Given f(x)=(2x+3)/(3x^2+7x-6) find the equations of any vertical, horizontal, or oblique asymptotes.
since the numerator has degree less than the denominator, y=0 is the horizontal asymptote.
Vertical asymptotes where the denominator is zero.
To find the equations of any vertical, horizontal, or oblique asymptotes of the given function f(x) = (2x + 3)/(3x^2 + 7x - 6), we can analyze the behavior of the function as x approaches positive or negative infinity and find the corresponding asymptotes.
1. Vertical asymptotes:
Vertical asymptotes occur when the denominator of the function approaches zero. To find the vertical asymptotes, we need to solve the equation 3x^2 + 7x - 6 = 0 for x.
The factorization of the quadratic equation is as follows:
(3x - 2)(x + 3) = 0.
Setting each factor equal to zero gives us:
3x - 2 = 0 => 3x = 2 => x = 2/3,
x + 3 = 0 => x = -3.
Therefore, there are two vertical asymptotes at x = 2/3 and x = -3.
2. Horizontal asymptote:
To find the horizontal asymptote, we need to compare the degrees of the numerator and denominator of the function.
In this case, the degree of the numerator is 1, and the degree of the denominator is 2. Since the degree of the denominator is greater, there is no horizontal asymptote at y = 0 (the x-axis).
However, we can determine the behavior of the function as x approaches positive or negative infinity by looking at the ratio of the leading terms of the numerator and denominator. In this case, the leading term of the numerator is 2x, and the leading term of the denominator is 3x^2.
As x approaches positive or negative infinity, the 2x term in the numerator becomes relatively insignificant compared to the 3x^2 term in the denominator. Therefore, the function approaches y = 0 as x approaches positive or negative infinity. Hence, y = 0 is the horizontal asymptote.
3. Oblique asymptote:
To determine if there is an oblique (slant) asymptote, we need to check if the degrees of the numerator and denominator differ by exactly 1.
In this case, the degree of the numerator is 1, and the degree of the denominator is 2. Since the degrees differ by more than 1, there is no oblique asymptote.
In summary, the equations of the asymptotes for the given function f(x) = (2x + 3)/(3x^2 + 7x - 6) are:
- Two vertical asymptotes: x = 2/3 and x = -3.
- One horizontal asymptote: y = 0 (the x-axis).
- No oblique asymptote.