https://file.aiquickdraw.com/mathbot/file/1725408333243q5cvk1dg.jpg

Let's analyze the content of the image to solve the problem.

The image contains a table with values of \( x \) and corresponding \( f(x) \), followed by three multiple-choice questions asking about the limit of \( f(x) \) as \( x \) approaches 3 from different directions:

### Table Data:
- \( x = 2.9 \), \( f(x) = 6.2 \)
- \( x = 2.99 \), \( f(x) = 6.03 \)
- \( x = 2.999 \), \( f(x) = 6.003 \)
- \( x = 3.001 \), \( f(x) = 5.997 \)
- \( x = 3.01 \), \( f(x) = 5.97 \)
- \( x = 3.1 \), \( f(x) = 5.7 \)

### Questions:
1. **\(\lim\limits_{x \to 3^-} f(x)\)** (Left-hand limit)
   - Options: \( 6 \), \( 7 \), \( 9 \), Undefined.
2. **\(\lim\limits_{x \to 3^+} f(x)\)** (Right-hand limit)
   - Options: \( 6 \), \( 7 \), \( 9 \), Undefined.
3. **\(\lim\limits_{x \to 3} f(x)\)** (Two-sided limit)
   - Options: \( 6 \), \( 7 \), \( 9 \), Undefined.

### Solving the Limits:

1. **Left-hand limit \(\lim\limits_{x \to 3^-} f(x)\)**:
   - As \( x \) approaches 3 from the left (\( x = 2.999 \), \( x = 2.99 \), \( x = 2.9 \)), \( f(x) \) values are getting closer to \( 6 \).
   - **Answer: \( 6 \)**

2. **Right-hand limit \(\lim\limits_{x \to 3^+} f(x)\)**:
   - As \( x \) approaches 3 from the right (\( x = 3.001 \), \( x = 3.01 \), \( x = 3.1 \)), \( f(x) \) values are getting closer to \( 6 \) as well.
   - **Answer: \( 6 \)**

3. **Two-sided limit \(\lim\limits_{x \to 3} f(x)\)**:
   - Since both the left-hand limit and right-hand limit as \( x \) approaches 3 are equal to \( 6 \), the two-sided limit exists and is \( 6 \).
   - **Answer: \( 6 \)**

### Summary of Answers:
1. **\(\lim\limits_{x \to 3^-} f(x)\) = 6**
2. **\(\lim\limits_{x \to 3^+} f(x)\) = 6**
3. **\(\lim\limits_{x \to 3} f(x)\) = 6**

Would you like any further details or have any questions?

### Related Questions:
1. What is the significance of the left-hand and right-hand limits in determining the overall limit?
2. How would the answers change if the left-hand and right-hand limits were different?
3. What is the relationship between continuity and limits?
4. Can a function have different left-hand and right-hand limits? If so, what does it imply?
5. How does the concept of limits apply to real-world scenarios?

### Tip:
When evaluating limits from a table, always observe how the function's value behaves as it approaches the point of interest from both sides to ensure accuracy.

Suppose f(x) has the following values in a table. Determine the left-hand limit \(\lim\limits_{x \to 3^-} f(x)\), the right-hand limit \(\lim\limits_{x \to 3^+} f(x)\), and the two-sided limit \(\lim\limits_{x \to 3} f(x)\) from the table.

Learn how to determine the left-hand limit, right-hand limit, and two-sided limit of a function using a table of values. This problem focuses on understanding the concept of limits and how to evaluate them from a table. Suitable for students in Grades 10-12.

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