Determining Horizontal Asymptotes in Math: Step-by-Step Guide for Rational Functions and Illustrative Example

Look for horizontal asymptotes

To find horizontal asymptotes, there are a few key steps you can follow:

1

To find horizontal asymptotes, there are a few key steps you can follow:

1. Determine the degree of the numerator and the denominator of the rational function. The degree is the highest power of the variable in the polynomial expression.

2. Compare the degrees of the numerator and the denominator. There are three possible scenarios:

a) If the degree of the numerator is less than the degree of the denominator, then the horizontal asymptote is y = 0 (the x-axis).

b) If the degree of the numerator is equal to the degree of the denominator, then the horizontal asymptote can either be y = 0 or a horizontal line determined by the ratio of the leading coefficients of the numerator and the denominator. This can be found by dividing the leading coefficient of the numerator by the leading coefficient of the denominator. If the ratio is a constant, then that constant is the equation of the horizontal asymptote. If the ratio is infinity or undefined (such as when the leading coefficient of the denominator is 0), then there is no horizontal asymptote.

c) If the degree of the numerator is greater than the degree of the denominator, then there is no horizontal asymptote. Instead, the graph will have an oblique (slant) asymptote.

Let’s illustrate these steps with an example:

Consider the rational function f(x) = (3x^2 – 2x + 1) / (2x^2 + 5x – 4).

Step 1: Determine the degree of the numerator and the denominator. The degree of the numerator is 2 (highest power of x) and the degree of the denominator is also 2.

Step 2: Compare the degrees.
Since the degrees of the numerator and denominator are equal, we move on to calculating the ratio of the leading coefficients.
The ratio of the leading coefficients is 3/2.

Therefore, the equation of the horizontal asymptote is y = 3/2.

In this example, we have a horizontal asymptote at y = 3/2 and no oblique asymptote.

Keep in mind that these steps apply to rational functions, which are functions expressed as the ratio of two polynomials. Other types of functions may not necessarily have horizontal asymptotes.

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