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This graph asks to find a number \( \delta \) such that for \( 0 < |x - 3| < \delta \), it satisfies the condition \( |f(x) - 2| < 0.5 \). This problem deals with the concept of limits, specifically the \(\epsilon-\delta\) definition of a limit.

### Step-by-step approach:

1. **Understanding the condition \( |f(x) - 2| < 0.5 \)**:
   - This inequality implies that the function values \( f(x) \) should stay within the range \( 1.5 < f(x) < 2.5 \) (since \( 2 - 0.5 = 1.5 \) and \( 2 + 0.5 = 2.5 \)).

2. **Identifying the \(x\)-range for \( f(x) \) in \( 1.5 < f(x) < 2.5 \)**:
   - Looking at the graph, the function reaches the value \( f(x) = 2 \) at \( x = 3 \).
   - From the graph, observe that when \( f(x) \) is between 1.5 and 2.5, the corresponding \( x \)-values seem to range from approximately \( 2.6 \) to \( 3.8 \).

3. **Determining \( \delta \)**:
   - The condition \( 0 < |x - 3| < \delta \) means that the value of \( x \) is within a distance \( \delta \) from 3.
   - From the graph, we see that the function values are between 1.5 and 2.5 when \( x \) is between 2.6 and 3.8.
   - Therefore, \( \delta \) would be the smallest of the distances from 3 to 2.6 or 3 to 3.8.
   - The distances are \( |3 - 2.6| = 0.4 \) and \( |3 - 3.8| = 0.8 \). Hence, the smallest \( \delta \) is \( 0.4 \).

### Conclusion:
The value of \( \delta \) that satisfies the condition is \( \delta = 0.4 \).

Would you like a more detailed breakdown of the concept or have any further questions? 

Here are five relative questions that can expand your understanding:
1. How does the \(\epsilon-\delta\) definition of limits work in general?
2. What is the significance of the \(\epsilon\) value in these problems?
3. How would the result change if we asked for \( |f(x) - 2| < 0.1 \)?
4. What are the key differences between left-hand and right-hand limits?
5. How do we handle \(\epsilon-\delta\) problems where the function is not continuous?

**Tip**: Always visualize or sketch the neighborhood of the limit point when dealing with \(\epsilon-\delta\) problems—it can greatly help with interpreting the conditions.

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

Learn how to find δ in an epsilon-delta limit problem using a graph of f(x). Understand the concept of limits and how to interpret function behavior to satisfy the condition |f(x) - L| < ε.

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