https://file.aiquickdraw.com/mathbot/file/17267368232830hwvfruq.jpg

The image you uploaded contains calculus exercises, particularly focusing on limits and the \(\varepsilon\)-\(\delta\) definition of a limit. Let's tackle problem 14, which asks you to:

"Given that \(\lim_{{x \to 2}} (5x - 7) = 3\), illustrate Definition 2 by finding \(\delta\)'s that correspond to \(\varepsilon = 0.1\), \(\varepsilon = 0.05\), and \(\varepsilon = 0.01\)."

### Solution:
We are dealing with the formal \(\varepsilon\)-\(\delta\) definition of a limit, which states that:

\[
\lim_{{x \to c}} f(x) = L
\]

if, for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < |x - c| < \delta\), then \(|f(x) - L| < \varepsilon\).

Here, \(f(x) = 5x - 7\), \(L = 3\), and \(c = 2\). To apply this, we need to find \(\delta\) for the given values of \(\varepsilon\).

1. **Start with** \(|f(x) - L| < \varepsilon\):
   \[
   |(5x - 7) - 3| < \varepsilon
   \]
   Simplify:
   \[
   |5x - 10| < \varepsilon
   \]
   Factor:
   \[
   5|x - 2| < \varepsilon
   \]
   Divide by 5:
   \[
   |x - 2| < \frac{\varepsilon}{5}
   \]

2. **Find the corresponding \(\delta\):**
   From the inequality, we can see that \(\delta = \frac{\varepsilon}{5}\).

3. **Calculate \(\delta\) for each \(\varepsilon\):**
   - For \(\varepsilon = 0.1\), \(\delta = \frac{0.1}{5} = 0.02\).
   - For \(\varepsilon = 0.05\), \(\delta = \frac{0.05}{5} = 0.01\).
   - For \(\varepsilon = 0.01\), \(\delta = \frac{0.01}{5} = 0.002\).

Thus, the corresponding \(\delta\) values are:
- \(\varepsilon = 0.1\) \(\Rightarrow \delta = 0.02\)
- \(\varepsilon = 0.05\) \(\Rightarrow \delta = 0.01\)
- \(\varepsilon = 0.01\) \(\Rightarrow \delta = 0.002\)

---

Would you like more details or clarifications on this explanation? Here are five follow-up questions to expand your understanding:

1. What would change in the solution if the limit involved a quadratic function instead of linear?
2. How do you geometrically interpret the \(\varepsilon\)-\(\delta\) definition of a limit?
3. How would you prove a limit does not exist using the \(\varepsilon\)-\(\delta\) method?
4. What happens if \(\delta\) depends on both \(\varepsilon\) and \(x\)?
5. Can you apply this approach to limits approaching infinity?

### Tip:
When solving \(\varepsilon\)-\(\delta\) limit problems, always manipulate \(|f(x) - L|\) to find an expression involving \(|x - c|\) and then relate this to \(\varepsilon\).

Given that \(\lim_{{x \to 2}} (5x - 7) = 3\), illustrate the \(\varepsilon\)-\(\delta\) definition of a limit by finding the corresponding \(\delta\)'s for \(\varepsilon = 0.1\), \(\varepsilon = 0.05\), and \(\varepsilon = 0.01\).

Learn how to illustrate the \(\varepsilon\)-\(\delta\) definition of limits with a detailed example involving the function \(f(x) = 5x - 7\), limit \(\lim_{{x \to 2}} f(x) = 3\), and calculating corresponding \(\delta\)'s for different \(\varepsilon\) values.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation