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We are given the limit:

\[
\lim_{x \to 3} \left(\frac{1}{3}x - 9\right) = -8
\]

To prove this formally, we need to express \(\delta\) as a function of \(\epsilon\). Here's the step-by-step process to find \(\delta\).

### Step 1: Understand the definition of a limit
For the limit to hold, for every \(\epsilon > 0\), there must be a \(\delta > 0\) such that whenever \(0 < |x - 3| < \delta\), it follows that \(\left| \left(\frac{1}{3}x - 9\right) + 8 \right| < \epsilon\).

### Step 2: Manipulate the function
Start with the expression:

\[
\left|\left(\frac{1}{3}x - 9\right) + 8\right| = \left|\frac{1}{3}x - 1\right|
\]

We want to make this less than \(\epsilon\), so:

\[
\left|\frac{1}{3}x - 1\right| < \epsilon
\]

### Step 3: Solve for \(\delta\)
We can manipulate the inequality:

\[
\left|\frac{1}{3}x - 1\right| = \frac{1}{3}|x - 3| < \epsilon
\]

Multiply both sides by 3:

\[
|x - 3| < 3\epsilon
\]

Thus, we can choose \(\delta = 3\epsilon\).

### Final Answer:
\[
\delta = 3 \cdot \text{epsilon}
\]

---

Would you like more detailed steps or further clarification on any part of this solution?

Here are 5 relative questions to expand your understanding:
1. How does the definition of limits work when proving limits using \(\epsilon\)-\(\delta\)?
2. How do you manipulate inequalities when solving for \(\delta\) in limit problems?
3. Why is multiplying by 3 important in the last step of solving for \(\delta\)?
4. Can you explain the role of absolute value in \(\epsilon\)-\(\delta\) proofs?
5. What does it mean for a function to approach a limit as \(x\) tends to a certain value?

**Tip:** Always check the form of the function you're working with before applying the \(\epsilon\)-\(\delta\) definition to ensure you manipulate it correctly.

In formally proving that lim x→3 ((1/3)x − 9) = −8, let ε > 0 be arbitrary. Determine δ as a function of ε.

This problem demonstrates how to formally prove the limit lim x→3 ((1/3)x − 9) = −8 using the epsilon-delta definition. Step-by-step guidance is provided to find δ as a function of ε, suitable for high school students studying calculus.

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