In[]:=
ClearAll["Global`*"]F[r_]:=r!+1G[v_]:=2v!+1(*Definec[r]tosearchexactly,butonlyinatightneighborhoodaroundr*)c[r_Integer]:=Min[Table[Abs[1-F[r]/G[v]],{v,Max[1,r-3],r+3}]](*Calculatethesequenceforthefirst20valuesofr*)(*WeuseN[]toviewthemasdecimalstoeasilyspotthelimit*)results=Table[N[c[r]],{r,1,20}](*Plottheresultstovisuallyseethelimit*)ListPlot[results,PlotRange->{0,1},AxesOrigin->{1,0},GridLines->{None,{0.5}},PlotLabel->"Sequence of c(r) as r increases"]
Out[]=
{0.333333,0.,0.4,0.489796,0.497925,0.499653,0.49995,0.499994,0.499999,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5,0.5}
Out[]=
