Reading Quiz – 2.2

In your own words, describe what a “End Behavior Model” is.

Limits involving infinity2.2

Goal

• I will be able to define vertical and horizontal asymptotes, and calculate limits to infinity.

Definition

•  

Example

•  

The line y=b is a horizontal asymptote if:

limx

f x b

or limx

f x b

Definition

1f xx

1lim 0x x

As the denominator gets larger, the value of the fraction gets smaller. So there is a horizontal asymptote at y=0.

Example

Find the horizontal asymptote of

Example: 5 sinlimx

x xx

Find:

5 sinlimx

x xx x

sinlim 5 limx x

xx

5 0

5

How to find the Horizontal asymptotes

• Divide the lead terms (ones with the biggest exponent)

Example

=

H.A. at y=0. If degree of numerator is smaller than denominator, then H.A is 0.

You try!

• Find the H.A of

• Answer: H.A at y=1/2

Example

==If degree is bigger in numerator, then there is no H.A, and the function goes on to infinity.

Finding the H.A also gives you the end behavior

• An End behavior model is a simple function that models what a more complex function does at infinity.

End Behavior Models:

Test to check if a function is a end behavior model.

A function g is:

a right end behavior model for f if and only if

lim 1x

f xg x

a left end behavior model for f if and only if

lim 1x

f xg x

5 4 2

2

2 13 5 7x x xf xx x

Example:

Right-end behavior models give us:

5

2

23xx

323x

dominant terms in numerator and denominator

Test ofmodel

Our modelis correct.

xf x x e Example:

As , approaches zero.x xe (The x term dominates.)

g x x becomes a right-end behavior model.

limx

x

x ex

lim1x

x

ex

1 0 1

xh x e becomes a left-end behavior model.

limx

xx

x ee

lim 1xx

xe

0 1 1

As , increases faster than x decreases,x xe

therefore is dominant.xe

Test ofmodel

Our modelis correct.

xf x x e Example 7:

g x x becomes a right-end behavior model.

xh x e becomes a left-end behavior model.

On your calculator, graph:

1

2

3

x

x

y x

y e

y x e

10 10x

1 9y

Use:

Don’t Forget…

• You can always look at the graph to see what is happening with the function if you are unsure.

Homework

• 2.2 AND read pages 78-83

• Reading quiz tomorrow!