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The question is asking to find a value of \( \delta \) such that for all \( x \), if \( 0 < |x - 3| < \delta \), then \( |f(x) - 2| < 0.5 \). This is related to the definition of continuity, specifically in terms of the epsilon-delta definition of limits.

### Analyzing the graph:

- The horizontal line \( y = 2 \) is relevant because we are interested in how the function \( f(x) \) deviates from 2.
- The condition \( |f(x) - 2| < 0.5 \) means we are looking for where the function lies between 1.5 and 2.5 (as \( 2 - 0.5 = 1.5 \) and \( 2 + 0.5 = 2.5 \)).
- From the graph, this means we are interested in the interval where the graph of \( f(x) \) lies between these two values of \( y \).

Now, we need to determine for which values of \( x \), \( f(x) \) is between 1.5 and 2.5:

- At \( f(2.6) \), the graph is approximately at 1.5.
- At \( f(3.8) \), the graph reaches 2.5.

Therefore, \( |f(x) - 2| < 0.5 \) holds when \( x \) is between approximately 2.6 and 3.8. Since the problem asks for \( |x - 3| < \delta \), we can infer the distance from \( x = 3 \) to these boundary points.

### Solution:
- The distance from 3 to 2.6 is \( 3 - 2.6 = 0.4 \).
- The distance from 3 to 3.8 is \( 3.8 - 3 = 0.8 \).

Therefore, \( \delta \) can be the smaller of these two values, which is \( \delta = 0.4 \).

Thus, \( \delta = 0.4 \).

Would you like a deeper explanation of how this ties into the epsilon-delta definition of limits?

### Related questions:
1. What is the epsilon-delta definition of limits?
2. How do you determine delta given epsilon in a limit problem?
3. Why is \( \delta = 0.4 \) chosen instead of 0.8 in this case?
4. How do continuity and limits relate to each other in calculus?
5. How can graphing help to solve limit problems visually?

**Tip:** In epsilon-delta limit problems, always focus on the range for \( f(x) \) (based on epsilon) and find the corresponding interval for \( x \) around the given point.

Use the given graph of f to find a number δ such that if 0 < |x - 3| < δ then |f(x) - 2| < 0.5.

Use the epsilon-delta definition of limits to find δ based on a graph of f(x). Learn how to apply this concept visually to determine where |f(x) - 2| < 0.5 for |x - 3| < δ.

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