End BehaviorUnit 3 Lesson 2c

End Behavior

• End Behavior is how a function behaves as x approaches infinity ∞ (on the right) or negative infinity -∞ (on the left).

• We write “the limit as x goes to infinity” as

limx

Example 1: Describe the End Behavior

• Describe the end behavior of 2y x

2limxx

2limx

x

4

2

-2

f x = x2

Example 2 : Describe the End Behavior

• Describe the end behavior of

2xy

lim 2xx

lim 2xx

4

2

f x = 2x

01. If the limit approaches a constant as x goes

to + or – infinity, we say it has a horizontal asymptote at y = that constant.

2. Does the function above have a horizontal asymptote?

• Examine the end behavior and asymptotes of

Example 3: Describe the End Behavior

1( )

2f x

x

lim ( )x

f x

lim ( )x

f x

6

4

2

-2

-4

-5 5

f x = 1

x-2

0

0

yes, 0y Any horizontal asymptotes?

• Examine the end behavior of

Conclusion:

( ) cos( )f x x2

-2

-5 5

f x = cos x

lim ( )x

f x

limx

Does Not Exist (DNE) Does Not Exist (DNE)

Example 4: Describe the End Behavior

• Determine the end behavior of the odd function

Example 5: Describe the End Behavior

3( )g x xlim ( )xg x

lim ( )x

g x

8

6

4

2

-2

-4

-6

-8

f x = x3

lim ( )xg x

lim ( )x

g x

• Determine the end behavior of the function

Example 6: Describe the End Behavior

( ) sing x x

lim ( )xg x

lim ( )

xg x

0 0

6.28 12.56–6.28–12.56 x

3

–3

y

• Determine the end behavior of the function

Example 7: Describe the End Behavior

sin( )

xg x

x

lim ( )xg x

lim ( )

xg x

1

-10 -5 5 10 15

f x = sin x

x

0 0

Closure: Possible End Behaviors:

There are three possible end behaviors: 1. The values of f(x) can increase or decrease without bound (to or - ) 2. The values of f(x) can approach some number L 3. The values of f(x) can follow neither of these patterns (such as

oscillating between two values).

Which functions on the previous slides match to the types described above?