Answer:

While vertical asymptotes describe the behavior of a graph as the output gets very large or very small, horizontal asymptotes help describe the behavior of a graph as the input gets very large or very small. Recall that a polynomial’s end behavior will mirror that of the leading term. Likewise, a rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions.

There are three distinct outcomes when checking for horizontal asymptotes:

Case 1: If the degree of the denominator > degree of the numerator, there is a horizontal asymptote at  

y

=

0

.

Example:  

f

(

x

)

=

4

x

+

2

x

2

+

4

x

−

5

In this case the end behavior is  

f

(

x

)

≈

4

x

x

2

=

4

x

. This tells us that, as the inputs increase or decrease without bound, this function will behave similarly to the function  

g

(

x

)

=

4

x

, and the outputs will approach zero, resulting in a horizontal asymptote at  

y

=

0

. Note that this graph crosses the horizontal asymptote.