https://file.aiquickdraw.com/mathbot/file/17343811392204awr19mx.jpg

To solve this problem and determine a value of **δ** that satisfies the given condition:

### Given:
1. \(\lim_{x \to 2} f(x) = 2\).
2. The condition: If \( 0 < |x - 2| < \delta \), then \( |f(x) - 2| < 1 \).

We need to find a **δ** such that this condition holds true.

---

### Steps:
1. The graph shows \(f(x)\) as increasing and passing through \( (2, 2) \), and the function is continuous.
2. The inequality \( |f(x) - 2| < 1 \) means that \( f(x) \) must be between \( 1 \) and \( 3 \):
   \[
   1 < f(x) < 3.
   \]
3. We now examine the graph around \(x = 2\) to determine where \(f(x)\) remains in this range.

From the graph:
- When \(x\) is close to \(2\), \(f(x)\) remains between \(1\) and \(3\) approximately within the interval \(1 \leq x \leq 3\).

Thus, the distance \( |x - 2| \) must satisfy:
\[
0 < |x - 2| < 1.
\]

---

### Solution:
\[
\delta \leq 1.
\]

---

### Final Answer:
\[
\delta \leq 1.
\]

Would you like a detailed explanation of the epsilon-delta concept used here? Let me know!

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### Related Questions:
1. What is the definition of a limit using the epsilon-delta approach?
2. How does the graph of a function affect the choice of delta (\(\delta\))?
3. Can the value of \(\delta\) vary for different values of epsilon (\(\epsilon\))?
4. What happens if a function is discontinuous at \(x = 2\)?
5. How can we generalize this approach for a more complex function?

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### Tip:
When working with limits and the epsilon-delta definition, always look closely at the behavior of the function near the limit point!

The graph of the function f satisfies lim(x→2)f(x)=2. Determine a value of δ>0 that satisfies: If 0 < |x-2| < δ, then |f(x)-2| < 1.

Solve a limit problem involving the epsilon-delta definition for the function where lim(x→2)f(x)=2. Determine δ such that if 0 < |x-2| < δ, then |f(x)-2| < 1. Step-by-step solution and explanation included.

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