Question

Find the vertical, horizontal, and oblique asymptotes, if any, of each rational function. $$ R(x)=\frac{3 x+5}{x-6} $$

Short answer

Vertical asymptote: x = 6Horizontal asymptote: y = 3No oblique asymptote.

Step-by-step solution

Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator equals zero and the numerator is not zero. Set the denominator of the function equal to zero and solve for x: \[ x - 6 = 0 \] \[ x = 6 \] So, the vertical asymptote is at \( x = 6 \).

Identify Horizontal Asymptotes

To find the horizontal asymptotes, compare the degrees of the numerator and the denominator. The degree of the numerator \( 3x + 5 \) is 1, and the degree of the denominator \( x - 6 \) is 1. Since the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients: \[ \frac{3}{1} = 3 \] So, the horizontal asymptote is at \( y = 3 \).

Check for Oblique Asymptotes

Oblique asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. Since the degrees of the numerator and the denominator are equal, there is no oblique asymptote for this function.

Key concepts

horizontal asymptotes

Horizontal asymptotes help describe the behavior of rational functions as x approaches ±∞. A rational function is of the form \( \frac{P(x)}{Q(x)} \), where P(x) and Q(x) are polynomials. To identify horizontal asymptotes, compare the degrees (highest power of x) of the numerator and the denominator.

If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at y = 0.
If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is at y = \( \frac{a}{b} \), where a and b are the leading coefficients of the numerator and the denominator respectively.
If the degree of the numerator is greater than the degree of the denominator, there isn't a horizontal asymptote.

In our example, since the degrees are both 1, and the leading coefficients are 3 and 1 respectively, the horizontal asymptote is y = 3.

oblique asymptotes

Oblique asymptotes, also known as slant asymptotes, occur when the degree of the numerator is exactly one higher than the degree of the denominator. They describe the behavior of rational functions where x becomes very large or very small, but the function doesn't approach a horizontal line.
To find an oblique asymptote, divide the numerator by the denominator using polynomial long division. The quotient (ignoring the remainder) gives the equation of the oblique asymptote.

In the current example, the degrees of the numerator and the denominator are both 1, so there is no oblique asymptote.

rational functions

Rational functions are the quotient of two polynomials. They are expressed in the form \( \frac{P(x)}{Q(x)} \), where P(x) and Q(x) are polynomial functions. These functions can have vertical, horizontal, and oblique asymptotes, which help describe their behavior.

The key aspects to study in rational functions are:

• Vertical asymptotes, which occur when the denominator equals zero, provided the numerator is not zero at that point.
• Horizontal asymptotes, which describe the limits of the function as x approaches ±∞.
• Oblique asymptotes, which occur when the degree of the numerator is exactly one more than the degree of the denominator.

These asymptotes help sketch the behavior of the function and make it easier to understand its long-term trends and discontinuities.