when demonstrating that lim(x->2) 2x+2=6 with e =0.2 which of the following values suffices

To demonstrate that \(\lim_{x \to 2} (2x + 2) = 6\) with \(\epsilon = 0.2\), we need to find a value of \(\delta\) that satisfies the definition of a limit:

\[
\text{For all } x, \text{ if } 0 < |x - 2| < \delta, \text{ then } |(2x + 2) - 6| < 0.2.
\]

### Step-by-step:

1. **Start with the inequality**:
   \[
   |(2x + 2) - 6| < 0.2.
   \]

2. **Simplify**:
   \[
   |2x + 2 - 6| = |2x - 4| = 2|x - 2|.
   \]

   So we need:
   \[
   2|x - 2| < 0.2.
   \]

3. **Solve for \(|x - 2|\)**:
   \[
   |x - 2| < \frac{0.2}{2} = 0.1.
   \]

Thus, a value of \(\delta = 0.1\) suffices to ensure that \(|(2x + 2) - 6| < 0.2\).

### Conclusion:
For \(\epsilon = 0.2\), \(\delta = 0.1\) is sufficient to demonstrate that \(\lim_{x \to 2} (2x + 2) = 6\).

---

Do you want further details or have any questions? 

Here are 5 related questions:

1. How would you find \(\delta\) for a different \(\epsilon\) value, such as \(\epsilon = 0.1\)?
2. Can you explain how the formal definition of a limit works?
3. What happens if you use a larger \(\delta\) value, like \(\delta = 0.2\)?
4. How can this limit be visualized on a graph of the function \(f(x) = 2x + 2\)?
5. What are the key steps in finding \(\delta\) for any general limit problem?

**Tip**: Always simplify the inequality after substituting the limit into the function to find \(\delta\).

Explore a step-by-step guide to proving lim(x→2)(2x + 2) = 6 using the epsilon-delta definition for ε = 0.2. Learn key mathematical concepts including limits and the formal epsilon-delta limit theorem. Suitable for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation