To find the equation of a rational function $f(x)$ that satisfies the given conditions, we need to use the provided information about limits, intercepts, and asymptotes. Here’s a step-by-step approach:

Conditions:

1. $\lim_{x \to \infty} f(x) = 0$: As $x \to \infty$, the function approaches 0, indicating a horizontal asymptote at $y = 0$.
2. $\lim_{x \to 0} f(x) = -\infty$: As $x \to 0$, the function goes to $-\infty$, implying a vertical asymptote at $x = 0$.
3. $f(5) = 0$: The function has a root at $x = 5$.
4. $\lim_{x \to 8^-} f(x) = \infty$ and $\lim_{x \to 8^+} f(x) = -\infty$: This indicates a vertical asymptote at $x = 8$, with the behavior differing on either side of 8.

Step-by-Step Analysis:

• Since there is a vertical asymptote at $x = 0$ and $x = 8$, the function will have terms of the form $\frac{1}{x}$ and $\frac{1}{x-8}$.
• The function has a root at $x = 5$, meaning $(x - 5)$ should be a factor in the numerator.

Given this information, we can propose a rational function of the form: $f(x) = \frac{(x - 5)}{x(x - 8)}$ This function satisfies the following:

• $f(5) = 0$, as required by the root at $x = 5$.
• As $x \to \infty$, the function tends to 0, which satisfies the horizontal asymptote.
• As $x \to 0$, the function goes to $-\infty$, and as $x \to 8^+$, the function goes to $-\infty$, showing the correct asymptotic behavior at $x = 0$ and $x = 8$.

Verification:

• For $\lim_{x \to 8^-} f(x) = \infty$ and $\lim_{x \to 8^+} f(x) = -\infty$, the vertical asymptote at $x = 8$ is satisfied with differing behavior on either side of the asymptote.

Thus, the rational function is: $f(x) = \frac{(x - 5)}{x(x - 8)}$

Would you like to explore more details about any of the conditions or the derivation of this function?

Here are 5 related questions to deepen your understanding:

1. What is the behavior of $f(x)$ near $x = 0$?
2. How does the function change as $x \to \infty$?
3. What other rational functions could satisfy similar asymptotic behaviors?
4. How does changing the degree of the numerator or denominator affect the function?
5. Can you graph $f(x)$ to visually verify its behavior?

Tip: When dealing with rational functions, always look at the behavior at infinity and near vertical asymptotes to guide you towards the correct form of the function.