lim 
x→±∞ 
f(x) = 0,   
lim 
x→0 
f(x) = −∞,  f(4) = 0
 
lim 
x→5− 
f(x) = ∞,   
lim 
x→5+ 
f(x) = −∞


Given the conditions for the function \( f(x) \), we can interpret the behavior of \( f(x) \) as follows:

1. **\( \lim_{x \to \pm\infty} f(x) = 0 \):**  
   This tells us that as \( x \) approaches both positive and negative infinity, the function \( f(x) \) tends towards zero. This suggests that the function has horizontal asymptotes at \( y = 0 \) for large values of \( x \) in both directions.

2. **\( \lim_{x \to 0} f(x) = -\infty \):**  
   As \( x \) approaches zero, the function tends to negative infinity. This suggests that there is a vertical asymptote at \( x = 0 \) with the function approaching \( -\infty \) as \( x \) gets closer to zero.

3. **\( f(4) = 0 \):**  
   This tells us that the function crosses the x-axis at \( x = 4 \). The point \( (4, 0) \) is on the graph of \( f(x) \).

4. **\( \lim_{x \to 5^-} f(x) = \infty \):**  
   As \( x \) approaches 5 from the left (i.e., \( x \to 5^- \)), the function tends to positive infinity. This indicates that there is a vertical asymptote at \( x = 5 \), with the function approaching \( +\infty \) from the left side of this asymptote.

5. **\( \lim_{x \to 5^+} f(x) = -\infty \):**  
   As \( x \) approaches 5 from the right (i.e., \( x \to 5^+ \)), the function tends to negative infinity. This further supports the existence of a vertical asymptote at \( x = 5 \), with the function approaching \( -\infty \) from the right side.

### Graphical Implications:
- The function has horizontal asymptotes at \( y = 0 \) for \( x \to \pm\infty \).
- There is a vertical asymptote at \( x = 0 \), where the function approaches \( -\infty \).
- There is another vertical asymptote at \( x = 5 \), with the function tending to \( +\infty \) from the left and \( -\infty \) from the right.
- The function crosses the x-axis at \( x = 4 \).

This information suggests a piecewise or rational function with asymptotic behavior at \( x = 0 \) and \( x = 5 \), and with the function approaching zero at extreme values of \( x \).

Would you like a detailed function model or explanation of how such functions can be constructed?

### Related Questions:
1. What is the general form of a function that has horizontal and vertical asymptotes?
2. How do you determine the vertical asymptotes of a rational function?
3. How does the behavior near asymptotes affect the shape of a graph?
4. What types of functions have both positive and negative infinity limits near vertical asymptotes?
5. How do you graph a function with multiple asymptotes?

**Tip:** When analyzing limits, vertical asymptotes typically occur where the function is undefined (often division by zero), while horizontal asymptotes describe behavior at extreme values of \( x \).

lim x→±∞ f(x) = 0, lim x→0 f(x) = −∞, f(4) = 0, lim x→5− f(x) = ∞, lim x→5+ f(x) = −∞

Explore the behavior of the function f(x) given specific limit conditions as x approaches infinity, zero, and five. This problem highlights the concepts of vertical and horizontal asymptotes, and how they shape the graph of a function, suitable for pre-calculus or calculus students.

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