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We are given the graph of the function \( f(x) = \sqrt{x} \) and asked to find a number \( \delta \) such that if \( |x - 4| < \delta \), then \( |\sqrt{x} - 2| < 0.4 \).

### Steps to solve this problem:
1. We know from the problem that:
   - \( \sqrt{4} = 2 \)
   - The inequality \( |\sqrt{x} - 2| < 0.4 \) means we want \( \sqrt{x} \) to stay within the range \( (1.6, 2.4) \) since \( 2 - 0.4 = 1.6 \) and \( 2 + 0.4 = 2.4 \).

2. Using the graph, we can check where \( y = \sqrt{x} \) takes values close to 1.6 and 2.4:
   - On the graph, for \( \sqrt{x} = 2.4 \), the corresponding \( x \)-value is about 5.76.
   - For \( \sqrt{x} = 1.6 \), the corresponding \( x \)-value is about 2.56.

3. Now, we need to find the difference between these \( x \)-values and 4 (since we are focusing on \( |x - 4| < \delta \)):
   - \( 5.76 - 4 = 1.76 \)
   - \( 4 - 2.56 = 1.44 \)

4. The smallest of these differences will be the value of \( \delta \), as we want to ensure the inequality holds for the entire interval. Thus, \( \delta = 1.44 \).

### Conclusion:
The value of \( \delta \) is approximately 1.44.

Would you like further clarification or additional details on this problem? Here are five related questions you might find helpful:

1. How does the choice of \( \delta \) ensure the function values stay within the specified range?
2. Can we derive the same \( \delta \) without using the graph, just algebraically?
3. What does this problem teach us about continuity and limits?
4. How would the solution change if the inequality \( |\sqrt{x} - 2| < 0.5 \) was given instead?
5. What is the general procedure for finding \( \delta \) in problems involving limits and square root functions?

**Tip**: When analyzing limit problems graphically, always check for where the function value deviates by the given amount, and then relate it back to the \( x \)-values.

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

Learn how to find a δ such that |x - 4| < δ ensures |√x - 2| < 0.4 using graphical analysis. This problem focuses on limit concepts, inequalities, and square root functions. Suitable for students in Grades 11-12 studying calculus.

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