Find the vertical, horizontal, and oblique asymptotes, if any, of the given rational function.
[tex]H(x)=\frac{x^3-27}{x^2-7 x+12}[/tex]
Answer (2)
The rational function H ( x ) = x 2 − 7 x + 12 x 3 − 27 has a vertical asymptote at x = 4 and an oblique asymptote at y = x + 7 . There is no horizontal asymptote. Additionally, x = 3 is a removable discontinuity.
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Factor the numerator and denominator: H ( x ) = ( x − 3 ) ( x − 4 ) ( x − 3 ) ( x 2 + 3 x + 9 ) .
Simplify the function: H ( x ) = x − 4 x 2 + 3 x + 9 for x = 3 .
Identify the vertical asymptote: x = 4 .
Determine the oblique asymptote by polynomial division: y = x + 7 .
The vertical asymptote is x = 4 and the oblique asymptote is y = x + 7 .
Explanation
Problem Analysis We are given the rational function H ( x ) = x 2 − 7 x + 12 x 3 − 27 . Our goal is to find any vertical, horizontal, and oblique asymptotes.
Factoring Numerator and Denominator First, we factor the numerator and denominator. The numerator is a difference of cubes, so x 3 − 27 = ( x − 3 ) ( x 2 + 3 x + 9 ) . The denominator is a quadratic, which factors as x 2 − 7 x + 12 = ( x − 3 ) ( x − 4 ) .
Simplifying the Function Now we simplify the rational function: H ( x ) = ( x − 3 ) ( x − 4 ) ( x − 3 ) ( x 2 + 3 x + 9 ) = x − 4 x 2 + 3 x + 9 , x = 3.
Finding Vertical Asymptotes To find vertical asymptotes, we set the denominator of the simplified function equal to zero: x − 4 = 0 , which gives x = 4 . Thus, there is a vertical asymptote at x = 4 . Note that x = 3 is a removable singularity (a hole) and not a vertical asymptote because the factor ( x − 3 ) cancels out.
Checking for Horizontal Asymptotes To determine if a horizontal asymptote exists, we compare the degrees of the numerator and denominator of the simplified function. The degree of the numerator ( x 2 + 3 x + 9 ) is 2, and the degree of the denominator ( x − 4 ) is 1. Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
Finding Oblique Asymptotes To determine if an oblique asymptote exists, we check if the degree of the numerator is exactly one greater than the degree of the denominator. In this case, it is. We perform polynomial long division to divide x 2 + 3 x + 9 by x − 4 . The result is x + 7 with a remainder of 37 . Therefore, x − 4 x 2 + 3 x + 9 = x + 7 + x − 4 37 . The oblique asymptote is given by the quotient, which is y = x + 7 .
Conclusion In summary, the vertical asymptote is x = 4 , there is no horizontal asymptote, and the oblique asymptote is y = x + 7 .
Examples
Understanding asymptotes is crucial in various real-world applications. For instance, in physics, the velocity of an object approaching the speed of light gets closer and closer to the speed of light but never quite reaches it, demonstrating a horizontal asymptote. In economics, cost functions might approach a certain minimum cost without ever reaching it, illustrating a horizontal asymptote. Similarly, in population modeling, population growth might be limited by environmental factors, causing the growth rate to approach an asymptote.
