Limit of a sequence given by $x_n = \sqrt{3 + x_{n-1}}$ as n approaches infinity

Question

Let $x_1 = 1$ and for $n \ge2, x_n = \sqrt{3 + x_{n-1}}$, I just want to know how to determine the limit.

Answer 1

Hint: prove that your sequence is increasing and bounded, hence converging to its supremum, then notice that there is only one real number $r$ such that $r=\sqrt{r+3}$.

Answer 2

Hint: If you know the Banach fixed-point theorem, then $f(x)=\sqrt{3+x}$ has Lipschitz constant $1/4$ on $[1,\infty)$.