Fibonacci-numbers/Fraction/Convergence and limit/Exercise

Let ${}{\left(f_{n}\right)}_{n\in \mathbb {N} }$ be the sequence of the Fibonacci numbers and

${}x_{n}:={\frac {f_{n}}{f_{n-1}}}\,.$

Prove that this sequence converges in ${}\mathbb {R}$ and that its limit ${}x$ satisfies the relation

${}x=1+x^{-1}\,.$

Calculate this ${}x$.

Create a solution