Taking the limit of a defined function

Question

The Fibonacci function I have constructed is:

f[n_] := f[n] = f[n - 1] + f[n - 2]

f[0] = 0; f[1] = 1;

\[ScriptF][n_] := \[ScriptF][n] = f[n] + f[n - 1]

Is there a way to take the limit of the ratios of

\[ScriptF][n_] := \[ScriptF][n] = f[n] + f[n - 1] / f[n_] := f[n] = f[n - 1] + f[n - 2]

as n tends to infinity?

Any help is greatly appreciated.

Answer 1

RSolve andLimit will do the equivalent:

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])  *)

Answer 2

Or use the built in function Fibonnaci:

Limit[Fibonacci[n + 1]/Fibonacci[n], n -> Infinity]
(*  1/2 (1 + Sqrt[5])  *)