https://file.aiquickdraw.com/mathbot/file/17297880094554duodjgu.jpg

To address the questions from the graph in Figure 14, we need to analyze the function \( f(x) \) at \( x = 4 \) and its behavior around this point.

### Estimation from the graph:
1. **Left-hand limit \( \lim_{x \to 4^-} f(x) \)**: 
   As \( x \) approaches 4 from the left, the graph seems to approach the value \( y = 2 \). So,
   \[
   \lim_{x \to 4^-} f(x) = 2
   \]

2. **Right-hand limit \( \lim_{x \to 4^+} f(x) \)**: 
   As \( x \) approaches 4 from the right, the graph also appears to approach \( y = 2 \). Thus,
   \[
   \lim_{x \to 4^+} f(x) = 2
   \]

3. **Two-sided limit \( \lim_{x \to 4} f(x) \)**: 
   Since both the left-hand and right-hand limits are equal to 2, the two-sided limit exists and is:
   \[
   \lim_{x \to 4} f(x) = 2
   \]

4. **Value of \( f(4) \)**:
   From the graph, there is a solid point at \( x = 4 \), with \( y = 3 \), indicating that the function's value at \( x = 4 \) is:
   \[
   f(4) = 3
   \]

### Final answers:
- \( \lim_{x \to 4^-} f(x) = 2 \)
- \( \lim_{x \to 4^+} f(x) = 2 \)
- \( \lim_{x \to 4} f(x) = 2 \)
- \( f(4) = 3 \)

Would you like further details or explanations?

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Here are 5 related questions to deepen your understanding:
1. What happens if the left-hand and right-hand limits are not equal?
2. How does the graph's continuity relate to the limits and function value at \( x = 4 \)?
3. What is the difference between a limit and the actual value of the function?
4. How do you calculate limits when a graph is not available?
5. Can a function have a limit at a point even if the function is not defined there?

**Tip**: A function is continuous at a point \( x = a \) if the left-hand limit, right-hand limit, and the function's value at \( a \) are all equal.

Estimate the functional values and the limits from the graph of the function f provided in Figure 14. Evaluate limits at x = 4 from both sides, the overall limit, and the function value at x = 4.

Learn how to estimate the left-hand and right-hand limits, two-sided limit, and function value at x = 4 from a graph. This problem uses key concepts of limits and continuity, suitable for high school students.

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