From the graph provided, let's analyze the points and limits to answer the questions.

Step-by-Step Analysis:

1. $f(9)$:

• From the graph, at $x = 9$, the corresponding $y$-value (or $f(9)$) is clearly marked at 5.
• Answer: $f(9) = 5$

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

• We need to examine the behavior of $f(x)$ as $x$ approaches 5 from both sides (left and right).
• From the left ($x \to 5^-$), $f(x)$ approaches 3.
• From the right ($x \to 5^+$), $f(x)$ also approaches 3.
• Since both limits agree, the two-sided limit exists and equals 3.
• Answer: $\lim_{x \to 5} f(x) = 3$

3. $\lim_{x \to 1^+} f(x)$:

• This asks for the right-hand limit as $x$ approaches 1.
• Approaching from the right ($x \to 1^+$), $f(x)$ approaches 1 (since the value approaches the point at $y = 1$).
• Answer: $\lim_{x \to 1^+} f(x) = 1$

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

• At $x = 2$, there is a jump discontinuity in the graph.
• The limit from the left ($x \to 2^-$) approaches 2, while the limit from the right ($x \to 2^+$) approaches 3.
• Since the left and right limits do not match, the limit does not exist.
• Answer: $\lim_{x \to 2} f(x) = \text{DNE (Does Not Exist)}$

Summary of Answers:

1. $f(9) = 5$
2. $\lim_{x \to 5} f(x) = 3$
3. $\lim_{x \to 1^+} f(x) = 1$
4. $\lim_{x \to 2} f(x) = \text{DNE}$

Would you like more details on any of these? Here are some related questions you might find interesting:

1. What is the difference between one-sided and two-sided limits?
2. How do you determine continuity of a function using limits?
3. Can a function be differentiable if it has a jump discontinuity?
4. How do you calculate the limit at a point of removable discontinuity?
5. Why are limits important in calculus?

Tip: When analyzing limits, always check the behavior from both sides of the point. If the left-hand and right-hand limits are unequal, the overall limit does not exist!