How to find the horizontal asymptote

Have you ever stared at a complex function graphed on paper, trying to decipher its long-term behavior? Imagine you’re a student preparing for a math exam and grappling with the concept of horizontal asymptotes. You’ve spent hours studying but still feel unsure about how to find these critical lines that describe the end behavior of rational functions. In the midst of your study session, you pause and think, “How do I find the horizontal asymptote?”

To find the horizontal asymptote of a rational function, compare the degrees of the polynomial in the numerator and the polynomial in the denominator.

When finding the horizontal asymptote for a rational function of the form \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials, you’ll primarily focus on the degrees of these polynomials, denoted as \(n\) for the numerator and \(m\) for the denominator. Here are the key steps to determine the horizontal asymptote:

1. Compare Degrees:

– Degree \(n < m\): If the degree of the numerator is less than that of the denominator, the horizontal asymptote is \(y = 0\).

– Degree \(n = m\): If the degrees are equal, the horizontal asymptote can be found by taking the ratio of the leading coefficients of \(P(x)\) and \(Q(x)\). So, if the leading coefficient of \(P(x)\) is \(a\) and of \(Q(x)\) is \(b\), the horizontal asymptote is \(y = \frac{a}{b}\).

– Degree \(n > m\): If the degree of the numerator is greater than that of the denominator, there is no horizontal asymptote. Instead, the function may have an oblique (slant) asymptote.

By following these steps, you can confidently determine the horizontal asymptote of any given rational function, providing valuable insight into its behavior as \(x\) approaches infinity or negative infinity.