Calculus: Early Transcendentals
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ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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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 -](https://content.bartleby.com/qna-images/question/c965ec1f-6ecd-467d-b5f1-022eb1dfa0a4/8caecf32-8673-41c6-a021-50248ed0a66c/k3zdvk_processed.jpeg)
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 -
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