Question

Find the vertical, horizontal, and oblique asymptotes, if any, of each rational function. $$ R(x)=\frac{6 x^{2}+19 x-7}{3 x-1} $$

Short answer

Vertical asymptote: \( x = \frac{1}{3} \), No horizontal asymptote, Oblique asymptote: \( y = 2x + 7 \).

Step-by-step solution

Identify the vertical asymptote

To find the vertical asymptote, set the denominator equal to zero and solve for x. The rational function given is \( R(x) = \frac{6x^2 + 19x - 7}{3x - 1} \). Set the denominator \(3x - 1\) to zero: \( 3x - 1 = 0 \) Solving for x, we get: \( x = \frac{1}{3} \) Therefore, the vertical asymptote is \( x = \frac{1}{3} \).

Identify the horizontal asymptote

To find the horizontal asymptote, compare the degrees of the numerator and the denominator. The degree of the numerator \(6x^2 + 19x - 7\) is 2, and the degree of the denominator \(3x - 1\) is 1. Because the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Identify the oblique asymptote

To find the oblique asymptote when the degree of the numerator is exactly one more than the degree of the denominator, perform polynomial long division of the numerator by the denominator. Divide \(6x^2 + 19x - 7\) by \(3x - 1\): After performing the division, the quotient is \(2x + 7\). Hence, the oblique asymptote is \(y = 2x + 7\).

Key concepts

vertical asymptote

Let's understand vertical asymptotes. Vertical asymptotes occur in rational functions where the function tends to infinity as the input approaches a specific value. They are found by setting the denominator of the function to zero and solving for the variable. For the function given, \( R(x) = \frac{6x^2 + 19x - 7}{3x - 1} \) we set the denominator equal to zero: \( 3x - 1 = 0 \). Solving this equation for \( x \), we get: \( x = \frac{1}{3} \). Thus, the vertical asymptote for this function is \( x = \frac{1}{3} \). This means as \( x \) approaches \( \frac{1}{3} \), the function's value will tend to infinity or negative infinity.

horizontal asymptote

Horizontal asymptotes describe the behavior of a rational function as \( x \) approaches positive or negative infinity. To determine the horizontal asymptote, compare the degrees of the numerator and the denominator. Recall, the degree is the highest power of \( x \) in the polynomial. For \( R(x) = \frac{6x^2 + 19x - 7}{3x - 1} \), the numerator has a degree of 2 (\(6x^2\)) and the denominator has a degree of 1 (\(3x\)). Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote for this function. This often indicates the presence of an oblique asymptote instead.

oblique asymptote

An oblique asymptote occurs in a rational function when the degree of the numerator is exactly one more than the degree of the denominator. To find the oblique asymptote, polynomial long division is used. For the given function \( R(x) = \frac{6x^2 + 19x - 7}{3x - 1} \), we divide the numerator \( 6x^2 + 19x - 7 \) by the denominator \( 3x - 1 \). After performing the division, the quotient is \( 2x + 7 \). Therefore, the oblique asymptote for this function is \( y = 2x + 7 \). This means as \( x \) tends to infinity, the function \( R(x) \) behaves like the line \( y = 2x + 7 \).

polynomial long division

Polynomial long division is used to divide one polynomial by another, similar to numerical long division. It helps find the quotient and remainder when dividing. To divide \( 6x^2 + 19x - 7 \) by \( 3x - 1 \), follow these steps:

• Divide the leading term of the numerator \( 6x^2 \) by the leading term of the denominator \( 3x \) to get \( 2x \).
• Multiply \( 3x - 1 \) by \( 2x \) to get \( 6x^2 - 2x \).
• Subtract \( 6x^2 - 2x \) from the original numerator to get a new polynomial \( 21x - 7 \).
• Divide \( 21x \) by \( 3x \) to get \( 7 \).
• Repeat the multiplication and subtraction to eventually reach a remainder of zero.

The quotient \( 2x + 7 \) indicates the oblique asymptote.