http://www.ck12.org Chapter 2. Polynomials and Rational Functions
Sometimes functions flatten out and other times functions increase or decrease without bound. There are basically
three cases.
Case 1:The first case is the function flattens out to 0 asxgets infinitely large or infinitely small. This happens when
the degree of the numerator is less than the degree of the denominator. The degree is determined by the greatest
exponent ofx.
f(x) =^2 x(^8) + 3 x (^2) + 100
x^9 − 12
One way to reason through why this makes sense is because whenxis a ridiculously large number then most parts
of the function hardly make any impact. The 100 for example is nothing in comparison and neither is the 3x^2. The
two important terms to compare arex^8 andx^9. The 2 isn’t even important now because ifxis even just a million
than thex^9 will be a million times bigger than thex^8 and the 2 hardly matters again. Essentially, whenxgets big
enough, this function acts like^1 xwhich has a horizontal asymptote of 0.
Case 2:The degree of the numerator is equal to the degree of the denominator. In this case, the horizontal asymptote
is equal to the ratio of the leading coefficients.
f(x) =^6 x
(^4) − 3 x (^3) + 12 x (^2) − 9
3 x^4 + 144 x− 0. 001
Notice how the degree of both the numerator and the denominator is 4. This means that the horizontal asymptote is
y=^63 =2. One way to reason through why this makes sense is because whenxgets to be a very large number all the
smaller powers will not really make much of an impact. The biggest contributors are only the biggest powers. Then
the value of the numerator will be about twice that of the denominator. Asxgets even bigger, then the function will
get even closer to 2.
Case 3: The degree of the numerator is greater than the degree of the denominator. In this case there does not exist
a horizontal asymptote and you must determine if the function increases or decreases without bound in both the left
and right directions.
Example A
Identify the horizontal asymptotes of the following 3 functions:
a. f(x) =^4 x
(^3) + 99
3 x^4 − 99
b.h(x) =^234 x
(^45) − 45 x (^234) + 100
x^235
c.g(x) =x
(^3) + 3 x 6
x^3 − 6 x^6
Solution:
a.y=0 because the degree of the numerator is 3 and the degree of the denominator is 4, so the denominator gets
bigger eventually and the fraction approaches 0.
b.y=0 because the degree of the numerator is 234 which is smaller than the degree of the denominator (235).
c.y=−^12 because the degree of both the numerator and the denominator is 6 so the horizontal asymptote is the
ratio of the leading coefficients. Note that leading refers to the coefficients of the greatest powers ofxnot
the coefficients that happen to be written out front. Convention usually tells you to write the powers ofxin
descending order.