Answer
See explanation
Work Step by Step
We want to compute the limit
\[
L \;=\; \lim_{n \to \infty}\,\frac{F_{n+1}}{F_n},
\]
assuming it exists. Recall that \(\{F_n\}\) is the Fibonacci sequence defined by
\[
F_0=0,\quad F_1=1,\quad \text{and for }n\ge2:\; F_n=F_{n-1}+F_{n-2}.
\]
---
## 1. Use the Fibonacci Recurrence
If the limit \(L\) exists, then for large \(n\),
\[
\frac{F_{n+1}}{F_n} \;\approx\; \frac{F_n + F_{n-1}}{F_n}
\;=\; 1 + \frac{F_{n-1}}{F_n}.
\]
Taking limits on both sides, we would have
\[
L \;=\; 1 + \frac{1}{L},
\]
because \(\lim_{n \to \infty} \frac{F_{n-1}}{F_n} = \frac{1}{L}\) if \(L\) exists.
---
## 2. Solve the Resulting Equation
We get the algebraic equation
\[
L = 1 + \frac{1}{L},
\]
which is the same as
\[
L^2 = L + 1.
\]
This is a standard quadratic equation \(L^2 - L - 1=0\). The solutions are
\[
L = \frac{1 \pm \sqrt{5}}{2}.
\]
Since the ratio \(\tfrac{F_{n+1}}{F_n}\) is positive for all \(n\), we must take the positive root:
\[
L = \frac{1 + \sqrt{5}}{2}.
\]
This number is commonly denoted by \(\phi\), the **golden ratio**, approximately \(1.618\ldots\).
---
## 3. Conclusion
Hence,
\[
\boxed{\lim_{n \to \infty} \frac{F_{n+1}}{F_n}
\;=\; \frac{1 + \sqrt{5}}{2}.
}
\]
