The golden question

Calculus Level 2

Let \(\{u_n\}\) be a sequence satisfying the recurrence relation \(u_1=1\), \(u_2=2\) and \( u_n=u_{n-1} + u_{n-2}\) for \(n \ge 3\).

Prove that \(\displaystyle \lim_{n\to\infty} \frac{u_{n+1}}{u_n} \) exists. Calculate this limit.

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