A step behind. During accretion, matter falls towards a central object. This matter heats up in the process, and this heat is freed in form of radiation. This radiation in turn "pushes" on the matter that keeps falling in, and a point is reached when the luminosity produced is comparable to the push by the infalling matter and so no higher luminosity can be achieved. This is called the Eddington luminosity and is a well known quantity in accretion studies. This luminosity scales with the mass of the central object, so that supermassive black holes (billions of times the mass of the Sun) will be able to radiate at much larger luminosities than stellar-mass black holes (mass several times the Sun)
Well, ULXs radiate at much more than the Eddington luminosity for stellar-mass black holes, so the first things that comes to mind is that they are bigger than stellar-mass black holes, and so they are members of the evasive class of Intermediate-mass black holes. Right?
Not so fast, my friend. People like these japanese researchers have done a great deal of simulations to show that it is possible to overcome the Eddington limit some extent.
Only problem: it was impossible to tell which of the two hypotheses was more right until 2012, since the few X-ray satellites capable of observing ULXs were sensitive only up to 10 keV, where the models started to really give incompatible predictions.
In this old post we talked about the launch of the NuSTAR satellite. Well, that was the turning point.
NuSTAR observed several ULXs and was able to finally strongly point in one direction. See the animated GIF above: the blue points are NuSTAR data, and they clearly follow better two models (the ones cutting off above 10 keV) than the others, while data from the XMM satellite weren't able to make a difference. This cutoff is considered a signature of super-Eddington accretion and permitted to estimate the mass of these ULXs to be in the high-end of the stellar-mass range.
Here is a press release of these studies. Enjoy!