There are three rules that horizontal asymptotes follow depending on the degree of the polynomials involved in the rational expression. Before we begin, let's define our function like this:
horizontal asymptote
Our function has a polynomial of degree n on top and a polynomial of degree m on the bottom. Our horizontal asymptote rules are based on these degrees.
1. When n is less than m, the horizontal asymptote is y = 0 or the x-axis.
2. When n is equal to m, then the horizontal asymptote is equal to y = a/b, the leading coefficient of numerator/the leading coeffcient of denominator.
3. When n is greater than m, there is no horizontal asymptote.
The degrees of the polynomials in the function determine whether there is a horizontal asymptote and where it will be.
Vertical asymptotes are straight lines of the equation x = a, toward which a function r(x) approaches infinitesimally closely, but never reaches the line, as r(x) increases without bound.
Let's see how we can use these rules to figure out horizontal asymptotes and vertical Asymptote.
Steps:
1. Input a expression for f(x), input a expression for g(x). try all three cases. Guess the Asymptote(s)
2. check the Horizontal Asymptote box to see if you have correct answer
3. check the Vertical Asymptote.