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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Question
![### Problem Statement
Evaluate the limit:
\[ \lim_{x \to \infty} \frac{2x + 4}{3x^2 - 7x + 7} \]
---
This problem requires evaluating the limit of a rational function as \( x \) approaches infinity.
#### Solution Approach
1. **Leading Terms Analysis**:
The numerator is \( 2x + 4 \), and the leading term is \( 2x \).
The denominator is \( 3x^2 - 7x + 7 \), and the leading term is \( 3x^2 \).
2. **Divide by the Highest Power of \( x \)**:
To simplify the expression, divide both the numerator and the denominator by \( x^2 \) (the highest power in the denominator):
\[
\frac{2x + 4}{3x^2 - 7x + 7} = \frac{\frac{2x}{x^2} + \frac{4}{x^2}}{\frac{3x^2}{x^2} - \frac{7x}{x^2} + \frac{7}{x^2}}
\]
Simplifying the fractions inside the limit yields:
\[
= \frac{\frac{2}{x} + \frac{4}{x^2}}{3 - \frac{7}{x} + \frac{7}{x^2}}
\]
3. **Evaluate the Limit**:
As \( x \) approaches infinity, the terms \( \frac{2}{x} \), \( \frac{4}{x^2} \), \( \frac{7}{x} \), and \( \frac{7}{x^2} \) all approach 0. Hence, the expression simplifies to:
\[
\lim_{x \to \infty} \frac{\frac{2}{x} + \frac{4}{x^2}}{3 - \frac{7}{x} + \frac{7}{x^2}}= \frac{0 + 0}{3 - 0 + 0} = \frac{0}{3} = 0
\]
#### Final Answer
\[
\lim_{x \to \infty} \frac{2x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F19a44d99-9126-45f2-b575-2ff09f428fba%2Facf7f77c-e964-4069-b821-7e8a2c1517d6%2Fu42hvdf.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Evaluate the limit:
\[ \lim_{x \to \infty} \frac{2x + 4}{3x^2 - 7x + 7} \]
---
This problem requires evaluating the limit of a rational function as \( x \) approaches infinity.
#### Solution Approach
1. **Leading Terms Analysis**:
The numerator is \( 2x + 4 \), and the leading term is \( 2x \).
The denominator is \( 3x^2 - 7x + 7 \), and the leading term is \( 3x^2 \).
2. **Divide by the Highest Power of \( x \)**:
To simplify the expression, divide both the numerator and the denominator by \( x^2 \) (the highest power in the denominator):
\[
\frac{2x + 4}{3x^2 - 7x + 7} = \frac{\frac{2x}{x^2} + \frac{4}{x^2}}{\frac{3x^2}{x^2} - \frac{7x}{x^2} + \frac{7}{x^2}}
\]
Simplifying the fractions inside the limit yields:
\[
= \frac{\frac{2}{x} + \frac{4}{x^2}}{3 - \frac{7}{x} + \frac{7}{x^2}}
\]
3. **Evaluate the Limit**:
As \( x \) approaches infinity, the terms \( \frac{2}{x} \), \( \frac{4}{x^2} \), \( \frac{7}{x} \), and \( \frac{7}{x^2} \) all approach 0. Hence, the expression simplifies to:
\[
\lim_{x \to \infty} \frac{\frac{2}{x} + \frac{4}{x^2}}{3 - \frac{7}{x} + \frac{7}{x^2}}= \frac{0 + 0}{3 - 0 + 0} = \frac{0}{3} = 0
\]
#### Final Answer
\[
\lim_{x \to \infty} \frac{2x
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