# A new method for determining the limits of recursion relations

This is meant as a note. I will definitely expand this article later.

I hope this method has not been discovered before :(.

Say $x_n=f(x_{n-1})$ is a recursion relation. Then $\lim\limits_{n\to\infty} x_n$ is the intersection point of $y=f(x)$ and $x=f(y)$.

For example, if we have $x_n=\sqrt{5 x_{n-1}+6}$, then $\lim\limits_{n\to\infty} x_n$ is the y-coordinate of the intersection of $y=\sqrt{5x+6}$ and $x=\sqrt{6y+5}$.

Nice post, but it will be more illuminating if you can give an example where solving the simultaneous equations $x=f(y)$ and $y=f(x)$ is easier than solving $x=f(x)$