The condition that guarantees that the line $y = L$ is a horizontal asymptote of the graph of the function $y = f(x)$ is:

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

Explanation:

• A horizontal asymptote occurs when the value of the function approaches a certain constant $L$ as $x$ goes to infinity or negative infinity. This means that as $x$ increases or decreases without bound, $f(x)$ gets closer and closer to $L$, even though it might never reach it.
• So, the condition $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$ indicates that as $x$ becomes very large (positive or negative), the function approaches the value $L$, making $y = L$ a horizontal asymptote.

Why the other options are incorrect:

1. $\lim_{x \to L} f(x) = \infty$ or $\lim_{x \to L} f(x) = -\infty$:
   This condition implies the function has a vertical asymptote at $x = L$, not a horizontal one. It describes the function's behavior near a specific value of $x$, rather than as $x$ goes to infinity or negative infinity.

2. The line $y = L$ never intersects the graph of the function $y = f(x)$:
   This condition does not necessarily imply a horizontal asymptote. Some functions with horizontal asymptotes might intersect the asymptote at some points before tending toward it as $x \to \infty$ or $x \to -\infty$.

3. The line $y = L$ intersects the graph of the function $y = f(x)$ infinitely many times:
   If a line intersects the graph of the function infinitely many times, it cannot be a horizontal asymptote. A horizontal asymptote suggests that the function approaches the line but does not cross it infinitely many times as $x \to \infty$ or $x \to -\infty$.

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Here are five related questions to expand on this:

1. How can we distinguish between horizontal and vertical asymptotes graphically?
2. Can a function have more than one horizontal asymptote? If so, how?
3. How does the behavior of rational functions relate to horizontal asymptotes?
4. What is the difference between horizontal asymptotes and oblique asymptotes?
5. How do limits at infinity help in finding asymptotes?

Tip: When analyzing limits for asymptotes, always pay attention to whether $x \to \infty$ or $x \to c$ (some finite value) to determine if it's a horizontal or vertical asymptote.