https://file.aiquickdraw.com/mathbot/file/172631414733079z18qwa.jpg

The problem you've shared involves determining the values or existence of limits for a function \( g(t) \), whose graph is provided. Let's break down the different parts of the question:

### Problem:

For the function \( g(t) \), state the value of each quantity or explain if it does not exist:

(a) \( \lim_{t \to 0^-} g(t) \)  
(b) \( \lim_{t \to 0^+} g(t) \)  
(c) \( \lim_{t \to 0} g(t) \)  
(d) \( \lim_{t \to -2^-} g(t) \)  
(e) \( \lim_{t \to -2^+} g(t) \)  
(f) \( \lim_{t \to -2} g(t) \)  
(g) \( g(2) \)  
(h) \( \lim_{t \to 4} g(t) \)

### Steps for Solving:

1. **Examine the graph** to observe the behavior of \( g(t) \) around the points 0, -2, 2, and 4.
2. **Determine the left-hand and right-hand limits** where applicable.
3. **Check for discontinuities** or sharp breaks in the function, which could indicate that the limit does not exist.

#### Graph observations (from what is visible):
- The function appears to pass through specific points, and shows notable changes around \( t = 0 \), \( t = -2 \), and \( t = 2 \). We will analyze the graph near these points for left and right limits.

Let me analyze and extract these values from the graph. One moment.I've displayed the image to review the graph more closely. Let me now provide the values or explanations for the limits based on the graph's behavior at the specified points.

### Analysis of Limits:

1. **(a) \( \lim_{t \to 0^-} g(t) \)**:  
   From the graph, as \( t \) approaches 0 from the left (negative side), it seems that \( g(t) \) approaches a value around 0. This suggests that:
   \[
   \lim_{t \to 0^-} g(t) = 0
   \]

2. **(b) \( \lim_{t \to 0^+} g(t) \)**:  
   As \( t \) approaches 0 from the right (positive side), it also looks like \( g(t) \) approaches 0. Hence:
   \[
   \lim_{t \to 0^+} g(t) = 0
   \]

3. **(c) \( \lim_{t \to 0} g(t) \)**:  
   Since both the left-hand and right-hand limits at \( t = 0 \) are equal, the two-sided limit exists and is:
   \[
   \lim_{t \to 0} g(t) = 0
   \]

4. **(d) \( \lim_{t \to -2^-} g(t) \)**:  
   As \( t \) approaches \( -2 \) from the left, the graph appears to approach a value of around 0. Therefore:
   \[
   \lim_{t \to -2^-} g(t) = 0
   \]

5. **(e) \( \lim_{t \to -2^+} g(t) \)**:  
   As \( t \) approaches \( -2 \) from the right, the function jumps to a value around 2. So:
   \[
   \lim_{t \to -2^+} g(t) = 2
   \]

6. **(f) \( \lim_{t \to -2} g(t) \)**:  
   Since the left-hand and right-hand limits at \( t = -2 \) are not equal, the limit does not exist:
   \[
   \lim_{t \to -2} g(t) \text{ does not exist}
   \]

7. **(g) \( g(2) \)**:  
   From the graph, the point \( t = 2 \) seems to have a defined value. It looks like \( g(2) = 2 \). Therefore:
   \[
   g(2) = 2
   \]

8. **(h) \( \lim_{t \to 4} g(t) \)**:  
   As \( t \) approaches 4, the function appears to approach a value of 4. Hence:
   \[
   \lim_{t \to 4} g(t) = 4
   \]

Would you like more detailed explanations for any of these limits or any clarifications?

### Related Questions:
1. How can we determine when a limit does not exist?
2. What is the difference between one-sided and two-sided limits?
3. How are jump discontinuities related to limits?
4. What is the significance of the graph's behavior at a point where the limit does not exist?
5. How can you visually identify discontinuities in a function's graph?

### Tip:
When evaluating limits graphically, always check the left-hand and right-hand behavior around the point of interest. If they are not equal, the limit does not exist at that point.

For the function g(t) whose graph is given, state the value of each quantity (limits at various points). If it does not exist, explain why.

Analyze the limits of the function g(t) at points t = 0, -2, 2, and 4 based on the given graph. Understand left-hand and right-hand limits, and discover when limits exist or do not exist due to discontinuities.

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