Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**Problem: Finding the Horizontal Asymptote**
**Objective:** Determine the horizontal asymptote of the function \( f(x) = \frac{-5x - 4x^3 - 3}{5x^3 + 3x^2 + 2} \).
**Solution:**
To find the horizontal asymptote of a rational function, compare the degrees of the numerator and the denominator:
1. **Numerator Degree:** The degree is 3 (from the term \(-4x^3\)).
2. **Denominator Degree:** The degree is also 3 (from the term \(5x^3\)).
Since the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients.
- Numerator's leading coefficient: \(-4\)
- Denominator's leading coefficient: \(5\)
Therefore, the horizontal asymptote is given by:
\[ y = \frac{-4}{5} \]
**Note:** The initial answer given in the box, \( y = -\frac{1}{2} \), is incorrect based on this analysis.
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Transcribed Image Text:**Problem: Finding the Horizontal Asymptote**
**Objective:** Determine the horizontal asymptote of the function \( f(x) = \frac{-5x - 4x^3 - 3}{5x^3 + 3x^2 + 2} \).
**Solution:**
To find the horizontal asymptote of a rational function, compare the degrees of the numerator and the denominator:
1. **Numerator Degree:** The degree is 3 (from the term \(-4x^3\)).
2. **Denominator Degree:** The degree is also 3 (from the term \(5x^3\)).
Since the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients.
- Numerator's leading coefficient: \(-4\)
- Denominator's leading coefficient: \(5\)
Therefore, the horizontal asymptote is given by:
\[ y = \frac{-4}{5} \]
**Note:** The initial answer given in the box, \( y = -\frac{1}{2} \), is incorrect based on this analysis.
**Additional Features:**
- **Question Help:** An option to watch a video for further explanation.
- **Submit Question Button:** Allows submission of the solution for evaluation.
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