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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 = 

To solve this problem, we need to find a number \(\delta\) such that:

\[
|x - 1| < \delta \quad \text{implies} \quad |f(x) - 1| < 0.2.
\]

### Steps:
1. **Identify the condition**: The condition \( |f(x) - 1| < 0.2 \) means that the values of \( f(x) \) must lie within the interval \( (1 - 0.2, 1 + 0.2) = (0.8, 1.2) \).

2. **Check the corresponding values of \(x\)**: From the graph, we need to find the \(x\)-values such that \(f(x)\) lies within this range. That is, we want to find the range of \(x\) where \( 0.8 \leq f(x) \leq 1.2 \).

   - From the graph, we observe that:
     - When \( x = 0.7 \), \( f(x) \approx 1.2 \).
     - When \( x = 1.1 \), \( f(x) \approx 0.8 \).

3. **Calculate \(\delta\)**: The value of \(\delta\) is the distance from \(x = 1\) to the boundary points of the interval \([0.7, 1.1]\).
   - So, \(\delta\) is the minimum of \( |1 - 0.7| \) and \( |1 - 1.1| \).
   - \( |1 - 0.7| = 0.3 \) and \( |1 - 1.1| = 0.1 \).

Thus, the smallest \(\delta\) that satisfies the condition is:

\[
\delta = 0.1.
\]

Would you like a more detailed explanation, or do you have any questions?

### Related Questions:
1. What does the \(\delta\)-\(\epsilon\) definition of a limit mean?
2. How can we find the value of \(\delta\) for other limits?
3. What happens if \(\epsilon\) is made smaller?
4. How do graphical approaches differ from algebraic approaches in limit problems?
5. Can we use this approach for discontinuous functions?

### Tip:
Always carefully examine the range of values on both axes when solving limit problems graphically. This will help you determine intervals for both \(x\) and \(f(x)\).

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 a delta (𝛿) value for the limit of a function using the epsilon-delta definition based on a graph. This problem involves limits, graphical analysis, and continuity for students in Grades 11-12 or early college level.

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