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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.

Find an equation for f that satisfies the given limits and conditions for the rational function.

Explore how to find the rational function f(x) that satisfies specific limits and conditions, including vertical asymptotes, horizontal asymptotes, and a root at x = 5. Learn about rational function behavior and limit analysis. Suitable for advanced high school or university-level students.

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