https://file.aiquickdraw.com/mathbot/file/1725920319181tktxirwi.jpg

The problem asks to find a rational function \( f(x) \) that satisfies the following conditions:

1. \( \lim_{x \to +\infty} f(x) = 0 \)
2. \( \lim_{x \to 0} f(x) = -\infty \)
3. \( f(5) = 0 \)
4. \( \lim_{x \to 8^-} f(x) = \infty \)
5. \( \lim_{x \to 8^+} f(x) = -\infty \)

### Step-by-step analysis of the conditions:
1. \( \lim_{x \to +\infty} f(x) = 0 \): This implies the function is a rational function with the degree of the denominator greater than the degree of the numerator.
   
2. \( \lim_{x \to 0} f(x) = -\infty \): This indicates there is a vertical asymptote at \( x = 0 \), suggesting the denominator has \( x \) as a factor.

3. \( f(5) = 0 \): This suggests that \( f(x) \) has a root at \( x = 5 \).

4. \( \lim_{x \to 8^-} f(x) = \infty \) and \( \lim_{x \to 8^+} f(x) = -\infty \): This suggests there is a vertical asymptote at \( x = 8 \), with different behaviors on the left and right of \( x = 8 \).

### Proposed function structure:
To satisfy these conditions, we need a rational function of the form:
\[
f(x) = \frac{N(x)}{D(x)}
\]
Where \( N(x) \) is the numerator and \( D(x) \) is the denominator.

- For the vertical asymptote at \( x = 0 \), the denominator should include \( x \) as a factor.
- For the root at \( x = 5 \), the numerator should include \( (x - 5) \).
- For the vertical asymptote at \( x = 8 \) with different behavior on either side, the denominator should include \( (x - 8)^2 \), because the sign change between \( x \to 8^- \) and \( x \to 8^+ \) suggests a factor of \( (x - 8)^1 \) rather than \( (x - 8)^2 \).

### Putting this together:
A possible rational function that satisfies these conditions is:
\[
f(x) = \frac{(x - 5)}{x(x - 8)}
\]

### Verifying the conditions:
1. \( \lim_{x \to +\infty} f(x) = 0 \):
   - As \( x \to +\infty \), both the numerator and denominator grow linearly, but the denominator grows faster due to the higher degree. Hence, \( \lim_{x \to +\infty} f(x) = 0 \).

2. \( \lim_{x \to 0} f(x) = -\infty \):
   - As \( x \to 0 \), the function behaves like \( \frac{1}{x} \), which tends to \( -\infty \).

3. \( f(5) = 0 \):
   - Substituting \( x = 5 \), we get \( f(5) = \frac{(5 - 5)}{5(5 - 8)} = 0 \).

4. \( \lim_{x \to 8^-} f(x) = \infty \) and \( \lim_{x \to 8^+} f(x) = -\infty \):
   - As \( x \to 8^- \), \( f(x) \) approaches \( +\infty \) because the numerator is negative and the denominator is negative, making the overall value positive.
   - As \( x \to 8^+ \), \( f(x) \) approaches \( -\infty \) because the numerator is negative and the denominator is positive, making the overall value negative.

Thus, the rational function \( f(x) = \frac{(x - 5)}{x(x - 8)} \) satisfies all the given conditions.

Would you like more details or have any questions? Here are some follow-up questions for deeper understanding:

1. How do vertical asymptotes affect the behavior of rational functions?
2. What is the role of the numerator in determining the zeros of a rational function?
3. How can we determine the behavior of a rational function near an asymptote?
4. What happens to a rational function as \( x \to \infty \) based on the degrees of the numerator and denominator?
5. How do limits help in analyzing the end behavior of functions?

**Tip:** Always check the degree of the numerator and denominator in rational functions to predict the end behavior (horizontal asymptotes).

Consider a rational function f that satisfies the following conditions:
1. \( \lim_{x \to +\infty} f(x) = 0 \)
2. \( \lim_{x \to 0} f(x) = -\infty \)
3. \( f(5) = 0 \)
4. \( \lim_{x \to 8^-} f(x) = \infty \)
5. \( \lim_{x \to 8^+} f(x) = -\infty \)

This problem explores finding a rational function f(x) that satisfies specific limits and asymptotic behavior. Learn how to analyze the function's behavior at different points and infinity. Suitable for advanced high school students studying calculus or precalculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation