Clear[f];
rsol = RSolve[{f[n] == f[n - 1] + f[n - 2], f[0] == 0, f[1] == 1}, f, n]
(*  {{f -> Function[{n}, Fibonacci[n]]}}  *)

Limit[f[n]/f[n - 1] /. First[rsol], n -> Infinity]
(*  1/2 (1 + Sqrt[5])  *)