Derivation of the Maximum Mass of a quantum particle:

This is a representation of the mass of an electron (we use the Compton wavelength of the electron).  Any particle mass can be used as long as we use its associated Compton wavelength.  We can take this diagram to the extreme and calculate the maximum possible mass of any quantum particle.
a. Take the Compton wavelength to its limit of the Planck length = (hG/c3)0.5 .
b. The quantum of energy in this shortest of Compton wavelengths is E = hc/λ
where λ is now the Planck length.

c. The energy in the mass of this short Compton wavelength is given by E = mc2
and we can equate the quantum energy and the mass energy and get:
hc/λ = mc
2 .

d. Solving for m we get that m = (hc/G)0.5. This by definition is the Planck mass.

We come to the conclusion that the maximum mass of a quantum particle as its velocity increases is the Planck mass not infinite mass. The reason for this is that we cannot make a Compton wavelength shorter than the Planck length. For more on this see the essay: An Elephant in the Room.