https://file.aiquickdraw.com/mathbot/file/1725409763210flgb5yqi.jpg

Let's analyze the graph and the associated questions.

### Graph Details:
- The graph shows the function \( y = f(x) \) plotted on the coordinate plane.
- There are several key points where the function changes direction, has jumps, or is undefined.

### Questions:
1. **\( f(2) \)**: This asks for the value of the function at \( x = 2 \).
   - At \( x = 2 \), the corresponding \( y \)-value is 4.
   - **Answer**: \( f(2) = 4 \).

2. **\( \lim_{x \to 2^-} f(x) \)**: This asks for the left-hand limit as \( x \) approaches 2.
   - As \( x \) approaches 2 from the left, the \( y \)-value approaches 1.
   - **Answer**: \( \lim_{x \to 2^-} f(x) = 1 \).

3. **\( \lim_{x \to 2^+} f(x) \)**: This asks for the right-hand limit as \( x \) approaches 2.
   - As \( x \) approaches 2 from the right, the \( y \)-value approaches 4.
   - **Answer**: \( \lim_{x \to 2^+} f(x) = 4 \).

4. **\( \lim_{x \to 2} f(x) \)**: This asks for the limit of \( f(x) \) as \( x \) approaches 2.
   - Since the left-hand limit (\( 1 \)) does not equal the right-hand limit (\( 4 \)), the overall limit does not exist.
   - **Answer**: \( \lim_{x \to 2} f(x) \) **does not exist**.

### Summary of Answers:
- \( f(2) = 4 \)
- \( \lim_{x \to 2^-} f(x) = 1 \)
- \( \lim_{x \to 2^+} f(x) = 4 \)
- \( \lim_{x \to 2} f(x) \) **does not exist**

Would you like further details on these concepts, or do you have any additional questions?

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Here are five relative questions that build upon this information:

1. What is the value of \( \lim_{x \to 4} f(x) \) based on the graph?
2. How would you determine \( f(5) \) using the graph?
3. What is the significance of a jump discontinuity at \( x = 2 \) in the context of limits?
4. How does the right-hand limit at \( x = 7 \) compare to the left-hand limit at the same point?
5. If \( \lim_{x \to 3} f(x) = 2 \), what does this tell us about the behavior of \( f(x) \) around \( x = 3 \)?

**Tip**: When evaluating limits graphically, always compare the left-hand and right-hand limits to determine if the overall limit exists.

Determine the value of f(2), the left-hand limit as x approaches 2 (lim x -> 2^-), the right-hand limit as x approaches 2 (lim x -> 2^+), and the limit as x approaches 2 (lim x -> 2) from the given graph of y = f(x).

Analyze the graph of the function y = f(x) to determine the value of f(2), the left-hand limit as x approaches 2, the right-hand limit as x approaches 2, and the overall limit as x approaches 2. This problem covers key calculus concepts such as limits, continuity, and graphical analysis, suitable for students in Grades 11-12.

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