Finding Vertical, Horizontal and Oblique Asymptotes of Rational Functions Vertical Asymptotes (1) Vertical asymptotes exist when the rational function is in lowest terms and its denominator polynomial can be equal to 0 for some real values of x. How to find them? Set the denominator = 0, and solve for x. Whatever x equal to (2), are the vertical asymptotes. Example: Horizontal Asymptotes (3) A horizontal asymptote exists only when the degree of numerator polynomial is less than or equal to that of denominator polynomial. How to find them? 1. If the degree of the numerator is less than that of the denominator (i.e., n < m) , then y = 0 (the x-axis) is the horizontal asymptote (always!) 2. If the degree of the numerator is equal to that of the denominator (i.e., n = m), then y = an/bm (the quotient of the leading coefficients) is the horizontal asymptote. Example 1: Example 2: Oblique Asymptotes (3) An oblique (or slant) asymptote exists only when the degree of numerator polynomial is exactly 1 higher than that of denominator polynomial (i.e., n = m + 1). How to find them? Divide the numerator polynomial by the denominator polynomial using long division, the quotient polynomial is the oblique asymptote. Example: Notes: A rational function can have more than one vertical asymptotes: R(x) = 1/(x2 – 1); or may have no vertical asymptotes at all: R(x) = 1/(x2 + 1). Real solutions only. A rational function can only have at most one horizontal or oblique asymptote, but not both. If it has a horizontal asymptote, then it doesn’t have a oblique asymptote, and vice versa. It may have no horizontal or oblique asymptotes at all, that is, when the degree of the numerator is at least 2 higher than that of denominator.

Vertical, Horizontal and Oblique Asymptotes of Rational Functions—In Depth We’ve learned when vertical, horizontal and oblique asymptotes will exist and how to find them (if they do exist). We will discuss now what exactly they are and what they do. This goes back to the aged-old question: What is an asymptote? Most people will say: An asymptote is a line that a curve is approaching, but never touch it. This definition is not exactly accurate. The curve can actually touch the asymptote, and as many times as it wants. See graph on the right. Fortunately, the graph above isn’t a rational function since the graph of a rational function 1. can never touch its vertical asymptote(s), and 2. will not touch its horizontal (or oblique) asymptote as it’s approaching it, but it can touch it when it isn’t approaching it. x y Vertical Asymptotes A vertical asymptote is a vertical line that determines the behavior of the graph of a rational function as it is approaching to an x-value at which the function is undefined. Example 1: x 0.9 0.99 1 1.01 1.1 2 R(x) Horizontal Asymptotes and Oblique Asymptotes A horizontal asymptote is a horizontal line (and an oblique asymptote is a slant line) that determines the end behavior of the graph of a rational function (i.e., as x becomes very large (approaching ) or very small (approaching –)). Example 1: Example 2: x – –1000 –100 –10 10 100 1000  R(x) x – –1000 –100 –10 10 100 1000  R(x) x + 2

How to plot the graph of a rational function? Find and draw the asymptotes (vertical, horizontal, and/or oblique) first, because they serve as guidelines/boundaries for the graph of rational functions. Example 1: Example 2: Example 3: “Punctured” Holes A rational function can also have a hole (or holes) in it. It occurs when the rational function is undefined at an x-value which happens to be a zero for both the numerator and the denominator. Example: We see that R(x) is undefined at x = 1 and x = –1. However, since x = –1 is not a zero of the numerator, therefore it becomes a vertical asymptote. On the other hand, x = 1 is also a zero of the numerator, therefore it becomes a hole. x 0.9 0.99 1 1.01 1.1 2 R(x) x –2 –1.1 –1.01 –1 –0.99 –0.9 R(x)