Please consider entering the code using the editor-spikey, otherwise it gets tousled. The following code also produces myLim = 1. (Deleted my earlier notes, which did not apply to a sum to infinity).
(* iabraham, Mathematica 7.0.1.0, Win-7-64b *)
Clear["Global`*"];
myLim[lim_] := Limit[n n! (E - SetPrecision[Sum[(1/k!), {k, 0, n}], 10]), n -> lim]
