To prove a precise definition of the limit statement:

$\lim_{x \to 2} (x^2 + 5x - 5) = 9,$

we'll use the epsilon-delta definition of a limit.

Epsilon-Delta Definition of a Limit:

For every $\varepsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - 2| < \delta$, then $|f(x) - 9| < \varepsilon$.

Here, the function is $f(x) = x^2 + 5x - 5$, and we are proving that as $x$ approaches 2, $f(x)$ approaches 9.

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Step-by-Step Proof:

1. Start with the inequality: $|f(x) - 9| < \varepsilon.$

   Plug $f(x) = x^2 + 5x - 5$ into the inequality: $|(x^2 + 5x - 5) - 9| = |x^2 + 5x - 14|.$

   We want to find $\delta$ such that for any $\varepsilon > 0$, if $|x - 2| < \delta$, then $|x^2 + 5x - 14| < \varepsilon$.

2. Factor the quadratic expression: $x^2 + 5x - 14 = (x - 2)(x + 7).$

   Thus, we need to make sure: $|(x - 2)(x + 7)| < \varepsilon.$

3. Find bounds for $|x + 7|$: Since we are taking $x \to 2$, we can assume that $x$ is close to 2. Let’s choose a small bound for $\delta$, say $\delta \leq 1$, so that when $x$ is within 1 unit of 2, we have: $|x - 2| < 1 \quad \text{which implies} \quad 1 < x < 3.$

   Then: $8 \leq x + 7 \leq 10,$ so $|x + 7|$ is bounded by 10. Therefore, we can estimate: $|(x - 2)(x + 7)| < 10|x - 2|.$

4. Choose $\delta$ based on $\varepsilon$: To make $|(x - 2)(x + 7)| < \varepsilon$, we need: $10|x - 2| < \varepsilon.$

   Dividing both sides by 10: $|x - 2| < \frac{\varepsilon}{10}.$

   Thus, we can choose $\delta = \frac{\varepsilon}{10}$.

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Conclusion:

For any $\varepsilon > 0$, if we choose $\delta = \frac{\varepsilon}{10}$, then whenever $|x - 2| < \delta$, it follows that $|f(x) - 9| < \varepsilon$. This proves that: $\lim_{x \to 2} (x^2 + 5x - 5) = 9.$

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Would you like further details on any step, or have any questions?

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Here are 5 related questions for deeper exploration:

1. How does the epsilon-delta definition apply to different types of functions (e.g., linear, cubic)?
2. Can the epsilon-delta method be used to prove limits at infinity?
3. What is the relationship between continuity and limits using the epsilon-delta definition?
4. How do you prove limits involving absolute value functions using epsilon-delta?
5. What are some common misconceptions when applying the epsilon-delta definition?

Tip: Always check the range of values around the point you're approaching to simplify the bounds on your expression!