https://file.aiquickdraw.com/mathbot/file/1725925399319d62gww82.jpg

The problem involves finding the largest value of \( \delta \) such that for all \( x \), the inequality \( |f(x) - L| < \epsilon \) holds whenever \( 0 < |x - c| < \delta \).

From the image, the key details are:
- Function \( f(x) = \sqrt{x} \).
- \( L = 3 \).
- \( c = 9 \).
- \( \epsilon = \frac{1}{8} \).

### Steps to solve the problem:

1. **Understanding the conditions:**
   The goal is to find \( \delta \) such that \( |f(x) - L| < \epsilon \) when \( 0 < |x - c| < \delta \).

2. **Apply the function:**
   Given \( f(x) = \sqrt{x} \), \( L = 3 \), and \( c = 9 \), we know that:
   \[
   |f(x) - L| = |\sqrt{x} - 3|
   \]
   The inequality becomes:
   \[
   |\sqrt{x} - 3| < \frac{1}{8}
   \]

3. **Solve the inequality:**
   For \( |\sqrt{x} - 3| < \frac{1}{8} \), square both sides to eliminate the square root:
   \[
   \left(\sqrt{x} - 3\right)^2 < \left(\frac{1}{8}\right)^2
   \]
   Simplify the inequality:
   \[
   0 < x - 9 < \delta
   \]
   Based on the graph, \( x \) ranges from \( \frac{529}{64} \) to \( \frac{625}{64} \).

4. **Find the largest \( \delta \):**
   The difference between the bounds of \( x \) gives \( \delta = \frac{625}{64} - 9 \).

   Simplifying \( \delta \):
   \[
   \delta = \frac{625}{64} - \frac{576}{64} = \frac{49}{64}
   \]

Hence, the largest value of \( \delta \) is \( \frac{49}{64} \).

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If you have any questions or need further clarification, feel free to ask!

### Related Questions:
1. How do you determine \( \epsilon \) in a general limit problem?
2. What role does \( \delta \) play in ensuring a limit condition is met?
3. How do you interpret the graph of \( f(x) \) in relation to \( \delta \) and \( \epsilon \)?
4. Can the value of \( \delta \) be adjusted based on different values of \( \epsilon \)?
5. What is the significance of choosing the correct value for \( c \) in a limit problem?

### Tip:
Always double-check your calculations, especially when working with fractions or square roots, to avoid small mistakes that could lead to incorrect results.

Use the graph to find the largest value of delta such that for all x, |f(x) - L| < epsilon whenever 0 < |x - c| < delta.

Solve a limit problem by finding the largest value of delta using a graph. The function involves square roots and inequalities, with the given function f(x) = sqrt(x), L = 3, c = 9, and epsilon = 1/8. Learn how to calculate the largest delta step by step.

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