We are asked to prove the limit:

$\lim_{x \to 7} (3x - 17) = 4$

Preliminary analysis

We need to prove the limit using the formal $\epsilon$-$\delta$ definition of a limit. Specifically, given any $\epsilon > 0$, we want to find a corresponding $\delta > 0$ such that if $0 < |x - 7| < \delta$, then:

$| (3x - 17) - 4 | < \epsilon$

Step 1: Simplify the expression $|(3x - 17) - 4|$

We start by simplifying the expression inside the absolute value:

$| (3x - 17) - 4 | = | 3x - 21 |$

Factoring out a 3 from the absolute value:

$| 3x - 21 | = 3 |x - 7|$

Step 2: Set the inequality $|3(x - 7)| < \epsilon$

Now, for the limit to hold, we need:

$3 |x - 7| < \epsilon$

Dividing both sides of the inequality by 3:

$|x - 7| < \frac{\epsilon}{3}$

This suggests that we should choose $\delta = \frac{\epsilon}{3}$.

Proof

To formally prove the limit, we proceed as follows:

Let $\epsilon > 0$ be given. Choose $\delta = \frac{\epsilon}{3}$. Now, suppose $0 < |x - 7| < \delta$. Then:

$|x - 7| < \delta = \frac{\epsilon}{3}$

Multiplying both sides by 3:

$3 |x - 7| < 3 \times \frac{\epsilon}{3} = \epsilon$

Thus, we have:

$|3x - 21| = 3 |x - 7| < \epsilon$

Therefore, if $0 < |x - 7| < \delta$, then $|(3x - 17) - 4| = |3x - 21| < \epsilon$.

This confirms that the limit holds, and by the definition of a limit, we conclude:

$\lim_{x \to 7} (3x - 17) = 4$

Do you need any further details or have any questions?

Here are some related questions to expand on this topic:

1. How does the $\epsilon$-$\delta$ definition of limits apply to nonlinear functions?
2. Can you explain the geometrical interpretation of the $\epsilon$-$\delta$ definition of limits?
3. How would the $\epsilon$-$\delta$ proof change if the function were quadratic instead of linear?
4. What is the difference between one-sided and two-sided limits in terms of $\epsilon$-$\delta$?
5. How can you use $\epsilon$-$\delta$ limits to prove the continuity of a function at a point?

Tip: For any linear function $f(x) = ax + b$, the $\epsilon$-$\delta$ proof follows a similar pattern, where $\delta$ is typically $\frac{\epsilon}{|a|}$.