AP CALCULUS

1004 Limits pt.3Limits at Infinity and End Behavior

REVIEW:

ALGEBRA is a ________________________ machine that ___________________ a function ___________ a point.

CALCULUS is a ________________________ machine that ___________________________ a function ___________ a point

END BEHAVIOR

lim x

LIMITS AT INFINITY

Part 2: End Behavior GENERAL IDEA: The behavior of a function as x gets very large ( in a positive or negative direction)

ALREADY KNOW many functions:

Polynomial:

Trigonometric:

Exponential:

Logarithmic:

NOW primarily RATIONAL and COMPOSITE.

( )x

END Behavior: Limit

Layman’s Description:

Notation:

Horizontal Asymptotes:

Note:

( )x

m

GNAW: Graphing

EX:

EX:

2

2

3( )1

xf xx

2

2( )1

xf xx

( )

( )x

x

Lim f x

Lim f x

( )

( )x

x

Lim f x

Lim f x

GNAW: Algebraic

Method: DIRECT SUBSTITUTION gives a second INDETERMINANT FORM

Theorem:

End Behavior Models

EX: (with Theorem)

2 11x

xLimx

3

2

2 53 1x

xLimx

End Behavior Models

2

2

2 53 1x

xLimx

Summary: ________________________________________

A).

B).

C).

3

2

2 53 1x

xLimx

2

2 53 1x

xLimx

Special Case

Special Note: If the function involves an ABSOLUTE VALUE it may have TWO Horizontal Asymptotes.

REM also:

EX: EX:

2x x

4( )

1x

f xx

2

2( )1

xf xx

Review

Last Update:08/09/10