Fibonacci's function

We note that $F(x) = F_{x-1}$ , where $F_n$ is the $n$ th Fibonacci number defined as $F_0 = 0$ , $F_1=1$ and $F_n=F_{n-1}+F_{n+2}$ for $n \ge 2$ . Then we have:

$\begin{aligned} L & = \lim_{x \to \infty} \frac {F(x+1)}{F(x)} \\ & = \lim_{x \to \infty} \frac {F_x}{F_{x-1}} & \small \color{#3D99F6} \text{Since }F_n = \frac {\varphi^n - (-\varphi)^{-n}}{\sqrt 5} \text{, where} \\ & = \lim_{x \to \infty} \frac {\varphi^x -(-\varphi)^{-x}}{\varphi^{x-1} -(-\varphi)^{-(x-1)}} & \small \color{#3D99F6} \varphi = \frac {1+\sqrt 5}2 \text{ is the golden ratio.} \\ & = \lim_{x \to \infty} \frac {\varphi -(-\varphi)^{-2x+1}}{1 -(-\varphi)^{-2(x-1)}} & \small \color{#3D99F6} \text{Divide up and down by }\varphi^{x-1}. \\ & = \varphi = \frac {1+\sqrt 5}2 \approx \boxed{1.618} \end{aligned}$

It converges to the golden ratio $\phi=\frac{1+\sqrt{5}}2$