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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