Evaluate the limit 2x + 4 lim I→ 0 3x2 - 7x + 7

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

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
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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