Finding a Horizontal Asymptote of a Rational Function (Precalculus - College Algebra 40)

The video begins by introducing the topic of horizontal asymptotes in rational functions. The presenter explains that horizontal asymptotes are similar to vertical asymptotes but are horizontal lines that the function never touches. The focus is on determining when a rational function has a horizontal asymptote by examining the leading terms of the numerator and denominator. If the degree of the denominator is greater than the degree of the numerator, there is a horizontal asymptote at y=0. If the degrees are equal, the horizontal asymptote is at the ratio of the leading coefficients.

This paragraph delves into the concept of end behavior in rational functions, explaining how it is determined by the leading terms of the numerator and denominator. The presenter clarifies that there are three types of end behavior: a horizontal asymptote, a slant (or oblique) asymptote, and a cubic or quadratic end behavior. It is emphasized that a function cannot have both a horizontal and a slant asymptote simultaneously. The importance of understanding the degree of the denominator and numerator in relation to the horizontal asymptote is discussed, along with a practical example to illustrate the concept.

The presenter explains how to simplify rational functions by focusing on the leading terms and their degrees. It is shown that power functions with the same variable can always be simplified, and this simplification helps in understanding the end behavior of the function. The video illustrates why a horizontal asymptote at y=0 occurs when the degree of the denominator is greater than the numerator. The concept of limits is introduced to explain the behavior of the function as x approaches positive or negative infinity.

This section further explores horizontal asymptotes, emphasizing that they are different from vertical asymptotes because they can be crossed by the function. The video explains that while a function cannot touch a vertical asymptote, it can cross a horizontal asymptote. The presenter also discusses the behavior of the function as x approaches large positive or negative values and how this relates to the horizontal asymptote. The concept of limits is again used to describe the function's behavior as it approaches the asymptote.

The video continues with an example to show how to determine the type of horizontal asymptote present in a rational function. It is explained that if the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients. The presenter simplifies the example and uses it to demonstrate how the function approaches the horizontal asymptote as x approaches positive and negative infinity. The importance of understanding the direction in which the function approaches the asymptote is also discussed.

In the final paragraph, the presenter summarizes the key points about horizontal asymptotes and their determination through the comparison of degrees in the numerator and denominator. It is reiterated that if the degree of the denominator is larger, the horizontal asymptote is at y=0, and if the degrees are equal, the asymptote is at the ratio of the leading coefficients. The video concludes with a preview of upcoming topics, which include oblique (or slant) asymptotes and more complex end behaviors. The presenter emphasizes the importance of grasping the basics before moving on to more advanced concepts.