A Continued Reciprocal Function

A function on $[1, +\infty)$ is defined by

Determine, with proof, which of the following are true:

1. $f(2024) = \dfrac{1}{2024}$; 2. for $x > 1$, $\dfrac{1}{x} \leqslant f(x) < \dfrac{1}{x-1}$; 3. if $f(x) \leqslant \dfrac{k}{x+1}$ for all $x$, then the minimum value of $k$ is $2$; 4. if $f(x) \geqslant a^x$ (with $a > 0$, $a \neq 1$) for all $x$, then the maximum value of $a$ is $\dfrac{\sqrt[3]{9}}{3}$.