Let $\displaystyle F(x)=x-\frac{f(x)}{g(x)}$, where $\displaystyle g(x)=\frac{f(x+f(x))-f(x)}{f(x)}$ and $\displaystyle f$ is smooth.

Suppose a sequence $\displaystyle \{x_n\}$ satisfies $\displaystyle x_{n+1}=F(x_n)$ and converges to $\displaystyle r$, where $\displaystyle r$ is a simple zero of $\displaystyle f$. We define "order of convergence" to be the largest real number $\displaystyle q$ such that

$\displaystyle \lim_{n\to\infty}\frac{|x_{n+1}-r|}{|x_n-r|^q}$

exists and is nonzero. Determine the order of convergence.