A problem on my dynamical-systems course assignment reads:
Consider the usual exponential function $\exp$ : $\mathbb{R}\rightarrow\mathbb{R}$ as a discrete time dynamical system. Find the asymptotic behavior of each orbit.
I immediately ruled out the possibility of a fixed point. Then I determined the intervals of the iterations of $\exp(x)$ (for $x_0\in\mathbb{R}$):
$$ \exp (x_0) \in (0,\infty),\\ \exp^2(x_0) \in (1,\infty),\\ \exp^3(x_0) \in (e,\infty),\\ ...,\\ \exp^n(x_0) \in (e^{e^{e^{...}}},\infty).$$
But the lower bounds increase infinitely as $n\rightarrow\infty$. I also noticed that the horizontal asymptotes, as $x\rightarrow-\infty$, of each iteration is equal to the lower bound of the interval (as follows with the exponential function).
Maybe this is a question of clarity, but by "asymptotic behavior," is the problem simply asking to identify that the horizontal asymptotes rise with each iteration? Or is it referring to asymptotic stability? Furthermore, could this system exhibit asymptotic stability if (as I assume) there exist no periodic orbits?
Your interval approach is already slightly overkill, if you ask me. My interpretation of the question would be to investigate how the dynamics behaves for $t→∞$ for each initial condition, i.e., does the state approach infinity, some fixed point, is there a periodic behaviour, or something else. This question is relatively easy to answer if you consider the monotonicity and lower bounds of the states over time.