2.10. Horizontal Asymptotes http://www.ck12.org
Example B
Identify the vertical and horizontal asymptotes of the following rational function.
f(x) =((xx−− 12 )()( 44 xx++^33 )()(xx−−^46 ))Solution: There is factor that cancels that is neither a horizontal or vertical asymptote. The vertical asymptotes
occur atx=1 andx=6. To obtain the horizontal asymptote you could methodically multiply out each binomial,
however since most of those terms do not matter, it is more efficient to first determine the relative powers of the
numerator and the denominator. In this case they both happen to be 3. Next determine the coefficient of the cubic
terms only. The numerator will have 4x^3 and the denominator will have 4x^3 and so the horizontal asymptote will
occur aty=^44 =1.
Example C
Describe the right hand end behavior of the following function.
Solution: Notice how quickly this dampening wave function settles down. There seems to be an obvious horizontal
axis on the right aty= 1
Concept Problem Revisited
As in Example C, functions may touch and pass through horizontal asymptotes without limit. This is a difference
between vertical and horizontal asymptotes. In calculus, there are rigorous proofs to show that functions like the
one in Example C do become arbitrarily close to the asymptote.
Vocabulary
Ahorizontal asymptoteis a flat dotted line that indicates where a function goes asxget infinitely large or infinitely
small.
End behavioris a term that asks you to describe the horizontal asymptotes.
Avertical asymptoteis a dashed vertical line that indicates that as a function approaches, it shoots off to positive or
negative infinity without ever actually touching the line.
Arational functionis a function with at least one rational expression.
Arational expressionis a ratio of two polynomial expressions.