https://file.aiquickdraw.com/mathbot/file/1726795789647aofjua3k.jpg

Prove the statement using the epsilon, delta definition of a limit

To prove the limit using the epsilon-delta definition, we need to demonstrate that for every \( \epsilon > 0 \), there exists a corresponding \( \delta > 0 \) such that:

\[
0 < |x - (-2)| < \delta \implies |(-2x + 1) - 5| < \epsilon
\]

### Step 1: Simplify the expression \( |(-2x + 1) - 5| \)

We start by simplifying the expression inside the absolute value:

\[
|(-2x + 1) - 5| = |-2x + 1 - 5| = |-2x - 4| = 2|x + 2|
\]

Thus, we have:

\[
|(-2x + 1) - 5| = 2|x + 2|
\]

### Step 2: Relate \( \epsilon \) and \( \delta \)

To make the expression \( |(-2x + 1) - 5| < \epsilon \), we now have:

\[
2|x + 2| < \epsilon
\]

Dividing both sides by 2:

\[
|x + 2| < \frac{\epsilon}{2}
\]

So, we can choose \( \delta = \frac{\epsilon}{2} \).

### Step 3: Conclusion

Given \( \delta = \frac{\epsilon}{2} \), we have shown that:

\[
0 < |x + 2| < \delta \implies |(-2x + 1) - 5| < \epsilon
\]

Thus, the limit is proven using the epsilon-delta definition.

### Would you like further details or clarification on any part of this proof?

Here are 5 related questions to expand your understanding:
1. How does the epsilon-delta definition apply to more complex functions?
2. What happens if the function is discontinuous at the limit point?
3. How can we extend this definition to limits approaching infinity?
4. What are the key differences between one-sided and two-sided limits?
5. How would you prove this limit if it was a piecewise function?

**Tip:** When using the epsilon-delta definition, simplifying the absolute value expression is a crucial step to finding the right \( \delta \).

Learn how to prove limits using the epsilon-delta definition with this example: lim (x -> -2) of (-2x + 1) = 5. Step-by-step guide, simplifying absolute value expressions, and finding δ in terms of ε.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation