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Let's break down the problem step-by-step:

### Part (a): Precise Definition of a Limit
The precise (epsilon-delta) definition of a limit states:
- "We say that \(\lim_{x \to a} f(x) = L\) if for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that \(0 < |x - a| < \delta\) implies \(|f(x) - L| < \varepsilon\)."

Filling in the blanks:
- **Blank 1**: \(0 < |x - a| < \delta\)
- **Blank 2**: \(|f(x) - L| < \varepsilon\)

So, the complete statement should be:
"We say \(\lim_{x \to a} f(x) = L\) if for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that \(0 < |x - a| < \delta\) whenever \(|f(x) - L| < \varepsilon\)."

### Part (b): Proving the Limit Using the Epsilon-Delta Definition
Now, let's prove that \(\lim_{x \to 3} (5 - 2x) = -1\) using the epsilon-delta definition.

1. **Given:**
   - \(f(x) = 5 - 2x\)
   - \(a = 3\)
   - \(L = -1\)

2. **Epsilon-Delta Definition:** We need to show that for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < |x - 3| < \delta\), then \(|(5 - 2x) - (-1)| < \varepsilon\).

3. **Simplify the Expression Inside the Absolute Value:**
   \[
   |(5 - 2x) - (-1)| = |5 - 2x + 1| = |6 - 2x|
   \]

4. **Factor Out 2:**
   \[
   |6 - 2x| = 2|3 - x|
   \]

5. **Set Up the Inequality:**
   We need \(2|3 - x| < \varepsilon\). Divide both sides by 2:
   \[
   |3 - x| < \frac{\varepsilon}{2}
   \]

6. **Choose \(\delta\):**
   Let \(\delta = \frac{\varepsilon}{2}\). This choice ensures that whenever \(0 < |x - 3| < \delta\), we have:
   \[
   |6 - 2x| = 2|3 - x| < 2 \times \frac{\varepsilon}{2} = \varepsilon
   \]

7. **Conclusion:**
   Therefore, by the epsilon-delta definition of a limit, \(\lim_{x \to 3} (5 - 2x) = -1\).

Would you like more details or have any questions about this proof?

### Relative Questions:
1. How do you choose \(\delta\) in more complex functions for the epsilon-delta proof?
2. What does the epsilon-delta definition tell us about the continuity of a function?
3. Can the limit of a function exist at a point where the function is not defined?
4. How does the epsilon-delta definition relate to the graphical interpretation of limits?
5. What are the implications if we cannot find a suitable \(\delta\) for a given \(\varepsilon\)?

### Tip:
Always start by simplifying the expression \(|f(x) - L|\) when using the epsilon-delta definition. This will guide you in choosing the appropriate \(\delta\).

Complete the precise definition of a limit and prove using the epsilon-delta definition that the limit of (5 - 2x) as x approaches 3 is -1.

Explore the epsilon-delta definition of limits with a detailed solution proving that the limit of (5 - 2x) as x approaches 3 is -1. This page covers core calculus concepts and includes step-by-step explanations to enhance understanding.

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