Could the Limit be preformed in pieces? (Mathematicians might correct this.)
Here I take the innermost section, take its limit as s-> Infinity, then stick in in
in place of the original counterpart.
In[2]:= Limit[-(2 Gamma[m, ((m*(\[Alpha]/s)^(1/n))/\[CapitalOmega])]),
s -> Infinity,
Assumptions -> {n > 0, m > 0 , 0 < \[Alpha] < s,
M > 0, \[CapitalOmega] > 0}]
Out[2]= -2 Gamma[m]
In[3]:= d=Limit[-(Log[(2-((2 Gamma[m])/Gamma[m]))^M])/Log[s], s->Infinity,
Assumptions->{n>0, m>0 , 0<\[Alpha] <s, \[CapitalOmega]>0}]
Out[3]= Indeterminate
It appears that one ends up with
Log[0] / Log[Infinity]
