It seems you're referencing a proof related to the continuity of a function $f(x) = 3x + 5$ at $x = 5$, using the epsilon-delta definition of continuity. Let me clarify the key steps involved, and how to correctly write out the proof.

Problem:

We want to show that the function $f(x) = 3x + 5$ is continuous at $x = 5$ using the epsilon-delta definition of continuity.

Solution Outline:

To prove that $f(x) = 3x + 5$ is continuous at $x = 5$, we use the epsilon-delta definition of continuity: $\lim_{x \to 5} f(x) = f(5).$ This is equivalent to showing that for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $|x - 5| < \delta$, then $|f(x) - f(5)| < \epsilon$.

Step-by-Step Solution:

1. Define the function: The function is given by $f(x) = 3x + 5$, and we want to show that it is continuous at $x = 5$.

2. Compute $f(5)$: $f(5) = 3(5) + 5 = 15 + 5 = 20.$

3. Consider the difference $|f(x) - f(5)|$: We need to analyze the expression: $|f(x) - f(5)| = |(3x + 5) - 20| = |3x - 15| = 3|x - 5|.$ So, we have: $|f(x) - f(5)| = 3|x - 5|.$

4. Set up the condition: We want $|f(x) - f(5)| < \epsilon$. From the previous step, we know: $3|x - 5| < \epsilon.$ Dividing both sides by 3: $|x - 5| < \frac{\epsilon}{3}.$

5. Choose $\delta$: To satisfy the condition $|x - 5| < \delta$ whenever $|f(x) - f(5)| < \epsilon$, we set: $\delta = \frac{\epsilon}{3}.$ This ensures that if $|x - 5| < \delta$, then $|f(x) - f(5)| < \epsilon$.

6. Conclusion: We have shown that for every $\epsilon > 0$, there exists $\delta = \frac{\epsilon}{3}$ such that if $|x - 5| < \delta$, then $|f(x) - f(5)| < \epsilon$. Therefore, $f(x) = 3x + 5$ is continuous at $x = 5$.

Summary:

The function $f(x) = 3x + 5$ is continuous at $x = 5$, and the epsilon-delta argument shows that for any $\epsilon > 0$, we can find a corresponding $\delta = \frac{\epsilon}{3}$ such that $|f(x) - f(5)| < \epsilon$ when $|x - 5| < \delta$.

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Would you like any further clarification on the steps, or any specific questions about the method used? Here are a few related questions to explore more:

1. How does the epsilon-delta definition of continuity work in other types of functions?
2. What is the geometric interpretation of the epsilon-delta proof for continuity?
3. How does the proof change for more complex functions (e.g., quadratic or piecewise functions)?
4. What happens if we try to prove continuity at a point where the function is not continuous?
5. Can you prove the continuity of $f(x) = x^2$ at $x = 3$ using a similar method?

Tip: For functions that are simple polynomials (like $f(x) = 3x + 5$), the proof is often straightforward. But for more complicated functions, always start by simplifying the expression $|f(x) - f(c)|$ before attempting to find $\delta$.