https://file.aiquickdraw.com/mathbot/file/1726394878362gsax9iry.jpg

Let's evaluate each of the expressions using the graph provided.

### a. \( f(-2) \)
- From the graph, at \( x = -2 \), the closed dot indicates \( f(-2) = 0 \).

### b. \( \lim_{x \to -2^+} f(x) \)
- The right-hand limit as \( x \) approaches \(-2\) from the right is approaching \( 2 \), based on the graph.

### c. \( \lim_{x \to -2} f(x) \)
- For the two-sided limit, \( \lim_{x \to -2} f(x) \), we must check both the left-hand and right-hand limits. 
  - Left-hand limit: as \( x \) approaches \( -2 \) from the left, the value approaches 0.
  - Right-hand limit: as \( x \) approaches \( -2 \) from the right, the value approaches 2.
- Since the left-hand and right-hand limits do not agree, the two-sided limit **does not exist**.

### d. \( \lim_{x \to -1^+} f(x) \)
- From the right, as \( x \to -1 \), the value of \( f(x) \) approaches 0. So, \( \lim_{x \to -1^+} f(x) = 0 \).

### e. \( \lim_{x \to -1^-} f(x) \)
- From the left, as \( x \to -1 \), the value of \( f(x) \) approaches 0 as well. Hence, \( \lim_{x \to -1^-} f(x) = 0 \).

### f. \( \lim_{x \to -1} f(x) \)
- Since both the left-hand and right-hand limits are equal to 0, \( \lim_{x \to -1} f(x) = 0 \).

### g. \( \lim_{x \to 1^+} f(x) \)
- As \( x \) approaches 1 from the right, the value of \( f(x) \) approaches 0. Hence, \( \lim_{x \to 1^+} f(x) = 0 \).

### h. \( \lim_{x \to 1^-} f(x) \)
- As \( x \) approaches 1 from the left, the value of \( f(x) \) approaches 0. Hence, \( \lim_{x \to 1^-} f(x) = 0 \).

### i. \( \lim_{x \to 1} f(x) \)
- Since both left-hand and right-hand limits are equal, \( \lim_{x \to 1} f(x) = 0 \).

### j. \( f(3) \)
- At \( x = 3 \), there is a closed dot at \( f(3) = 5 \). So, \( f(3) = 5 \).

### k. \( \lim_{x \to 3^+} f(x) \)
- As \( x \) approaches 3 from the right, the graph approaches \( 0 \). Hence, \( \lim_{x \to 3^+} f(x) = 0 \).

### l. \( \lim_{x \to 3^-} f(x) \)
- As \( x \) approaches 3 from the left, the graph approaches \( 5 \). Hence, \( \lim_{x \to 3^-} f(x) = 5 \).

### m. \( \lim_{x \to 3} f(x) \)
- Since the left-hand limit (\( 5 \)) and right-hand limit (\( 0 \)) do not agree, \( \lim_{x \to 3} f(x) \) **does not exist**.

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Would you like more details or clarification on any of these points?

Here are some related questions to expand understanding:
1. How do you determine the existence of a limit at a point?
2. What is the significance of the right-hand and left-hand limits in relation to two-sided limits?
3. What role do discontinuities play in the behavior of limits?
4. Can a function have different values for left-hand and right-hand limits but still be continuous?
5. How would the graph change if the limit at \( x = 3 \) existed?

**Tip**: When evaluating limits from a graph, always check the behavior of the function from both sides of the point to see if they match.

Use the graph to evaluate the expressions below.

Learn how to evaluate function values and limits from a graph, including right-hand and left-hand limits, as well as determining the existence of a limit. This problem is suitable for students in Grades 10-12 and covers core calculus concepts.

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