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The problem asks us to use the graph of \( f(x) = \sqrt{x} \) to find a value of \( \delta \) such that \( |\sqrt{x} - 2| < 0.4 \) whenever \( |x - 4| < \delta \).

### Steps to solve:

1. **Understanding the question**:
   - The condition \( |\sqrt{x} - 2| < 0.4 \) means the value of \( f(x) = \sqrt{x} \) is close to 2 (the function's value at \( x = 4 \), since \( \sqrt{4} = 2 \)).
   - We need to find a range for \( x \) around \( x = 4 \) such that the function value \( f(x) \) remains within 0.4 units of 2. 
   - Specifically, we want \( \delta \) such that when \( |x - 4| < \delta \), we get \( |\sqrt{x} - 2| < 0.4 \).

2. **Interpreting the graph**:
   - From the graph, we observe points where \( f(x) = \sqrt{x} \). The graph indicates the behavior of the function for different values of \( x \).
   - From the second image (or based on the graph interpretation), we see that \( \sqrt{4} = 2 \), and the goal is to find where \( f(x) \) stays between 1.6 and 2.4 (i.e., \( 2 \pm 0.4 \)).

3. **Finding the interval for \( x \)**:
   - The points on the graph corresponding to \( f(x) = 1.6 \) and \( f(x) = 2.4 \) help us identify the range of \( x \) values.
   - From the graph, it appears that \( f(x) = 1.6 \) when \( x = 2 \), and \( f(x) = 2.4 \) when \( x = 6 \).

4. **Conclusion**:
   - This means \( f(x) \) lies between 1.

Use the given graph of f(x) = √x to find a number δ such that |√x - 2| < 0.4 whenever |x - 4| < δ.

Explore how to determine the δ-value for the function f(x) = √x such that the function value remains within 0.4 units of 2 when x is close to 4. This analysis helps understand the behavior of square root functions and is suitable for high school level students learning limits.

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