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The image shows the graph of a rational function \( f(x) \), and below it, there are several statements asking you to complete limits as \( x \) approaches certain values. Let's break this down based on the graph:

1. The graph seems to have vertical asymptotes at \( x = -1 \) and \( x = 3 \).
2. It also appears that as \( x \to -\infty \) or \( x \to \infty \), the graph approaches \( y = 0 \) (a horizontal asymptote).

Now, let's address the specific statements:

a) As \( x \to -1^- \), \( f(x) \to -\infty \).
   - This means as \( x \) approaches -1 from the left, the graph goes downward towards negative infinity.

b) As \( x \to -1^+ \), \( f(x) \to \infty \).
   - As \( x \) approaches -1 from the right, the graph goes upward towards positive infinity.

c) As \( x \to 3^- \), \( f(x) \to \infty \).
   - As \( x \) approaches 3 from the left, the graph goes upwards towards positive infinity.

d) As \( x \to 3^+ \), \( f(x) \to -\infty \).
   - As \( x \) approaches 3 from the right, the graph goes downward towards negative infinity.

These statements describe the behavior of the function at its vertical asymptotes.

Do you need more details or have any questions?

Here are five questions to expand on this information:
1. What is the significance of vertical asymptotes in rational functions?
2. How can you determine horizontal asymptotes of a rational function from its equation?
3. What happens to the function as \( x \to \infty \) and \( x \to -\infty \)?
4. What types of rational functions produce graphs with vertical asymptotes?
5. How can the behavior near an asymptote help you understand the overall behavior of the function?

**Tip**: When analyzing the limits at vertical asymptotes, always check the direction of the approach (from the left or right) to determine if the function tends to positive or negative infinity.

Consider the graph of a rational function f and complete the following limits: As x → -1^-, As x → -1^+, As x → 3^-, As x → 3^+.

Learn how to analyze the behavior of a rational function near vertical asymptotes by solving limits as x approaches -1 and 3. This problem involves understanding limits and the behavior of functions near vertical asymptotes, suitable for students in Grades 10-12.

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