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Use the given graph of f to find a number 𝛿 such that if 
|x − 1| < 𝛿
 then 
|f(x) − 1| < 0.2.  𝛿 =

From the given graph, we need to find a value of \( \delta \) such that if \( |x - 1| < \delta \), then \( |f(x) - 1| < 0.2 \).

### Key conditions to understand:
1. \( |f(x) - 1| < 0.2 \) means that the function value \( f(x) \) must lie between 0.8 and 1.2 (since 1 - 0.2 = 0.8 and 1 + 0.2 = 1.2).
2. \( |x - 1| < \delta \) defines the range around \( x = 1 \) such that the function values satisfy the condition for \( f(x) \).

### Steps:
1. Observe the graph around \( x = 1 \). We can see the behavior of \( f(x) \) as it moves away from \( x = 1 \).
2. When \( x = 1 \), \( f(x) = 1 \), so we are looking for how far \( x \) can deviate from 1 such that \( f(x) \) stays within the bounds 0.8 and 1.2.

From the graph:
- At \( x \approx 0.9 \), \( f(x) \) is close to 1.2.
- At \( x \approx 1.1 \), \( f(x) \) is close to 0.8.

Thus, for \( f(x) \) to stay between 0.8 and 1.2, \( \delta \) is approximately 0.1, since \( |x - 1| < 0.1 \) keeps \( f(x) \) within the desired range.

### Conclusion:
\( \delta = 0.1 \).

Let me know if you have further questions or need more details! Here are some related questions to explore:
1. How would the value of \( \delta \) change if the condition were \( |f(x) - 1| < 0.1 \)?
2. What is the formal definition of the epsilon-delta criterion in limits?
3. Can you calculate the slope of the curve \( f(x) \) at \( x = 1 \)?
4. How would you find the value of \( \delta \) if the graph was more complex?
5. How do we use limits to formally prove continuity?

**Tip**: When dealing with limit problems, always check the graph for critical points where the function crosses specific bounds, as this helps in determining appropriate \( \delta \) values.

Use the given graph of f to find a number 𝛿 such that if |x − 1| < 𝛿 then |f(x) − 1| < 0.2. 𝛿 = ?

Learn how to find δ such that |x − 1| < δ implies |f(x) − 1| < 0.2 using the epsilon-delta criterion. This solution uses graphical analysis to determine the value of δ for the given function f(x) and explains the epsilon-delta definition of limits.

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