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 of a rational function, we need to analyze the behavior of the function as x approaches positive and negative infinity.
1. Horizontal Asymptotes:
To find the horizontal asymptote(s), we compare the degrees of the numerator and the denominator of the rational function.
For the given function G(x) = (x^4 - 1) / (3x^2 - 3x):
- The degree of the numerator is 4 (highest power of x).
- The degree of the denominator is 2 (highest power of x).
Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
2. Oblique Asymptotes:
To determine whether there are oblique asymptotes, we can perform polynomial long division to divide the numerator by the denominator.
Performing polynomial long division, we have:
(x^4 - 1) / (3x^2 - 3x)
= (1/3)x^2 + (1/3)x - (1/3) + remainder((2/27)x - (15/27))
From the division, we can see that there is a quotient term, (1/3)x^2 + (1/3)x - (1/3), and a remainder term, (2/27)x - (15/27).
For oblique asymptotes, we only consider the quotient term. In this case, the quotient is (1/3)x^2 + (1/3)x - (1/3).
To determine the oblique asymptote(s), we evaluate the limit of the quotient term as x approaches positive and negative infinity.
As x approaches positive or negative infinity, the value of the quotient term tends towards the highest power of x, which is (1/3)x^2. So, the oblique asymptote is y = (1/3)x^2.
Therefore, the given rational function G(x) has an oblique asymptote y = (1/3)x^2.
In summary:
- The rational function G(x) = (x^4 - 1) / (3x^2 - 3x) has no horizontal asymptote.
- The oblique asymptote is y = (1/3)x^2.