Lucas meets Fibonacci

Calculus Level 3

The \(n\)th Lucas number is defined such that \(L_n = L_{n-1} + L_{n-2}\), where \(L_1 = 1\) and \(L_2 = 3.\)
Similarly, the \(n\)th Fibonacci number is defined such that \(F_n = F_{n-1} + F_{n-2}\), where \(F_1 = 1\) and \(F_2 = 1.\)
Let \(Q_n\) be the ratio \(\frac{L_n}{F_n}.\) With simple calculation we find that \(Q_1 = \frac{1}{1} = 1,\ Q_2 = \frac{3}{1} = 3, ...\)

Find \[\lim_{n\to\infty} Q_n.\]

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