The image presents a mathematical statement about the behavior of a function $f(x)$ with a domain $[0, \infty)$ and no horizontal asymptote. It concludes the following:

$\lim_{{x \to \infty}} f(x) = \infty \quad \text{or} \quad \lim_{{x \to \infty}} f(x) = -\infty.$

This suggests that since the function $f(x)$ has no horizontal asymptote, its limit as $x \to \infty$ must tend to either $\infty$ or $-\infty$, implying that $f(x)$ grows without bound in the positive or negative direction as $x \to \infty$.

Would you like further clarification or details on any part of this?

Here are five related questions you might consider:

1. What is the precise definition of a horizontal asymptote?
2. Can a function have no horizontal asymptote but still have bounded behavior?
3. How can the limit of a function at infinity be evaluated?
4. What kind of functions typically have behavior that leads to $\lim_{{x \to \infty}} f(x) = \infty$?
5. Could there be any other special cases where $\lim_{{x \to \infty}} f(x)$ exists, but the function lacks a horizontal asymptote?

Tip: When a function does not have a horizontal asymptote, it often means that the function's growth is unbounded, which is why the limit at infinity tends toward $\infty$ or $-\infty$.