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Let's solve the problem step by step.

### Part 1: Find the Limit
The function given is \( f(x) = -5x + 3 \). We need to find:
\[
\lim_{x \to 2} f(x)
\]
Since \( f(x) \) is a linear function, it's continuous everywhere, so we can find the limit by direct substitution:
\[
\lim_{x \to 2} f(x) = f(2) = -5(2) + 3 = -10 + 3 = -7
\]
Thus, the real number \( L \) is \( -7 \).

### Part 2: Prove the Equality Using an \( \varepsilon \)-\( \delta \) Proof
To prove \( \lim_{x \to 2} f(x) = -7 \) using an \( \varepsilon \)-\( \delta \) proof, we proceed as follows:

1. **Definition of the Limit**: We need to show that for every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x - 2| < \delta \), then \( |f(x) - (-7)| < \varepsilon \).

2. **Set Up the Inequality**:
   \[
   |f(x) - (-7)| = |(-5x + 3) + 7| = |-5x + 10|
   \]
   
   We can factor out the \( -5 \):
   \[
   |-5x + 10| = 5|x - 2|
   \]

3. **Choosing \( \delta \)**:
   We want \( 5|x - 2| < \varepsilon \). To make this true, set:
   \[
   |x - 2| < \frac{\varepsilon}{5}
   \]
   
   Therefore, let \( \delta = \frac{\varepsilon}{5} \). This choice ensures that if \( 0 < |x - 2| < \delta \), then:
   \[
   |f(x) - (-7)| = 5|x - 2| < 5 \cdot \frac{\varepsilon}{5} = \varepsilon
   \]

4. **Conclusion**:
   By the definition of the limit, if \( 0 < |x - 2| < \delta \), then \( |f(x) - (-7)| < \varepsilon \), confirming that:
   \[
   \lim_{x \to 2} f(x) = -7
   \]

This completes the \( \varepsilon \)-\( \delta \) proof.

Would you like further details or explanations on any step?

### Relative Questions
1. How do you find the limit of a polynomial function?
2. What makes a function continuous, and how does it relate to limits?
3. How do you approach proving a limit for more complex functions using the \( \varepsilon \)-\( \delta \) method?
4. Can you explain how to choose \( \delta \) when dealing with non-linear functions?
5. Why is it important that \( \delta \) depends on \( \varepsilon \) in an \( \varepsilon \)-\( \delta \) proof?

### Tip
When proving limits using \( \varepsilon \)-\( \delta \), focus on manipulating the absolute value expression to isolate \( |x - c| \). This step is crucial in finding a suitable \( \delta \).

Let f(x) = -5x + 3. 1. Find the real number L such that lim(x→2) f(x) = L. 2. Prove the equality from part (1) using an epsilon-delta proof.

Understand how to solve the limit problem for the function f(x) = -5x + 3 at x = 2 using an epsilon-delta proof. This step-by-step guide covers key concepts in calculus, including the definition of a limit, continuity, and epsilon-delta techniques. Suitable for college-level calculus students.

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