I have found a bug in Limit. I have Mathematica 10.0.1.0 on Windows 8.1.
Limit[(n!*
LaguerreL[
n, -1]/(n^(n + 1/4)/Sqrt[2]/E^(n - 2 Sqrt[n] + 1/2)) -
1) Sqrt[n], n -> 1000] // N
Output: 0.649562
Limit[(n!*
LaguerreL[
n, -1]/(n^(n + 1/4)/Sqrt[2]/E^(n - 2 Sqrt[n] + 1/2)) -
1) Sqrt[n], n -> Infinity]
Output:13/16 = 0.8125