$x_{n} = \sqrt{1+\dfrac{3}{x_{n-1}}}$ with initial condition $x_1 = 2$.
Prove that $\lim_{n\to\infty}{x_n}$ exist.
I've calculated the first few values and it seems to be an oscillating series with tending towards $L$, where $L$ is the real root to the cubic $x^3 - x - 3 = 0$.
I've tried expressing the sequence as two monotone sub-sequences that converges to the same limit, but to no avail. Would appreciate any help! Thank you!