How to show that the limit of a fibonacci sequence equates to 1 as n goes to infinity

Question

$$\lim_{n \to \infty} \frac{f_{n+1} f_{n-1}}{f_n^2} = 1$$

I tried expanding both the numerator and denominator to probably cancel out but that did not work... I also split it up into different fractions.

Answer 1

Take Cassini's identity: $$ f_{n-1}f_{n+1} - f_n^2 = (-1)^n $$ divide by $f_n^2$, and use that $f_n \to \infty$ as $n \to \infty$.

Answer 2

Hint: when $n\to\infty$, $$f_n \sim \frac 1{\sqrt 5}\phi^n$$ where $\phi$ is the golden ratio.