Prove the precise definition for each limit statement lim(x^2+5x-5)=9

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.

---

### **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 \(\delta = \frac{\varepsilon}{10}\).

---

### **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:
\[
\lim_{x \to 2} (x^2 + 5x - 5) = 9.
\]

---

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!

Learn how to prove the limit of the quadratic function lim(x^2 + 5x - 5) = 9 using the epsilon-delta definition. This detailed guide walks you through each step, including factorization and bounding values. Ideal for students in Grades 11-12 or those beginning calculus.

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