Consider a rational function f that satisfies the following conditions.
lim 
x→±∞ 
f(x) = 0,	   
lim 
x→0 
f(x) = −∞,	   f(8) = 0,

 
lim 
x→9− 
f(x) = ∞,	   
lim 
x→9+ 
f(x) = −∞,	  
Find an equation for f.
f(x) = 


We are given a rational function \( f(x) \) that satisfies the following conditions:

1. \( \lim_{x \to \pm \infty} f(x) = 0 \) (The function approaches zero as \( x \) tends to infinity or negative infinity.)
2. \( \lim_{x \to 0} f(x) = -\infty \) (As \( x \) approaches zero, the function tends to negative infinity, indicating a vertical asymptote at \( x = 0 \).)
3. \( f(8) = 0 \) (The function crosses the x-axis at \( x = 8 \), so \( x = 8 \) is a root of the function.)
4. \( \lim_{x \to 9^-} f(x) = \infty \) and \( \lim_{x \to 9^+} f(x) = -\infty \) (At \( x = 9 \), there is a vertical asymptote, with the function tending to \( +\infty \) from the left and \( -\infty \) from the right.)

### Step-by-Step Construction:

- From condition 1, \( f(x) \) must have the form of a rational function where the degree of the numerator is less than or equal to the degree of the denominator to ensure \( \lim_{x \to \pm \infty} f(x) = 0 \).

- From condition 2, \( x = 0 \) is a vertical asymptote. This suggests that the denominator has a factor of \( x \) to create this asymptote at \( x = 0 \).

- From condition 3, the function has a zero at \( x = 8 \). Thus, the numerator of the rational function must have a factor of \( (x - 8) \).

- From condition 4, there is a vertical asymptote at \( x = 9 \), with the behavior \( \lim_{x \to 9^-} f(x) = \infty \) and \( \lim_{x \to 9^+} f(x) = -\infty \). This suggests that the denominator must have a factor of \( (x - 9)^2 \), because a simple factor of \( (x - 9) \) would not create the required behavior at both sides of 9.

### General Form:

The rational function can be written as:

\[
f(x) = \frac{A(x - 8)}{x(x - 9)^2}
\]

where \( A \) is a constant to be determined.

- Since \( \lim_{x \to \infty} f(x) = 0 \), we don’t need any additional terms in the numerator.
- To determine \( A \), note that the behavior of the function around \( x = 0 \), \( x = 9 \), and \( x = 8 \) matches the given conditions without further adjustments to the degree.

Thus, the function is:

\[
f(x) = \frac{(x - 8)}{x(x - 9)^2}
\]

This satisfies all the given conditions.

Would you like more details on how the behavior of the function matches the given limits, or how the equation was derived?

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### Relative Questions:
1. Why does \( f(x) = \frac{(x - 8)}{x(x - 9)^2} \) approach 0 as \( x \to \infty \)?
2. What would change if the vertical asymptote at \( x = 9 \) behaved differently?
3. How does the factor \( (x - 8) \) in the numerator affect the graph of the function?
4. Could the behavior at \( x = 0 \) be explained by other forms of rational functions?
5. What role does the squared term \( (x - 9)^2 \) play in the behavior of the function at 9?

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### Tip:
When constructing rational functions from limits, focus on how asymptotic behavior and roots guide the choice of factors in the numerator and denominator.

Consider a rational function f that satisfies the following conditions: lim x→±∞ f(x) = 0, lim x→0 f(x) = −∞, f(8) = 0, lim x→9− f(x) = ∞, lim x→9+ f(x) = −∞. Find an equation for f. f(x) =

Find the equation of a rational function f(x) given specific asymptotic behavior and a root at x=8. The function approaches infinity near x=9 and negative infinity at x=0. Suitable for advanced high school students studying rational functions and limits.

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