It seems like you're describing a limit problem, possibly involving the epsilon-delta definition of a limit. The setup implies that the function in question involves square roots, and you want to show that the function's value gets arbitrarily close to a limit \( L \) as \( x \) approaches a certain value.

Let's break it down:

### Epsilon-Delta Definition:
The epsilon-delta definition of a limit states that for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x - c| < \delta \), then \( |f(x) - L| < \epsilon \).

In this case:
- You seem to be dealing with a function that includes a square root, possibly \( f(x) = \sqrt{x} \).
- You want to prove that as \( x \) approaches a certain value, the function approaches a limit \( L \).

### Analyzing Your Statement:
1. **Bounds on \( x \):** You are given \( 9 < x < 25 \), which suggests you're interested in \( x \) values between 9 and 25.

2. **Delta Condition:** You choose \( \delta = 7\epsilon \), which means the delta depends linearly on epsilon. This is often done to make the expression manageable.

3. **Inequality Setup:**
   - You have \( 0 < |x - ( ? )| < \delta \), where the missing part likely indicates the point \( x \) is approaching.
   - The expression \( |x - ( ? )| < \epsilon \cdot | \sqrt{x} + 4| \cdot | \sqrt{x} - 4| \) suggests that you are working with the difference between two square roots, which simplifies into a factored form, possibly involving the difference of squares.

   Specifically:
   \[
   |f(x) - L| = |\sqrt{x} - 4| < \epsilon
   \]

   Here, \( L = 4 \) might be the limit you are approaching, and the goal is to show that \( \sqrt{x} \) gets close to 4 as \( x \) gets close to 16.

4. **Conclusion:** This suggests you're working to prove that \( \lim_{x \to 16} \sqrt{x} = 4 \).

### Next Steps:

- To complete this, you would determine \( \delta \) based on \( \epsilon \) to satisfy the inequality \( |\sqrt{x} - 4| < \epsilon \) whenever \( 0 < |x - 16| < \delta \).

Would you like further details or clarifications on any part of this explanation?

### 5 Related Questions:
1. How do you formally define the epsilon-delta definition of a limit?
2. What is the significance of choosing \( \delta \) in terms of \( \epsilon \) in the epsilon-delta proof?
3. How do you handle limits that involve square roots or more complex expressions?
4. What are some common techniques for solving epsilon-delta problems involving inequalities?
5. Can you provide an example of applying the epsilon-delta definition for a different function, like \( f(x) = x^2 \)?

### Tip:
When proving limits with the epsilon-delta definition, always try to express \( \delta \) in terms of \( \epsilon \) to create a direct relationship between how close \( x \) is to the target value and how close \( f(x) \) is to the limit.

Learn how to solve limit problems using the epsilon-delta definition. Explore a detailed example involving square roots and the proof that the limit of sqrt(x) as x approaches 16 is 4. Suitable for advanced mathematics students and those studying calculus.

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