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

10
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 3 7 2 lim 2 3 x x x x

Transcript of Objectives: 1.Be able to find the limit of a function using direct substitution. 2.Be able to find...

Page 1: 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.

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

Page 2: 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.

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

Page 3: 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.

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

Page 4: 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.

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

Page 5: 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.

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

Page 6: 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.

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

MUST SHOW YOUR WORK TO GET CREDIT!!!!

Page 7: 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.

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)

Page 8: 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.

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)

Page 9: 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.

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

Page 10: 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.

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

MUST SHOW YOUR WORK TO GET CREDIT!!!!