How can we find this limit $\lim_{n\to\infty \\x\to\infty}f^n(x)$? Let $f:\mathbb{R}\rightarrow \mathbb{R}$ , $f(x)=\frac{ax+b}{cx+d}$ and $a,b,c,d>0$ then  $f^1(x)=f(x), f^2(x)=f(f(x)), f^3(x)=f(f(f(x)))$ and $f^n(x)={f(f(f\cdots f(x)\cdots )))}$, where $ f^n(x)$ is the $ n $ composition of the function $f(x).$

It's required finding this limit:
$$\lim_{n\to\infty \\x\to\infty}f^n(x)$$

It is possible to find $f^n(x)$ for finite number $n$. For example, I found $f^2(x) = \frac{a^2 x + a b + b c x + b d}{a c x + b c + c d x + d^2}$. I can't do more than that. What is the math level of this problem? In fact, I wanted to solve. But I could not.
Thank you very much!

You could define the limits by recursion. If $p_n = \lim_{x\rightarrow \infty} f^n(x)$, then
$$ p_{n+1}=\frac{a \cdot p_n+b}{c\cdot p_n+d}$$
And $p_0=a/c$
There should exist some results on this kind of sequences.
