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:

Let me know if you'd like further explanations!

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.