk about how an object's mass increases as its acceleration increases; however, I want to avoid talking about the increases in mass altogether, as it makes for a clearer explanation of why objects with mass can't reach light speed.

Now then, historically (before Einstein), people had already realized that electrons behaved quite peculiar when they approached the speed of light. They appeared to have much greater "mass" than at other times. Worse, if they were going fast in one direction and you tried to make it go faster, it would have a different "mass" than when you tried to change its direction of motion (i.e., it had a different apparent mass depending on whether you were trying to push it parallel to its direction of motion or perpendicular). It was crazy.

Then Einstein explained it all with Special Theory of Relativity (SR); however, some unfortunate (and rather erroneous) historical ideas stuck around. In short, contemporary physicists agree that relativistic mass (the assertion that mass changes as acceleration changes) should no longer be an accepted statement. Why? Let me explain...

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