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...
Related questions
Question
![**Title: Using the Squeeze Theorem to Find the Limit of a Sequence**
**Problem Statement:**
Use the Squeeze Theorem to find the limit of the sequence:
\[ 4n \sin\left(\frac{1}{n}\right) \]
**Explanation:**
The problem requires applying the Squeeze Theorem to determine the limit of the sequence as \( n \) approaches infinity.
**Correctness Check:**
A text box is provided for the answer. There is a red 'x' indicating an incorrect or incomplete solution attempt.
**Conceptual Approach:**
1. **Understanding the Squeeze Theorem:**
- The Squeeze Theorem states that if \( a_n \leq b_n \leq c_n \) for all \( n \) beyond a certain point, and if \(\lim_{n \to \infty} a_n = \lim_{n \to \infty} c_n = L\), then \(\lim_{n \to \infty} b_n = L\).
2. **Sequence Analysis:**
- As \( n \to \infty \), \(\frac{1}{n} \to 0\).
- We know the property of sine: \(-1 \leq \sin(x) \leq 1\) for any real number \( x \).
- Therefore, \(-\frac{1}{n} \leq \sin\left(\frac{1}{n}\right) \leq \frac{1}{n}\).
3. **Applying the Squeeze Theorem:**
- Multiply throughout by \( 4n \): \(-4 \leq 4n \sin\left(\frac{1}{n}\right) \leq 4\).
- The limits of \(-4\) and \(4\) as \( n \to \infty\) are both \(4\).
4. **Conclusion:**
- By the Squeeze Theorem, \(\lim_{n \to \infty} 4n \sin\left(\frac{1}{n}\right) = 4\).
This concept can be further illustrated with diagrams or examples to deepen understanding.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F829812ca-fcc6-4eb6-9db8-017261f61931%2F82ae7440-63c0-4c47-b5b8-76ed6d8dcc19%2F80ft5hh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Using the Squeeze Theorem to Find the Limit of a Sequence**
**Problem Statement:**
Use the Squeeze Theorem to find the limit of the sequence:
\[ 4n \sin\left(\frac{1}{n}\right) \]
**Explanation:**
The problem requires applying the Squeeze Theorem to determine the limit of the sequence as \( n \) approaches infinity.
**Correctness Check:**
A text box is provided for the answer. There is a red 'x' indicating an incorrect or incomplete solution attempt.
**Conceptual Approach:**
1. **Understanding the Squeeze Theorem:**
- The Squeeze Theorem states that if \( a_n \leq b_n \leq c_n \) for all \( n \) beyond a certain point, and if \(\lim_{n \to \infty} a_n = \lim_{n \to \infty} c_n = L\), then \(\lim_{n \to \infty} b_n = L\).
2. **Sequence Analysis:**
- As \( n \to \infty \), \(\frac{1}{n} \to 0\).
- We know the property of sine: \(-1 \leq \sin(x) \leq 1\) for any real number \( x \).
- Therefore, \(-\frac{1}{n} \leq \sin\left(\frac{1}{n}\right) \leq \frac{1}{n}\).
3. **Applying the Squeeze Theorem:**
- Multiply throughout by \( 4n \): \(-4 \leq 4n \sin\left(\frac{1}{n}\right) \leq 4\).
- The limits of \(-4\) and \(4\) as \( n \to \infty\) are both \(4\).
4. **Conclusion:**
- By the Squeeze Theorem, \(\lim_{n \to \infty} 4n \sin\left(\frac{1}{n}\right) = 4\).
This concept can be further illustrated with diagrams or examples to deepen understanding.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning