A limit regarding Fibonacci and Phi

It is known that $f_n = \frac{1}{\sqrt{5}} \left( \left(\frac{1+\sqrt{5}}{2} \right)^n -\left(\frac{1-\sqrt{5}}{2} \right)^n\right)$ and that $\phi = \frac{1+\sqrt{5}}{2}$ .

Now $\displaystyle L=\lim_{n \to \infty} \dfrac{f_n}{\phi^n} = \lim_{n \to \infty} \dfrac{\frac{1}{\sqrt{5}} \left( \left(\frac{1+\sqrt{5}}{2} \right)^n -\left(\frac{1-\sqrt{5}}{2} \right)^n\right)}{\left(\frac{1+\sqrt{5}}{2}\right)^n} = \lim_{n \to \infty} \frac{1}{\sqrt{5}} \left(1- \left( \frac{1-\sqrt{5}}{1+\sqrt{5}}\right)^n \right) = \frac{1}{\sqrt{5}} \approx 0.4472$ .