The symbol N is reserved. Use a different name. The following code gives an answer at once:

Subscript[\[Kappa]p, 1] = \[Kappa]1 + \[Mu]1/3;
Subscript[\[Kappa]p, 2] = \[Kappa]2 + \[Mu]2/3;
Subscript[\[Mu]p, 1] = \[Mu]1;
Subscript[\[Mu]p, 2] = \[Mu]2;
Subscript[c, 1] = 1 - Subscript[c, 2];
en = 2;
kp = (Sum[Subscript[c, r]/(
      Subscript[\[Kappa]p, r] + Subscript[\[Mu]p, 1]), {r, 1, 
       en}])^(-1) - Subscript[\[Mu]p, 1];
kp1 = Limit[
  kp, {\[Kappa]1 -> \[Infinity], \[Kappa]2 -> \[Infinity], \[Mu]1 -> 
    5, \[Mu]2 -> 30}, Assumptions -> Subscript[c, 2] > 1]