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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![### Problem Statement
**4. Find the limit of the sequence \(\left\{\frac{3e^n + 20n}{6e^n + 37}\right\}\) or determine that it diverges.**
### Instructions for Solving the Problem
1. **Identify Dominant Terms in the Sequence:**
- Compare the growth rates of the terms in the numerator and the denominator.
- The exponential terms \(e^n\) are likely to dominate for large \(n\).
2. **Simplify the Expression:**
- Factor out the dominant exponential term (\(e^n\)) from both the numerator and the denominator.
- Simplify the remaining expression to determine the behavior as \(n\) approaches infinity.
3. **Limit Calculation:**
- Use L'Hôpital's Rule if necessary, or alternatively, algebraic manipulation to find the limit of the simplified expression.
### Example Solution
To find the limit of the sequence \(\left\{\frac{3e^n + 20n}{6e^n + 37}\right\}\), we proceed as follows:
1. **Factor Out \(e^n\):**
- Numerator: \(3e^n + 20n\)
- Denominator: \(6e^n + 37\)
For large \(n\),
\[
\frac{3e^n + 20n}{6e^n + 37} \approx \frac{e^n (3 + \frac{20n}{e^n})}{e^n (6 + \frac{37}{e^n})}
\]
2. **Simplify the Fraction:**
- As \(n\) grows larger, \(\frac{20n}{e^n} \to 0\) and \(\frac{37}{e^n} \to 0\).
Thus, the expression simplifies to:
\[
\frac{3 + 0}{6 + 0} = \frac{3}{6} = \frac{1}{2}
\]
Therefore, the limit of the sequence \(\left\{\frac{3e^n + 20n}{6e^n + 37}\right\}\) as \(n\) approaches infinity is \(\frac{1}{2}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf5ca60f-6906-44bf-a492-477ae53fc791%2F5daa9d32-f5c6-4d0c-b944-e9a023394f1a%2Fruwswtc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**4. Find the limit of the sequence \(\left\{\frac{3e^n + 20n}{6e^n + 37}\right\}\) or determine that it diverges.**
### Instructions for Solving the Problem
1. **Identify Dominant Terms in the Sequence:**
- Compare the growth rates of the terms in the numerator and the denominator.
- The exponential terms \(e^n\) are likely to dominate for large \(n\).
2. **Simplify the Expression:**
- Factor out the dominant exponential term (\(e^n\)) from both the numerator and the denominator.
- Simplify the remaining expression to determine the behavior as \(n\) approaches infinity.
3. **Limit Calculation:**
- Use L'Hôpital's Rule if necessary, or alternatively, algebraic manipulation to find the limit of the simplified expression.
### Example Solution
To find the limit of the sequence \(\left\{\frac{3e^n + 20n}{6e^n + 37}\right\}\), we proceed as follows:
1. **Factor Out \(e^n\):**
- Numerator: \(3e^n + 20n\)
- Denominator: \(6e^n + 37\)
For large \(n\),
\[
\frac{3e^n + 20n}{6e^n + 37} \approx \frac{e^n (3 + \frac{20n}{e^n})}{e^n (6 + \frac{37}{e^n})}
\]
2. **Simplify the Fraction:**
- As \(n\) grows larger, \(\frac{20n}{e^n} \to 0\) and \(\frac{37}{e^n} \to 0\).
Thus, the expression simplifies to:
\[
\frac{3 + 0}{6 + 0} = \frac{3}{6} = \frac{1}{2}
\]
Therefore, the limit of the sequence \(\left\{\frac{3e^n + 20n}{6e^n + 37}\right\}\) as \(n\) approaches infinity is \(\frac{1}{2}\).
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