Let's analyze the uploaded image to address the questions regarding the graph of the function $f(x)$. I'll interpret each part and provide a detailed solution.

Graph Analysis:

1. (a) $\lim_{x \to 2^-} f(x)$:

   • This is the left-hand limit as $x$ approaches 2.
   • From the graph, as $x$ approaches 2 from the left, $f(x) \to 3$.
   • Answer: $3$.

2. (b) $\lim_{x \to 2^+} f(x)$:

   • This is the right-hand limit as $x$ approaches 2.
   • From the graph, as $x$ approaches 2 from the right, $f(x) \to 1$.
   • Answer: $1$.

3. (c) $\lim_{x \to 2} f(x)$:

   • The two-sided limit exists only if the left-hand limit equals the right-hand limit.
   • Since $\lim_{x \to 2^-} f(x) = 3 \neq 1 = \lim_{x \to 2^+} f(x)$, the two-sided limit does not exist.
   • Answer: $\text{DNE}$ (Does Not Exist).

4. (d) $f(2)$:

   • The value of $f(2)$ is represented by the filled or hollow circle at $x = 2$ on the graph.
   • At $x = 2$, the hollow circle at $(2, 3)$ and filled circle at $(2, 1)$ indicate $f(2) = 1$.
   • Answer: $1$.

5. (e) $\lim_{x \to 4} f(x)$:

   • As $x$ approaches 4, the graph clearly shows that $f(x) \to 4$.
   • Answer: $4$.

6. (f) $f(4)$:

   • At $x = 4$, the filled circle at $(4, 4)$ shows $f(4) = 4$.
   • Answer: $4$.

Final Answers:

• (a) $3$
• (b) $1$
• (c) $\text{DNE}$
• (d) $1$
• (e) $4$
• (f) $4$

Would you like further clarification on any of these parts or a detailed explanation of limits and continuity?

Relative Questions:

1. How do left-hand and right-hand limits affect the existence of two-sided limits?
2. What is the significance of filled and hollow circles in function graphs?
3. How can a function be discontinuous at a point but still have a defined value?
4. How do you graphically verify the existence of a limit?
5. What does the notation $\text{DNE}$ imply in terms of continuity and limits?

Tip:

Always check the behavior of a function from both sides of a point to determine if a two-sided limit exists!