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. Use the graph of the function *f* to decide whether the value of the given quantity. If it does, find it. If not, explain why.

Let's analyze the graph and the corresponding questions to determine the values or limits of the function \( f(x) \).

### 1. Understanding the Graph:
- The graph shows a piecewise function with two different segments.
- The first segment (on the left) appears to be a curve that ends at \( x = 1 \) with a hole (open circle).
- The second segment (on the right) is a straight line starting at \( x = 1 \) with a filled dot, indicating the function is defined there.
- The x-axis ranges from about \(-1\) to 4, and the y-axis from 0 to 5.

### 2. Analysis of Given Quantities:
a. **\( f(0) \)**:
   - At \( x = 0 \), the graph intersects the y-axis at \( y = 2 \).
   - **Answer**: \( f(0) = 2 \).

b. **\( \lim_{x \to 0} f(x) \)**:
   - As \( x \) approaches 0 from both sides, the function value approaches \( y = 2 \).
   - **Answer**: \( \lim_{x \to 0} f(x) = 2 \).

c. **\( f(2) \)**:
   - At \( x = 2 \), the graph has a filled circle at \( y = 3 \).
   - **Answer**: \( f(2) = 3 \).

d. **\( \lim_{x \to 2} f(x) \)**:
   - As \( x \) approaches 2 from both sides, the function values approach \( y = 3 \).
   - **Answer**: \( \lim_{x \to 2} f(x) = 3 \).

### Summary:
- **\( f(0) = 2 \)**
- **\( \lim_{x \to 0} f(x) = 2 \)**
- **\( f(2) = 3 \)**
- **\( \lim_{x \to 2} f(x) = 3 \)**

These values indicate that the function is continuous at \( x = 0 \) and \( x = 2 \).

Would you like any further details or have additional questions?

### Follow-up Questions:
1. What is the definition of a limit, and how is it evaluated graphically?
2. What does it mean when a graph has a hole (open circle) at a point?
3. How do you determine the continuity of a function from its graph?
4. What is the difference between a limit and the actual value of the function at a point?
5. How would you explain the concept of piecewise functions and their graphs?
6. What are the common types of discontinuities in a function?
7. How does one determine if a limit does not exist from a graph?
8. Can a function have a limit at a point where it is not defined?

### Tip:
When analyzing limits graphically, always check the behavior of the function as it approaches the point from both the left and right sides.

Understand how to analyze the graph of function f(x) to determine values and limits. Explore the concepts of piecewise functions, continuity, and limit evaluation at specific points. Detailed solutions provided for f(0), f(2), and their respective limits.

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