TITLE: Let $f(x)\triangleq\ln (1+x)$, and for all integer $n\ge 2$, let $f^n(x)\triangleq f(f^{n-1}(x))$. Calculate $\lim_{x\to\infty\\n\to\infty} f^n(x)=?$
QUESTION [5 upvotes]: Let $f\colon [0,\infty)\to [0,\infty)$ be defined by $f(x)\triangleq\ln (1+x)$. For all integer $n\ge 2$, let $f^n(x)\triangleq f(f^{n-1}(x))$, where $f^{1}(x)\triangleq f(x)$. How can we calculate the following limit?
\begin{equation}
\lim_{x\to\infty\\n\to\infty} f^n(x)=?
\end{equation}
Here is my attempt: Since $\ln x$ is much slower than $x$, I think this limit is 0 but I do not know how to calculate it.

REPLY [3 votes]: You have the right intuition. If there were a positive $a$ such that $f(a)=a$, then we would have $f^n(a)=a$, yielding the limit $a$ at the point $a$.
So it is important that $f(x)<x$ for all positive $x$: you should verify this claim by computing the derivative of $f(x)$ and comparing it to the derivative of $x$. (Note that the limit of both functions at $0$ is $0$.) If you show this, then you obtain that $f^n(x)$ is a strictly monotone decreasing sequence for any fixed $x$. So it has a limit for any fixed $x$. Show that this limit $a$ must satisfy $\ln(1+a)=a$ and $a\geq 0$, and conclude that $a$ must be 0.
Note that it is important to make the problem clear: this shows that $\lim\limits_{n\rightarrow \infty} f^n(x)=0$ for all $x$, and in particular, $\lim\limits_{x\rightarrow \infty}\lim\limits_{n\rightarrow \infty} f^n(x)=0$. If you change the limits, or you converge simultaneously by $x$ and $n$ in a weird way, then anything can happen.