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 $\frac{49}{64}$.

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.