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.