How to Find the Vertical and Horizontal Asymptotes of a Rational Function

To find the vertical and horizontal asymptotes of a rational function, you should follow these steps:

Finding Vertical Asymptotes

Vertical asymptotes occur where the function approaches infinity, typically where the denominator is equal to zero (and the numerator is not zero at those points). To find these points:

1. Identify the denominator of the rational function.
2. Set the denominator equal to zero.
3. Solve for the values of x. These values are where the vertical asymptotes are located.

For example, in the function f(x) = (2x)/(x^2 – 1), the denominator is x^2 – 1. Setting it to zero gives:

x^2 - 1 = 0  
=> x^2 = 1  
=> x = ±1

Thus, the vertical asymptotes are at x = 1 and x = -1.

Finding Horizontal Asymptotes

Horizontal asymptotes describe the behavior of the function as x approaches infinity or negative infinity. To find horizontal asymptotes, consider the degrees of the numerator and 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 = (leading coefficient of the numerator)/(leading coefficient of the denominator).
• If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (the function approaches infinity).

Continuing with our example, in f(x) = (2x)/(x^2 – 1), the degree of the numerator (1) is less than the degree of the denominator (2). Therefore, we have a horizontal asymptote at:

y = 0

Final Summary

In summary, to find vertical asymptotes, set the denominator to zero and solve for x. For horizontal asymptotes, compare the degrees of the numerator and denominator to determine the equation of the asymptote. Understanding these concepts can help you analyze the behavior of rational functions effectively.