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
Please show EVERY step for each problem.
Question: Evaluate each of the following limits:
![### Limits of Rational Functions as \( x \) Approaches Infinity
Understanding how to calculate the limit of rational functions as \( x \) approaches infinity is an essential skill in calculus. Here, we provide three examples illustrating this concept.
#### Example (a)
\[
\lim_{{x \to \infty}} \frac{7x^2 + x - 100}{2x^2 - 5x}
\]
#### Example (b)
\[
\lim_{{x \to \infty}} \frac{x^2 - 3x + 7}{x^3 + 10x - 4}
\]
#### Example (c)
\[
\lim_{{x \to \infty}} \frac{7x^2 - x + 11}{4 - x}
\]
To evaluate these limits, we analyze the degrees of the polynomials in the numerator and the denominator. This process involves comparing the highest power of \( x \) present in the numerator and denominator of each rational function.
### Detailed Steps:
1. **Identify the highest power of \( x \) (leading term) in both the numerator and the denominator.**
2. **If the degree of the numerator is higher than the degree of the denominator, the limit is \( \pm \infty \).**
3. **If the degree of the numerator is lower than the degree of the denominator, the limit is 0.**
4. **If the degrees are equal, the limit is the ratio of the leading coefficients.**
Let's illustrate these steps with the examples above.
#### Analysis of Example (a):
The numerator \( 7x^2 + x - 100 \) has a degree of 2, and the denominator \( 2x^2 - 5x \) also has a degree of 2. For rational functions where the degrees of the numerator and the denominator are the same, the limit as \( x \) approaches infinity is the ratio of the leading coefficients:
\[
\lim_{{x \to \infty}} \frac{7x^2 + x - 100}{2x^2 - 5x} = \frac{7}{2}
\]
#### Analysis of Example (b):
Here, the numerator \( x^2 - 3x + 7 \) has a degree of 2, while the denominator \( x^3 + 10x -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc965ec1f-6ecd-467d-b5f1-022eb1dfa0a4%2F8caecf32-8673-41c6-a021-50248ed0a66c%2Fk3zdvk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Limits of Rational Functions as \( x \) Approaches Infinity
Understanding how to calculate the limit of rational functions as \( x \) approaches infinity is an essential skill in calculus. Here, we provide three examples illustrating this concept.
#### Example (a)
\[
\lim_{{x \to \infty}} \frac{7x^2 + x - 100}{2x^2 - 5x}
\]
#### Example (b)
\[
\lim_{{x \to \infty}} \frac{x^2 - 3x + 7}{x^3 + 10x - 4}
\]
#### Example (c)
\[
\lim_{{x \to \infty}} \frac{7x^2 - x + 11}{4 - x}
\]
To evaluate these limits, we analyze the degrees of the polynomials in the numerator and the denominator. This process involves comparing the highest power of \( x \) present in the numerator and denominator of each rational function.
### Detailed Steps:
1. **Identify the highest power of \( x \) (leading term) in both the numerator and the denominator.**
2. **If the degree of the numerator is higher than the degree of the denominator, the limit is \( \pm \infty \).**
3. **If the degree of the numerator is lower than the degree of the denominator, the limit is 0.**
4. **If the degrees are equal, the limit is the ratio of the leading coefficients.**
Let's illustrate these steps with the examples above.
#### Analysis of Example (a):
The numerator \( 7x^2 + x - 100 \) has a degree of 2, and the denominator \( 2x^2 - 5x \) also has a degree of 2. For rational functions where the degrees of the numerator and the denominator are the same, the limit as \( x \) approaches infinity is the ratio of the leading coefficients:
\[
\lim_{{x \to \infty}} \frac{7x^2 + x - 100}{2x^2 - 5x} = \frac{7}{2}
\]
#### Analysis of Example (b):
Here, the numerator \( x^2 - 3x + 7 \) has a degree of 2, while the denominator \( x^3 + 10x -
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 5 steps with 5 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