Section 5.2.3 Day 2Limits Review

Lesson Objective:

Students will:

• Understand the necessary conditions for a limit to exist.

• Practice finding limits.• Predict limits from tables.

  (1). General Idea: Behavior of a function very near the point where

(2). Layman’s Description of Limit (Local Behavior)

 L If x approaches a, from both sides, then the

function approaches a single number, L.

a

(3). Notation

(4). Mantra

x ax a

lim ( )x a

f x L

, then .If x a y L

G N A W

4 Ways to Depict Limits

Graphically

Numerically

Analytically

Words

G N A W Graphically

2x

Lim f x

1x

Lim f x

FINDING LIMITS

2( ) 2 1f x x x 3 22 2 4( )

2x x xf x

x

• Graphically 2x

G N A W

4

28xLim

x

3.9 3.99 3.999 4 4.001 4.01 4.1

7

284

Mantra:

• Numerically

• Words

FINDING LIMITS

• Analytically

2

3

93x

xLimx

23

39x

xLimx

Rem: Always start with Direct Substitution

FINDING LIMITS

• Analytically

0( )

xLimf x

1 if 0 ( ) 3x-1 if x 0

x xIff x

  

(1). If a is in the domain: Use Substitution

x a

1lim( 3)x

22

lim(8 4)x

x

  (2). If a is not in the domain: Start with Substitution

x a

2

3

9lim 3x

xx

3, then 6.If x y

If the result is , ( this step must be shown),

then factor and substitute again.

00

3

( 3)( 3)lim 3x

x xx

A.

Graphically

Since the function is undefined at x = 3, this produces a “hole” in the graph.

x

y

  (2). If a is not in the domain: Start with Substitution

x a

23

3lim 9x

xx

3, then .If x y

If the result is , ( this step must be shown),

then factor and substitute again.

#0

3

3lim ( 3)( 3)x

xx x

B.

Graphically

Since the function is undefined at x = 3, this produces a “hole” and an asymptote in the graph.

x

y

  

, when x = odd #, then:ny x

Polynomials (Power Functions)

, when x = even #, then:ny x

lim ( )lim ( )

x

x

f xf x

®¥

®- ¥

=+¥=- ¥

lim ( )lim ( )

x

x

f xf x

®¥

®- ¥

=+¥=+¥

  Remember:

Rational Functions

1lim 0x x®¥

=

#lim 0x x®¥

=

#lim 0nx x®¥=

  Polynomial:Polynomial

Rational Functions

Divide all terms in the numerator & denominator by the largest degree in the denominator.

2

24 3 5) 7 5 1x xa yx x

  

Rational FunctionsDivide all terms in the numerator & denominator by the largest degree in the

denominator.2

33 1) 5 2xb yx x

34 5 1) 7 2x xc y

x

  

Leading Term Test

Take the ratio of the leading terms.

) If a number, then that number is the limit. The end behavior is an horizontal asymptote #.a

y

) If x is left in the denominator, then the limit is 0. The end behavior is and horizontal asymptote 0. b

y

c) If x is in the numerator, then there is no limit. The end behavior mimics the polynomial.

  

Examples

1) lim ( 1)( 2)x

xx x

  

Examples

2

28 26 152) lim 2 15x

x xx x

  

Examples

2 93) lim 2x

xx

AssignmentWS 13.1  #7-32