END BEHAVIOR MODELS

Section 2.2b

End Behavior ModelsFor large values of x, we can sometimes model the behavior ofa complicated function by a simpler one that acts in virtually thesame way…

4 3 23 2 3 5 6f x x x x x 43g x xEx: Given:

Show that while f and g are quite different for numerically smallvalues of x, they are virtually identical for |x| large.

First, graph in 2,2 by 5,20 What do you notice?

Next, graph in 20,20 by 100000,500000How about now?

End Behavior Models

4 3 23 2 3 5 6f x x x x x 43g x xShow that while f and g are quite different for numerically smallvalues of x, they are virtually identical for |x| large.

We can test our claim algebraically by examining the ratio of thetwo functions as x approaches positive or negative infinity:

limx

f x

g x

4 3 2

4

3 2 3 5 6lim

3x

x x x x

x

2 3 4

2 1 5 2lim 1

3 3x x x x x

1Convincing Evidence???

Definition: End Behavior ModelThe function g is

(a) a right end behavior model for f if and only if

lim 1x

f x

g x

(b) a left end behavior model for f if and only if

lim 1x

f x

g x

Note: A function’s right and left end behavior models need notbe the same function!!! (but when a function fits both, it issimply called an end behavior model)

Guided PracticeFor all polynomials, an end behavior model is simplythe leading term!

For all rational functions, an end behavior model issimply the ratio of the leading terms from thenumerator and denominator!

Then simply use some logical thinking to establish theexistence of any horizontal asymptotes…

Guided Practice

3 24 2 1f x x x x

For each of the following, (a) find a power function end behaviormodel for f, and (b) identify any horizontal asymptotes.

34y xE.B.M.: H.A.: None

Guided PracticeFor each of the following, (a) find a power function end behaviormodel for f, and (b) identify any horizontal asymptotes.

2

2

3 5

4

x xf x

x

2

2

33

xy

x E.B.M.: H.A.: 3y

Guided PracticeFor each of the following, (a) find a power function end behaviormodel for f, and (b) identify any horizontal asymptotes.

4 2

2

2 3

4

x x xf x

x

42

2

xy x

x

E.B.M.: H.A.: None

Guided PracticeFor each of the following, (a) find a power function end behaviormodel for f, and (b) identify any horizontal asymptotes.

3

7 3

2 4

xf x

x

3 2

7 7

2 2

xy

x x E.B.M.: H.A.: 0y

Guided PracticeFind the limit of f (x) as x approaches negative and positiveinfinity, and as x approaches zero from the left and from the right.

2

2, 01

1 , 0

xx

f x x

x x

limx

f x

2lim

1x

x

x

1

limx

f x 2

1limx x

0

0

limx

f x 0

2lim

1x

x

x

0 2

0 1

2

0

limx

f x 20

1limx x