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.