Math Problem Statement
y=(x^2+x)/(x^2-1) find the asymptote
Solution
To find the asymptotes of the function , we'll consider both vertical and horizontal asymptotes.
1. Vertical Asymptotes
Vertical asymptotes occur where the denominator of the function is zero, provided that the numerator is not zero at those points.
The denominator is:
Solve for :
These values correspond to potential vertical asymptotes. We check the numerator at these points:
The numerator is zero at , so is a removable discontinuity, not a vertical asymptote. However, at , the numerator is not zero, so is a vertical asymptote.
Vertical asymptote:
2. Horizontal Asymptote
To find the horizontal asymptote, we examine the behavior of the function as approaches infinity.
The degrees of the numerator and denominator are both 2 (since the highest power of in both the numerator and denominator is ). For rational functions, if the degrees are the same, the horizontal asymptote is given by the ratio of the leading coefficients.
The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is also 1. Therefore, the horizontal asymptote is:
Horizontal asymptote:
Summary
- Vertical asymptote:
- Horizontal asymptote:
Would you like more details on this process or have any questions?
Here are 5 related questions:
- What is the behavior of the function near the vertical asymptote ?
- How do you determine the presence of slant asymptotes?
- What happens to the function as approaches the removable discontinuity ?
- Can the function have oblique asymptotes?
- How would the asymptotes change if the numerator had a higher degree than the denominator?
Tip: When analyzing asymptotes, always start by factoring the numerator and denominator to check for common factors (removable discontinuities).
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Algebra
Rational Functions
Asymptotes
Formulas
Vertical Asymptote: Set denominator to 0
Horizontal Asymptote: Compare degrees of numerator and denominator
Theorems
Theorem on Rational Functions Asymptotes
Factorization and Removable Discontinuities
Suitable Grade Level
Grades 10-12
Related Recommendation
Finding Vertical and Horizontal Asymptotes for y = (x^2 + 4)/(x^2 - 1)
Finding Asymptotes of Rational Function x^2 + x - 1 / x^2 - 1
Finding Horizontal and Vertical Asymptotes for Rational Function (2x^2 + 1) / (x(x - 2))
Find Horizontal and Vertical Asymptotes for y = (2x^2 + x − 1) / (x^2 + x − 72)
Asymptotes of the Rational Function Y = (x-1)(x-3)/x^2