Limit of a Mobius-type recursion

Let $f:\mathbb R\setminus\mathbb Q\to\mathbb R\setminus\mathbb Q$ be the function defined as

$$f(x)=\frac{3x+2}{4x+3}.$$

Let $x_1\in\mathbb R\setminus\mathbb Q$. For $n\ge1$, define $x_{n+1}=f(x_n)$. Suppose that the sequence $(x_n)_{n\ge1}$ converges to a real number $\ell$. Which of the following statements is true?