1. End Behavior Models
Section 2.2b

2. End Behavior Models
For large values of x, we can sometimes model the behavior of
a complicated function by a simpler one that acts in virtually the
same way…
Ex: Given:
Show that while f and g are quite different for numerically small
values of x, they are virtually identical for |x| large.
First, graph in by
What do you notice?
Next, graph in by
How about now?

3. End Behavior Models
Show that while f and g are quite different for numerically small
values of x, they are virtually identical for |x| large.
We can test our claim algebraically by examining the ratio of the
two functions as x approaches positive or negative infinity:
Convincing Evidence???

4. Definition: End Behavior Model
The function g is
(a) a right end behavior model for f if and only if
(b) a left end behavior model for f if and only if
Note: A function’s right and left end behavior models need not
be the same function!!! (but when a function fits both, it is
simply called an end behavior model)

5. Guided Practice For all polynomials, an end behavior model is simply
the leading term!
For all rational functions, an end behavior model is
simply the ratio of the leading terms from the
numerator and denominator!
Then simply use some logical thinking to establish the
existence of any horizontal asymptotes…

6. Guided Practice E.B.M.: H.A.: None
For each of the following, (a) find a power function end behavior
model for f, and (b) identify any horizontal asymptotes.
E.B.M.:
H.A.: None

7. Guided Practice E.B.M.: H.A.:
For each of the following, (a) find a power function end behavior
model for f, and (b) identify any horizontal asymptotes.
E.B.M.:
H.A.:

8. Guided Practice E.B.M.: H.A.: None
For each of the following, (a) find a power function end behavior
model for f, and (b) identify any horizontal asymptotes.
E.B.M.:
H.A.: None

9. Guided Practice E.B.M.: H.A.:
For each of the following, (a) find a power function end behavior
model for f, and (b) identify any horizontal asymptotes.
E.B.M.:
H.A.:

10. Guided Practice
Find the limit of f (x) as x approaches negative and positive
infinity, and as x approaches zero from the left and from the right.