Question regarding maximum accretion of black holes

[can't.do.phys]

So I've got a question that I can't seem to figure out.

If the accretion efficiency of a black hole is L=εM(dot)c² with ε=0.1
I have to set this to the Eddington luminosity which is L_edd = 4πcGMm_p/σ_T to find the maximum rate at which a black hole can accrete by. I found that κ=m_p/σ_T so I substituted that but then I got stuck. Has it got something to do with the accretion disk being optically thin? the answer should be M(dot)=κM but i can't figure out how to get it. help?

 

[Chronos]

Try not assuming there is a maximum accretion rate.