What value(s) of 8 are an appropriate choice when proving the following limit lim (4x - 2) = 6 x-2 Enter your answer in terms of ε. Provide your answer below: d= or smaller. Content attribution
What value(s) of 8 are an appropriate choice when proving the following limit lim (4x - 2) = 6 x-2 Enter your answer in terms of ε. Provide your answer below: d= or smaller. Content attribution
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Understanding Limits: Choosing δ for Proving a Limit**
In this section, we will explore the process of selecting an appropriate value of δ (delta) in terms of ε (epsilon) when proving a limit. Consider the following limit problem:
\[ \lim_{{x \to 2}} (4x - 2) = 6 \]
### Task:
Determine the appropriate value(s) of δ that can be used to prove the given limit statement.
### Instructions:
1. **Enter your answer in terms of ε.**
2. **Provide your answer below:**
\[ \delta = \ \_\_\_ \text{ or smaller.} \]
### Explanation:
To prove the limit using the ε-δ definition, recall that for the limit:
\[ \lim_{{x \to 2}} (4x - 2) = 6 \]
We need to show that for every ε > 0, there exists a δ > 0 such that:
\[ 0 < |x - 2| < \delta \implies |(4x - 2) - 6| < \epsilon \]
### Simplifying the Inequality:
Consider the expression \(|(4x - 2) - 6|\):
\[ |(4x - 2) - 6| = |4x - 8| \]
Factor out the 4:
\[ |4x - 8| = 4|x - 2| \]
### Establishing the Relationship:
We want:
\[ 4|x - 2| < \epsilon \]
Hence,
\[ |x - 2| < \frac{\epsilon}{4} \]
### Conclusion:
Therefore, an appropriate choice for δ in terms of ε is:
\[ \delta = \frac{\epsilon}{4} \]
Substitute this into the provided answer field:
\[ \delta = \frac{\epsilon}{4} \text{ or smaller.} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4bc92d6f-8141-453b-b1aa-15dddf73f72d%2Fb28d815b-70cd-436d-a680-aef490b0ccf4%2Fx1ij5n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding Limits: Choosing δ for Proving a Limit**
In this section, we will explore the process of selecting an appropriate value of δ (delta) in terms of ε (epsilon) when proving a limit. Consider the following limit problem:
\[ \lim_{{x \to 2}} (4x - 2) = 6 \]
### Task:
Determine the appropriate value(s) of δ that can be used to prove the given limit statement.
### Instructions:
1. **Enter your answer in terms of ε.**
2. **Provide your answer below:**
\[ \delta = \ \_\_\_ \text{ or smaller.} \]
### Explanation:
To prove the limit using the ε-δ definition, recall that for the limit:
\[ \lim_{{x \to 2}} (4x - 2) = 6 \]
We need to show that for every ε > 0, there exists a δ > 0 such that:
\[ 0 < |x - 2| < \delta \implies |(4x - 2) - 6| < \epsilon \]
### Simplifying the Inequality:
Consider the expression \(|(4x - 2) - 6|\):
\[ |(4x - 2) - 6| = |4x - 8| \]
Factor out the 4:
\[ |4x - 8| = 4|x - 2| \]
### Establishing the Relationship:
We want:
\[ 4|x - 2| < \epsilon \]
Hence,
\[ |x - 2| < \frac{\epsilon}{4} \]
### Conclusion:
Therefore, an appropriate choice for δ in terms of ε is:
\[ \delta = \frac{\epsilon}{4} \]
Substitute this into the provided answer field:
\[ \delta = \frac{\epsilon}{4} \text{ or smaller.} \]
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