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Let's analyze the graph of \( f(x) \) to answer the questions in the table. We are tasked with filling in the values for \( f(a) \), \( \lim_{x \to a} f(x) \), and whether \( f(x) \) is continuous at the given values of \( a \).

### Step-by-step:

1. **For \( a = -5 \):**
   - **\( f(a) \)**: At \( x = -5 \), the graph has a point at \( y = 3 \), so \( f(-5) = 3 \).
   - **\( \lim_{x \to -5} f(x) \)**: As \( x \) approaches -5 from both sides, the value approaches \( 3 \), so \( \lim_{x \to -5} f(x) = 3 \).
   - **Continuity**: Since \( f(-5) = 3 \) and the limit exists and equals \( 3 \), the function is continuous at \( x = -5 \).

2. **For \( a = -2 \):**
   - **\( f(a) \)**: At \( x = -2 \), the graph shows a hole (open circle) at \( y = 0 \), so \( f(-2) \) does not exist, i.e., \( f(-2) = \text{DNE} \).
   - **\( \lim_{x \to -2} f(x) \)**: As \( x \) approaches -2 from both sides, the value approaches \( 0 \), so \( \lim_{x \to -2} f(x) = 0 \).
   - **Continuity**: Since \( f(-2) \) does not exist, the function is not continuous at \( x = -2 \).

3. **For \( a = 0 \):**
   - **\( f(a) \)**: At \( x = 0 \), the graph passes through \( y = 2 \), so \( f(0) = 2 \).
   - **\( \lim_{x \to 0} f(x) \)**: As \( x \) approaches 0 from both sides, the value approaches \( 2 \), so \( \lim_{x \to 0} f(x) = 2 \).
   - **Continuity**: Since \( f(0) = 2 \) and the limit exists and equals \( 2 \), the function is continuous at \( x = 0 \).

4. **For \( a = 1 \):**
   - **\( f(a) \)**: At \( x = 1 \), the graph has a point at \( y = -3 \), so \( f(1) = -3 \).
   - **\( \lim_{x \to 1} f(x) \)**: As \( x \) approaches 1 from both sides, the value approaches \( -3 \), so \( \lim_{x \to 1} f(x) = -3 \).
   - **Continuity**: Since \( f(1) = -3 \) and the limit exists and equals \( -3 \), the function is continuous at \( x = 1 \).

5. **For \( a = 3 \):**
   - **\( f(a) \)**: At \( x = 3 \), the graph shows a hole (open circle) at \( y = -4 \), so \( f(3) \) does not exist, i.e., \( f(3) = \text{DNE} \).
   - **\( \lim_{x \to 3} f(x) \)**: As \( x \) approaches 3 from both sides, the value approaches \( -4 \), so \( \lim_{x \to 3} f(x) = -4 \).
   - **Continuity**: Since \( f(3) \) does not exist, the function is not continuous at \( x = 3 \).

### Completed Table:

| \( a \) | \( f(a) \)  | \( \lim_{x \to a} f(x) \) | Is \( f(x) \) continuous at \( x = a \)? |
|:------:|:-----------:|:-------------------------:|:---------------------------------------:|
|  -5    | 3           | 3                         | Yes                                     |
|  -2    | DNE         | 0                         | No                                      |
|   0    | 2           | 2                         | Yes                                     |
|   1    | -3          | -3                        | Yes                                     |
|   3    | DNE         | -4                        | No                                      |

Let me know if you'd like further explanations!

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**5 Related Questions:**
1. What are the criteria for continuity at a point?
2. How do we calculate limits from a graph?
3. What is the relationship between a limit and the value of a function at a point?
4. How do removable discontinuities appear in graphs?
5. Can a function be continuous if the limit does not exist at a point?

**Tip:** Always check for holes or jumps in the graph when determining continuity.

The graph of the function f(x) is given. Answer the questions about f(a), lim(x→a) f(x), and continuity at different values of a.

Learn how to evaluate limits, determine function values, and assess continuity from a graph. This example explores key calculus concepts including the limit definition and continuity at various points, with detailed step-by-step explanations.

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