<s>[INST] Analysis - Find the Limit Let $P(x) = ax^3 + bx^2 + cx + d$, $a, b, c, d \in \mathbb{R}$, determine whether the sequence
$$
\left(
\frac{P(n)}
{P(n + 1)}
\right)
$$
it is convergent or not. If so, calculate the limit.
I managed to show that the sequence
$$\frac{an^3 + bn^2 + cn + d}{a(n+1)^3 + b(n+1)^2 + c(n+1) + d}$$
is limited $(0<x_n<1)$, but I can't prove that your limit is 1 by the definition.
 [/INST] We can use that, for $a\neq 0$
$$\frac{an^3 + bn^2 + cn + d}{a(n+1)^3 + b(n+1)^2 + c(n+1) + d}=\frac{a + b\frac1{n} + c\frac1{n^2} + d\frac1{n^3}}{a\left(1+\frac1n\right)^3 + \frac b n\left(1+\frac1n\right)^2 + \frac c {n^2}\left(1+\frac1n\right) + \frac d{n^3}} \to 1$$
For $a=0$ we can divide by $n^2$ and so on.
The assuming $a\neq 0$ by the definition we need to show that
$$\left|\frac{an^3 + bn^2 + cn + d}{a(n+1)^3 + b(n+1)^2 + c(n+1) + d}-1\right| \le \epsilon$$
$$\left|\frac{a(3n^2+3n+1) + b(2n+1) + c }{a(n+1)^3 + b(n+1)^2 + c(n+1) + d}\right| \le \epsilon$$
</s>