CK-12-Pre-Calculus Concepts

2.10. Horizontal Asymptotes http://www.ck12.org

a. The degrees of the numerator and the denominator are equal so the horizontal asymptote isy=3.

b. The degrees of the numerator and the denominators are equal again so the horizontal asymptote isy=af.

c. Asxgets infinitely large,

g(x) =hf((xx))=

(^3) xx (^66) +− 99972 x
ax^4 +f xbx (^4) +^3 +gxcx (^3) +^2 +hxdx 2 +e≈

3

af =

3 f

a

When you study calculus, you will learn the rigorous techniques that enable you to feel more confident about results
like this.

Practice

Identify the horizontal asymptotes, if they exist, for the following functions.

1. f(x) =^5 x

(^4) − 2 x
x^4 + 32
2.g(x) =^3 x
(^4) − 2 x 6
−x^4 + 2
3.h(x) =^3 x
(^4) − 5 x
8 x^3 + 3 x^4

4. j(x) =^2 x

(^3) − 15 x
−x^4 + 3
5.k(x) =^2 x
(^5) − 3 x
5 x^2 + 3 x^4 + 2 x− 7 x^5

6. f(x) =ax

(^14) +bx (^23) +cx (^12) +dx+e
f x^24 +gx^23 +hx^21
7.g(x) =|((xx−− 21 ))(|·x(x+−^4 ) 1 )

8. Write a function that fits the following criteria:
   
   
   • Vertical asymptotes atx=1 andx= 4
   
   
   • Zeroes at 3 and 5
   
   
   • Hole whenx= 6
   
   
   • Horizontal asymptote aty=^23
   
   
   
   


9. Write a function that fits the following criteria:
   
   
   • Vertical asymptotes atx=−2 andx= 2
   
   
   • Zeroes at 1 and 5
   
   
   • Hole whenx= 3
   
   
   • Horizontal asymptote aty= 1
   
   
   
   


10. Write a function that fits the following criteria:

• Vertical asymptotes atx=0 andx= 3


• Zeroes at 1 and 2


• Hole whenx= 8


• Horizontal asymptote aty= 2

11. Write a function that fits the following criteria:

• Vertical asymptotes at 2 and 6


• Zero at 5


• Hole whenx= 4