Define $f: \mathbb{R} \rightarrow \mathbb{R}$ by $f(x)=\left(3 x^{2}+1\right) /\left(x^{2}+3\right)$. Let $f^{\circ 1}=f$, and let $f^{\circ n}=f^{\circ(n-1)} \circ f$ for all integers $n \geq 2$. Which of the following statements is correct?
- $\displaystyle{}\lim _{n \rightarrow \infty} f^{\circ n}(1 / 2)=1$, and $\displaystyle{}\lim _{n \rightarrow \infty} f^{\circ n}(2)=1$.
- $\displaystyle{}\lim _{n \rightarrow \infty} f^{\circ n}(1 / 2)=1$, but $\displaystyle{}\lim _{n \rightarrow \infty} f^{\circ n}(2)$ does not exist.
- $\displaystyle{}\lim _{n \rightarrow \infty} f^{\circ n}(1 / 2)$ does not exist, but $\displaystyle{}\lim _{n \rightarrow \infty} f^{\circ n}(2)=1$.
- Neither $\displaystyle{}\lim _{n \rightarrow \infty} f^{\circ n}(1 / 2)$ nor $\displaystyle{}\lim _{n \rightarrow \infty} f^{\circ n}(2)$ exists.
