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 $x$.