I recently learned that relativistic mass isn't actually real or popular and that mass is actually invariant and not dependent on velocity. My entire sense of why we couldn't travel faster than light was because I thought that the faster we got the more massive we became. Now I'm lost and trying to understand why we can't travel faster than light.
$E = \gamma m_0 c^2,$
regardless. $m_0$ is the rest mass and $\gamma$ is the Lorentz factor. So, in the early days, people said let's have a relativistic mass:
$$ m = \gamma m_0, $$
so as to preserve the legendary:
$$ E= mc^2$$
at all velocities (an other formulae), not just at $v=0$. That's less attractive than working with invariants of the Lorentz transformation, such as:
$$ E^2-(cp)^2 = (m_0c^2)^2 $$.
At this point, it's safe to drop the $0$-subscript and consider mass to always be rest mass.
In summary, the difference is where you put the $\gamma$: in the so-called relativistic mass, or explicitly in the formula for total energy. Either way, total energy diverges at $v=c$.