Defining a function with certain regularity (e.g. smooth)

My goal is to understand how Mathematica computes limits that would require differentiability assumptions on functions (for, for instance, L'Hopital or Taylor Series). Consider, for instance, the following limit:

Limit[ n * (F[x - 1/Sqrt[n]] + F[x + 1/Sqrt[n]] - 2 F[x]),
n -> Infinity]


If F is a smooth (or just a C^2) function, this should converge to F''(x) by Taylor expansion. Is it possible to define assumptions on F[x] as a smooth function, so that Mathematica would be able to evaluate the above limit?