Let $x_{n+1} = x_n^2 - 6x_n + 10$. For what values of $x_0$ is $\{x_n\}$ convergent, and how does the value of the limit depend on $x_0$?
I know that if this recursion does converge, then it must converge to $2$ or $5$. I'm also pretty confident that $x_0 < 0$ and $x_0 > 5$ doesn't converge to these values. However, I'm having trouble making much progress past that. Some hints would be greatly apprecated!