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?