IMC 2010 · Problem 3

Define the sequence $x_1, x_2, \dots$ inductively by $x_1 = \sqrt{5}$ and $x_{n+1} = x_n^2 - 2$ for each $n \ge 1$. Compute

$$\lim_{n \to \infty} \frac{x_1 \cdot x_2 \cdot x_3 \cdots x_n}{x_{n+1}}.$$