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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See picture for question details. Need to know how this answer was reached.
![**Problem Statement:**
Find the horizontal asymptote of the graph of the function. (If an answer does not exist, enter DNE.)
\[ f(x) = \frac{x^3 - 5x^2 + 6x + 1}{x^2 - 6x + 5} \]
**Solution:**
The result is displayed in a grey box with "DNE" written inside it, indicating that the horizontal asymptote does not exist for this function.
**Explanation:**
For rational functions, horizontal asymptotes are determined by the degrees of the polynomials in the numerator and the denominator.
- If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\).
- If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is at \(y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}\).
- If the degree of the numerator is greater than the degree of the denominator, as is the case for this function where the numerator's degree is 3 and the denominator's is 2, there is no horizontal asymptote (denoted as DNE).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe68fb1ae-fc32-4d84-ad25-16ca36cebf4d%2Fde920cb7-cb28-4f69-a935-ac2cbed1c3c9%2F21ylxf_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the horizontal asymptote of the graph of the function. (If an answer does not exist, enter DNE.)
\[ f(x) = \frac{x^3 - 5x^2 + 6x + 1}{x^2 - 6x + 5} \]
**Solution:**
The result is displayed in a grey box with "DNE" written inside it, indicating that the horizontal asymptote does not exist for this function.
**Explanation:**
For rational functions, horizontal asymptotes are determined by the degrees of the polynomials in the numerator and the denominator.
- If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\).
- If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is at \(y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}\).
- If the degree of the numerator is greater than the degree of the denominator, as is the case for this function where the numerator's degree is 3 and the denominator's is 2, there is no horizontal asymptote (denoted as DNE).
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