Use Use the E-S definition of limit to establish the lim X+5 X->-12x+3=4
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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![**Title: Understanding Limits Using the ε-δ Definition**
**Objective:**
Learn how to apply the ε-δ (epsilon-delta) definition of limits to establish the following limit:
\[ \lim_{{x \to -1}} \frac{x+5}{2x+3} = 4 \]
**Instructions:**
1. **Recognize the Limit Expression:**
- We want to show that as \(x\) approaches \(-1\), the value of the function \(\frac{x+5}{2x+3}\) approaches 4.
2. **Apply the ε-δ Definition:**
- The ε-δ definition states that for every ε > 0, there exists a δ > 0 such that if \(0 < |x + 1| < δ\), then \(|\frac{x+5}{2x+3} - 4| < ε\).
3. **Steps to Solve:**
- Begin by manipulating the inequality \(|\frac{x+5}{2x+3} - 4| < ε\) to find a corresponding δ that works for each ε.
- Simplify the expression and solve for values of \(x\) where \(|x + 1| < δ\).
4. **Conclusion:**
- By demonstrating this relationship, you show that the limit as \(x\) approaches \(-1\) of the function \(\frac{x+5}{2x+3}\) is indeed 4.
**Further Reading:**
- Explore examples with varying limits to strengthen your understanding of the ε-δ definition.
- Review foundational calculus concepts such as continuity and the behavior of functions around specified points.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe57a7d70-87de-4a1f-8104-5b2578062c6c%2F8c57b57f-cd9b-42e0-b439-b6619d1d75ac%2Fpfjgb1_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Limits Using the ε-δ Definition**
**Objective:**
Learn how to apply the ε-δ (epsilon-delta) definition of limits to establish the following limit:
\[ \lim_{{x \to -1}} \frac{x+5}{2x+3} = 4 \]
**Instructions:**
1. **Recognize the Limit Expression:**
- We want to show that as \(x\) approaches \(-1\), the value of the function \(\frac{x+5}{2x+3}\) approaches 4.
2. **Apply the ε-δ Definition:**
- The ε-δ definition states that for every ε > 0, there exists a δ > 0 such that if \(0 < |x + 1| < δ\), then \(|\frac{x+5}{2x+3} - 4| < ε\).
3. **Steps to Solve:**
- Begin by manipulating the inequality \(|\frac{x+5}{2x+3} - 4| < ε\) to find a corresponding δ that works for each ε.
- Simplify the expression and solve for values of \(x\) where \(|x + 1| < δ\).
4. **Conclusion:**
- By demonstrating this relationship, you show that the limit as \(x\) approaches \(-1\) of the function \(\frac{x+5}{2x+3}\) is indeed 4.
**Further Reading:**
- Explore examples with varying limits to strengthen your understanding of the ε-δ definition.
- Review foundational calculus concepts such as continuity and the behavior of functions around specified points.
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Follow-up Questions
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Follow-up Question
Where did you get this min 2 from?
I get the ep/7....
But why not min 1
Transcribed Image Text:Now choose 8 such that,
8 = min {2₁ €// }
Solution
by Bartleby ExpertFollow-up Question
Why'd you choose 1/5?
I can't figure out what delta to use and how to determine
Solution
by Bartleby ExpertFollow-up Question
Where does -8x - 12 come from.... Why come up with that?
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