Objectives:1. Be able to find the limit of a function using direct substitution.2. Be able to find the limit of function that is in the indeterminate

form.

Critical Vocabulary:Limit, Direct substitution, Indeterminate Form

Warm Up: Find the limit numerically and graphically

3

372lim

2

3

x

xxx

Warm Up: Find the limit numerically and graphically

3

372lim

2

3

x

xxx

x

f(x)

-5

-9

-4

-7

-3.1

-5.2

-3.01

-5.02

-3.001

-5.002

-1

-1

-2

-3

-2.9

-4.8

-2.99

-4.98

-2.999

-4.998

-3

?

53

372lim

2

3

x

xxx

I. Direct Substitution

Example 1: Find the limit

1lim 2

1

x

xf(1) = (1)2 + 1 f(1) = 1 + 1

f(1) = 2

21lim 2

1

x

x

Example 2: Find the limit 3

4lim

2

2

x

xx

32

4)2()2(

2

f

1

0)2(

f

0)2( f

03

4lim

2

2

x

xx

I. Direct Substitution

Example 3: Find the limit 32

1lim

2

2

3

xx

x

x

323

23

2

123

2

32

f

323

49

2

123

2

3

f

323

29

123

2

3

f

321

2

3

f

6

1

2

3

f

6

1

32

1lim

2

2

3

xx

x

x

I. Direct Substitution

Example 4: Find the limit 1

32lim

2

1

x

xxx

11

31)1(2)1(

2

f

11

312)1(

f

0

0)1( f

Direct Substitution will not work on this because it yields an indeterminate form.

Undefined

Section 3.3 Assignment Part 1Page 226-228 #1-25 odd

MUST SHOW YOUR WORK TO GET CREDIT!!!!

II. Indeterminate Form

1. Factor and Cancel

2. Rationalizing Technique (Use when you have a radical)

3. Use Algebra (Common Denominators/Expand)

Then try to use direct substitution a second time!!!!!!

Example 4: Find the limit 1

32lim

2

1

x

xxx

1

)1)(32(lim

1

x

xxx

32lim1

xx

f(-1) = 2(-1) - 3 f(-1) = -5

532lim1

xx

Causes a hole in the graph

or an asymptote

This graph will have a hole at

(-1, -5)

II. Indeterminate Form

1. Factor and Cancel

2. Rationalizing Technique (Use when you have a radical)

3. Use Algebra (Common Denominators/Expand)

Then try to use direct substitution a second time!!!!!!

Example 5: Find the limit xx

xx 20lim

)1(lim

0 xx

xx

1

1lim

0 xx

f(0) = 1/(0-1)

f(0) = -1

1lim20

xx

xx

Causes a hole in the graph

or an asymptote

This graph will have a hole at

(0, -1)

II. Indeterminate Form

1. Factor and Cancel

2. Rationalizing Technique (Use when you have a radical)

3. Use Algebra (Common Denominators/Expand)

Then try to use direct substitution a second time!!!!!!

Example 6: Find the limit

x

xx

22lim

0

22

22

x

x

22lim

0

xx

xx

22

1lim

0

xx

202

1)0(

f

22

1

4

2

4

222lim

0

x

xx

Section 3.3 Assignment Part 2Page 226-228 #31-49 odd, 57

MUST SHOW YOUR WORK TO GET CREDIT!!!!