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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 \( -\infty \).

If f has domain [0, ∞) and has no horizontal asymptote, then lim_{x→∞} f(x) = ∞ or lim_{x→∞} f(x) = −∞.

Explore the behavior of a function as x approaches infinity without a horizontal asymptote. This mathematical problem covers limits and horizontal asymptotes for students in Grades 10-12.

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