$\newcommand{\N}{\mathbb{N}}$ $\newcommand{\F}{\mathbb{F}}$ $\newcommand{\coloneqq}{:=}$
If it exists find the value of: $$1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}$$ i.e. the limit of the sequence $(x_{n})_{n \in \N}$ with: $$x_{1} \coloneqq 1 \hspace{0.5cm}\text{and}\hspace{0.5cm} x_{n+1} \coloneqq 1 + \frac{1}{x_{n}}$$
May you help with hints to begin?
Hint: If the sequence has a limit, then it must hold that $$ \lim_{n \to \infty} x_{n+1} = \lim_{n \to \infty}\left(1 + \frac 1{x_n} \right) = 1 + \frac{1}{\lim_{n \to \infty} x_n}. $$