(E^-EulerGamma/EulerGamma) as limit to infinity

(*start*)
Clear[n];
(E^-EulerGamma/EulerGamma)
Limit[Exp[
   ProductLog[
    n]]/((Exp[HarmonicNumber[n]]*Log[HarmonicNumber[n]] -
     Exp[EulerGamma]*Exp[Log[n]]*Log[Log[n]])), n -> Infinity]
(*end*)


(*start*)
Clear[n];
Limit[(2*Pi*Exp[1]*
    Exp[ProductLog[(n - 11/8)/Exp[1]]])/(((HarmonicNumber[n] +
       Exp[HarmonicNumber[n]]*Log[HarmonicNumber[n]]) -
     Exp[EulerGamma]*(Log[n] + Exp[Log[n]]*Log[Log[n]]))),
 n -> Infinity]


(2 E^-EulerGamma \[Pi])/EulerGamma
(*end*)