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To solve the limits of the rational function \( g(x) \) as shown in the graph, let's analyze the behavior of \( g(x) \) at the given points.

### Analyzing the Graph
The graph shows a rational function with vertical asymptotes and horizontal asymptotes. Here's what we need to determine:

1. **Horizontal Asymptotes (as \( x \to \infty \) or \( x \to -\infty \))**:
   - The behavior of the function as \( x \) goes to positive or negative infinity will indicate the horizontal asymptotes.
2. **Vertical Asymptotes (as \( x \to \text{specific value} \))**:
   - The behavior of the function near vertical asymptotes will tell us the limit approaching specific points.

### Limits at Infinity
From the graph:
- As \( x \to \infty \), \( g(x) \) approaches 2.
- As \( x \to -\infty \), \( g(x) \) approaches -1.

### Limits Near Vertical Asymptotes
The vertical asymptotes occur at \( x = -2 \) and \( x = 2 \).
- As \( x \to -2^+ \), \( g(x) \to \infty \).
- As \( x \to -2^- \), \( g(x) \to -\infty \).
- As \( x \to 2^+ \), \( g(x) \to -\infty \).
- As \( x \to 2^- \), \( g(x) \to \infty \).

### Filling in the Blanks
Based on the above analysis:
- \( \lim_{x \to \infty} g(x) = 2 \)
- \( \lim_{x \to -\infty} g(x) = -1 \)
- \( \lim_{x \to -2^+} g(x) = \infty \) (Note: The limit near -2 is as \( x \to -2^+ \), but the closest match to our problem format is \( x \to 2 \)).
- \( \lim_{x \to 2^-} g(x) = \infty \)

Now, let's map these to the problem format:

a. \( \lim_{x \to \infty} g(x) = 2 \)  
b. \( \lim_{x \to -\infty} g(x) = -1 \)  
c. \( \lim_{x \to -2} g(x) = \text{does not exist (DNE, because the limits from both sides are not the same)} \)
d. \( \lim_{x \to 2} g(x) = \text{does not exist (DNE, because the limits from both sides are not the same)} \)

### Input the Values
Let's fill in the blanks based on this analysis:

1. \( \lim_{x \to \infty} g(x) = 2 \)
2. \( \lim_{x \to -\infty} g(x) = -1 \)
3. \( \lim_{x \to -2} g(x) = \text{DNE} \)
4. \( \lim_{x \to 2} g(x) = \text{DNE} \)

If you have any questions or need further details, please let me know! Here are some related questions to deepen your understanding:

1. What is the significance of horizontal asymptotes in rational functions?
2. How do you determine vertical asymptotes of a rational function?
3. What happens to the limit if the left-hand limit and right-hand limit are different at a point?
4. Can a function have more than one horizontal asymptote? Why or why not?
5. How do the degrees of the polynomial in the numerator and denominator of a rational function affect its asymptotic behavior?

**Tip:** When analyzing the behavior of a function at infinity or near vertical asymptotes, always consider the leading terms of the numerator and denominator to predict the end behavior accurately.

The graph of a rational function g(x) is shown, and the task is to determine the limits at infinity and near vertical asymptotes. The specific limits to find are lim x -> infinity g(x), lim x -> -infinity g(x), lim x -> -2 g(x), and lim x -> 2 g(x).

Learn how to solve limits of a rational function g(x) at infinity and near vertical asymptotes. Analyze the graph to determine behavior as x approaches infinity and at points where vertical asymptotes exist. Suitable for students in Grades 10-12.

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