Find the horizontal and oblique asymptotes, if any, for the given rational function G(x)=(x^4-1)/(3x^2-3x)
To find the horizontal and oblique asymptotes, if any, for the rational function G(x) = (x^4 - 1)/(3x^2 - 3x), we need to analyze the behavior of the function as x approaches both positive and negative infinity.
1. Finding the Horizontal Asymptotes:
a) Determine the degree of the numerator and the denominator.
The degree of the numerator is 4, as indicated by the highest power of x.
The degree of the denominator is 2, as indicated by the highest power of x.
b) Compare the degrees of 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 degrees are the same, there is a horizontal asymptote at the ratio of the leading coefficients.
- If the degree of the numerator is one higher than the degree of the denominator, there is no horizontal asymptote. Instead, there is a slant asymptote (oblique asymptote).
Since the degree of the numerator (4) is higher than the degree of the denominator (2) by 2, there is no horizontal asymptote. Instead, we need to find the slant asymptote.
2. Finding the Slant (Oblique) Asymptote:
To find the slant asymptote, perform polynomial long division of the numerator (x^4 - 1) by the denominator (3x^2 - 3x).
x^2
_________________________
3x^2 - 3x | x^4 + 0x^3 + 0x^2 - 1
-(x^4 - x^3)
0x^3 - 1x^2
-(0x^3 + 0x^2)
0x^2 - 1x
-(0x^2 + 0x)
- 1x - 1
-(- 1x + 1)
0
The slant asymptote is obtained from the quotient of the division, which is x^2. Therefore, the oblique asymptote is y = x^2.
In summary, for the given rational function G(x) = (x^4 - 1)/(3x^2 - 3x):
- There is no horizontal asymptote.
- The oblique asymptote is y = x^2.