<s>[INST] Find $\lim_{n\to \infty}r_n$ and $\lim_{n\to \infty}s_n$ 
A sequence of polynomials is given by $$p_n(x) =a_{n+2}x^2 +a_{n+1}x-a_n$$ for
  $n\geq 0$, where $a_0=a_1=1$ and for $n\geq 0$ $$a_{n+2}=a_{n+1} +a_n$$ Denote by $r_n$ and $s_n$ the roots of $p_n(x)=0$, with $r_n\leq s_n$. Find $\lim_{n\to \infty}r_n$ and $\lim_{n\to \infty}s_n$.


This is a Fibonacci sequence because for $n\geq 0$,$a_{n+2}=a_{n+1} +a_n$ and $a_0=a_1=1$ is a definition of Fibonacci sequence.
It is also clear that $x=-1$ is a root of the equation $$p_n(x) =a_{n+2}x^2 +a_{n+1}x-a_n$$
So it is clear that $\lim_{n\to \infty}r_n=-1$. But I cannot find $\lim_{n\to \infty}s_n.$ 
What should I do? I am stuck here.
 [/INST] We can use sum of roots and product of roots.
For a quadratic equation $ax^2+bx+c=0$ with roots $s, t$, we have $s+t=-\frac{b}{a}$ and $st=\frac{c}{a}$.
Since we already have $1$ root, we can substitute it into the product of roots equation: $s_nr_n=\frac{-a_n}{a_{n+2}}$
Hence $s_n=\frac{a_n}{a_{n+2}}$.
Since Fibonacci numbers increase exponentially, i.e.
$$\lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}=\frac{1+\sqrt5}{2}$$
we have:
$$\lim_{n\rightarrow\infty}s_n=\lim_{n\rightarrow\infty}\frac{a_{n}}{a_{n+1}}\times\frac{a_{n+1}}{a_{n+2}}=\left(\frac{1+\sqrt5}{2}\right)^{-2}=\left(\frac{3+\sqrt5}{2}\right)^{-1}=\frac{2}{3+\sqrt5}\times\frac{3-\sqrt{5}}{3-\sqrt{5}}=\frac{2(3-\sqrt{5})}{4}=\frac{3-\sqrt{5}}{2}$$
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