A mistake includes but is not limited to a false statement; a statement that is not a logic consequence of its proceeding statement; a statement that is not usefull for or related to the final results. Prove lim,-2 3x2 – - x = 14. Proof for e > 0, we need 3x2 – x – 14| < e. This is |(x +2)(3.x – 7)|< e SO |æ + 2| < |3x – 7| |3x| + 7 Therefore, we just need to pick 8 = $, then when |x + 2| < 8 = , we have |(x + 2)(3x – 7)|< e whch shows the limit=14.
A mistake includes but is not limited to a false statement; a statement that is not a logic consequence of its proceeding statement; a statement that is not usefull for or related to the final results. Prove lim,-2 3x2 – - x = 14. Proof for e > 0, we need 3x2 – x – 14| < e. This is |(x +2)(3.x – 7)|< e SO |æ + 2| < |3x – 7| |3x| + 7 Therefore, we just need to pick 8 = $, then when |x + 2| < 8 = , we have |(x + 2)(3x – 7)|< e whch shows the limit=14.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Understanding Limits: Proving a Limit Example
#### Introduction
In calculus, understanding and proving limits are fundamental skills. Here, we'll go through the process of proving a specific limit:
**Prove**
\[
\lim_{x \to -2} (3x^2 - x) = 14.
\]
#### Proof
For any \(\epsilon > 0\), we need to show that:
\[
|3x^2 - x - 14| < \epsilon.
\]
This equation can be rewritten as:
\[
|(x + 2)(3x - 7)| < \epsilon.
\]
#### Steps to Prove the Limit
1. **Inequality Manipulation:**
We need:
\[
|x + 2| < \frac{\epsilon}{|3x - 7|} < \frac{\epsilon}{|3x| + 7} < \frac{\epsilon}{7}.
\]
This leads us to pick our \(\delta\).
2. **Choosing \(\delta\):**
We choose \(\delta = \frac{\epsilon}{7}\).
3. **Final Step:**
Whenever \(|x + 2| < \delta = \frac{\epsilon}{7}\), we achieve:
\[
|(x + 2)(3x - 7)| < \epsilon.
\]
Thus, this confirms that the limit is indeed \(14\).
### Conclusion
Through this process, we demonstrated how to approach and prove the limit using the \(\epsilon\)-\(\delta\) definition. This structured method ensures precise understanding and accuracy in limit evaluations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fde096970-4fbf-4bcb-afca-efa41531f77d%2Fd3a70cb4-e4bd-4e33-b929-39e2a7a0c682%2Fyr8an8a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Limits: Proving a Limit Example
#### Introduction
In calculus, understanding and proving limits are fundamental skills. Here, we'll go through the process of proving a specific limit:
**Prove**
\[
\lim_{x \to -2} (3x^2 - x) = 14.
\]
#### Proof
For any \(\epsilon > 0\), we need to show that:
\[
|3x^2 - x - 14| < \epsilon.
\]
This equation can be rewritten as:
\[
|(x + 2)(3x - 7)| < \epsilon.
\]
#### Steps to Prove the Limit
1. **Inequality Manipulation:**
We need:
\[
|x + 2| < \frac{\epsilon}{|3x - 7|} < \frac{\epsilon}{|3x| + 7} < \frac{\epsilon}{7}.
\]
This leads us to pick our \(\delta\).
2. **Choosing \(\delta\):**
We choose \(\delta = \frac{\epsilon}{7}\).
3. **Final Step:**
Whenever \(|x + 2| < \delta = \frac{\epsilon}{7}\), we achieve:
\[
|(x + 2)(3x - 7)| < \epsilon.
\]
Thus, this confirms that the limit is indeed \(14\).
### Conclusion
Through this process, we demonstrated how to approach and prove the limit using the \(\epsilon\)-\(\delta\) definition. This structured method ensures precise understanding and accuracy in limit evaluations.
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