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Using the graph answer question 28 all parts. 28 (a) lim x-> -2^- g(x). 28 (b) lim x-> -2^+ g(x). 28 © lim x-> -2 g(x). 28 (d) g(2). Show all work

To answer question 28 using the graph you provided, let’s carefully evaluate each part step-by-step:

### 28 (a) \( \lim_{x \to -2^-} g(x) \)
This represents the left-hand limit as \( x \) approaches \(-2\). Observing the graph, as \( x \) approaches \(-2\) from the left side, we need to follow the values of the function. It appears that the function approaches the value 1 from the left side. 

Thus:
\[
\lim_{x \to -2^-} g(x) = 1
\]

### 28 (b) \( \lim_{x \to -2^+} g(x) \)
This is the right-hand limit as \( x \) approaches \(-2\). Following the curve from the right-hand side of \( x = -2 \), it seems that the function approaches the value \(-2\).

Thus:
\[
\lim_{x \to -2^+} g(x) = -2
\]

### 28 (c) \( \lim_{x \to -2} g(x) \)
This is the overall limit as \( x \) approaches \(-2\), which only exists if both the left-hand and right-hand limits are equal. Since we found that:
\[
\lim_{x \to -2^-} g(x) = 1 \quad \text{and} \quad \lim_{x \to -2^+} g(x) = -2
\]
the overall limit does not exist because the left-hand and right-hand limits are not equal.

Thus:
\[
\lim_{x \to -2} g(x) \quad \text{does not exist}.
\]

### 28 (d) \( g(-2) \)
To find \( g(-2) \), we check the value of \( g(x) \) at \( x = -2 \). Looking at the graph, there is a filled circle at \( x = -2 \), and its y-coordinate is \(-2\). This indicates that the function is defined at this point and takes the value \(-2\).

Thus:
\[
g(-2) = -2
\]

### Final Answers:
- 28 (a) \( \lim_{x \to -2^-} g(x) = 1 \)
- 28 (b) \( \lim_{x \to -2^+} g(x) = -2 \)
- 28 (c) \( \lim_{x \to -2} g(x) \) does not exist.
- 28 (d) \( g(-2) = -2 \)

Would you like more details or have any questions? Here are 5 related questions that might interest you:

1. What happens if the left-hand and right-hand limits are equal but the function is not defined at that point?
2. Can a function have different left-hand and right-hand limits at multiple points?
3. How would you determine the limit of a function at infinity using a graph?
4. What is the significance of open and closed circles in graphs for evaluating limits?
5. How does continuity at a point affect the existence of limits?

**Tip:** Always check both the left-hand and right-hand limits before concluding whether a limit exists at a specific point.

Using the graph answer question 28 all parts. 28 (a) lim x-> -2^- g(x). 28 (b) lim x-> -2^+ g(x). 28 (c) lim x-> -2 g(x). 28 (d) g(2). Show all work.

Answer the full solution to question 28 by using the graph to evaluate limits. This problem involves the left-hand and right-hand limits of g(x) as x approaches -2, as well as determining the value of g(-2). Suitable for high school students learning about limits and functions.

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