In this lesson, our instructor Vincent Selhorst-Jones teaches Horizontal Asymptotes in greater detail. Youll investigate a fundamental function, go over the idea of a horizontal asymptote, and answer the question: Whats going on? Vincent also goes over the definition of a slant asymptotes and how to find them on a graph. Four fully worked-out practice examples close out the lesson.
A horizontal asymptote is a way of asking what happens to a rational function in the "long run". Is there a vertical location which the function approaches as the horizontal location "slides to infinity"? Symbolically, we can express a horizontal
x → ±∞ ⇒ f(x) → b.
Informally, a horizontal asymptote is a vertical height that the function is "pulled" towards as it moves very far left or right.
Not all rational functions have horizontal asymptotes.
To find a horizontal asymptote, expand the polynomials of the numerator and denominator so you have something in the form
an xn + an−1 xn−1 + …+ a1 x + a0
bm xm + bm−1 xm−1 + …+ b1 x + b0
Notice that n is the numerator's degree and m is the denominator's degree. There are three possibilities:
If n < m, then there is a horizontal asymptote at y=0 (the x-axis).
If n=m, then there is a horizontal asymptote based on the ratio of the leading coefficients:
If n > m, there is no horizontal asymptote.
A slant asymptote is similar to a horizontal asymptote, but instead of approaching a horizontal line, it approaches some slanted line in the "long run". It doesn't settle down to a single value, but it does get "pulled" along the slanted line
as it moves very far left or right.
A rational function has a slant asymptote if the degree of the numerator is exactly one greater than the degree of the denominator (n = m+1 from the above). We find the asymptote by polynomial division. Break the function into two parts: a portion
with no denominator (the asymptote) and the remainder to the division still over the denominator (which goes to 0 in long term). [This method also works to find horizontal asymptotes.]
Graphically, we represent horizontal and slant asymptotes the same way we did vertical asymptotes: with a dashed line. However, unlike vertical asymptotes, the graph can cross over a horizontal or slant asymptote. Furthermore, there is only ever one horizontal/slant
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.