https://file.aiquickdraw.com/mathbot/file/1726181149762uohsd106.jpg

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. \( \lim_{x \to 2} f(x) = \text{DNE} \)

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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!

Determine whether a function is continuous and/or differentiable at a point using limits based on the given graph.

Understand how to determine the continuity and limits of a piecewise function at specific points on a graph. This example explores key calculus concepts, including limits from the left and right, and conditions for a function to be continuous. Suitable for students in Grades 11-12.

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