Drill Find f(2)f(x) =

f(x) =

f(x) =

f(x) =

452 23 xx

3

543

2

x

x

)2

sin(x

21

1213

2 xx

xx

day 1 homework p 66 1-6 8-14(even) 16-18 20-28 (even)

day 2 homework p 66-67 29-50

Lesson 21Rates of Changes and Limits

Average and Instantaneous Speed

A moving bodyrsquos average speed during an interval of time is found by dividing the distance covered by elapsed time

Example 1A rock breaks lose from the top of a tall cliff What

is its average speed during the first 2 seconds of fall

SolutionUse the equation y(t) = 16t2 where y = distance

and t = time in secondsFor the first two seconds of the fall starting time is

0 and ending time is 2Use the formula sec32

2

64

02

)0(16)2(16 22ft

timeoflength

travdist

t

y

Finding Instantaneous SpeedTo find instantaneous speed at a time t and some

time h seconds from t use the formula

Find the speed of the rock from example 1 at the instant t = 2 seconds

If we are looking for the speed to be instantaneous h = 0 but that can only be evaluate at the end so 64 + 16(0) = 64 ftsec

h

tyhty

tht

tyhty

t

y )()(

)(

)()(

hh

hh

h

hh

h

hh

h

h

h

tyhy

t

y

16641664

6416646464)44(16

)2(16)2(16

2)2(

)()2(

2

22

22

Definition of LimitThe limit is a method of evaluating an

expression as an argument approaches a value This value can be any point on the number line and often limits are evaluated as an argument approaches infinity or minus infinity The following expression states that as x approaches the value c the function approaches the value L

When looking at a graph the limit of a rational function (f(x)g(x)) is the HORIZONTAL asymptote

Lxfcx

)(lim

Determining the limit by substitution and confirm graphically (put in calc)

)12(3lim 2

21

xx

x)1)21(2()21(3 2

)11)(41(3 51)2(43

Determine the limit graphically Then prove algebraically(Always FACTOR with rational expressions)

1

1lim

21

x

xx

the limit appears to

be 12

2

1

1

1

)1)(1(

1

1

1lim

21

xxx

x

x

xx

Determine the limit graphically Then prove algebraically(Always FACTOR with rational expressions)

24

23

0 163

85lim

xx

xxx

the limit appears to

be -12

516)0(3

8)0(5

163

85

)163(

)85(

163

85lim

22

22

2

24

23

0

x

x

xx

xx

xx

xxx

Properties of LimitsKnown limits

limit of the function with the constant k

limit of the identity function of x = c

Properties of Limits If L M c and k are real number and

kkcx

)(lim

cxcx

)(lim

Lxfcx

)(lim Mxgcx

)(lim

Properties of LimitsSum Rule

The limit of the sum of 2 functions is the sum of their limits

Difference RuleThe limit of the difference of 2 functions is

the difference of their limits

Product RuleThe limit of a product of 2 functions is the

product of their limits

MLxgxfcx

))()((lim

MLxgxfcx

))()((lim

MLxgxfcx

))()((lim

Properties of LimitsConstant Multiple Rule

The limit of a constant times a function is the constant times the limit of the function

Quotient RuleThe limit of a quotient of two functions is the

quotient of their limits provided that the denominator is not zero

Power RuleIf r and s are integers and s ne 0 then

Lkxfkcx

))((lim

0)(

)(lim

MM

L

xg

xfcx

srsr

cxLxf ))((lim

Using Properties of LimitsUse the observations that and

and the properties of limits to evaluate the following

limxc (x4 + 4x2 -3)limxc x4 + limxc 4x2 ndash limxc 3c4 + 4c2 -3

limxc x4 + x2 -1 x2 + 5 limxc x4 + limxc x2 - limxc 1

limxc x2 + limxc 5

c4 + c2 -1 c2 + 5

kkcx

)(lim cxcx

)(lim

DrillEvaluate the following limits

1 limx3 x2 (2 ndash x)

2 limx2 x2 + 2x + 4 x + 2

3 limx0 tanx you will want to use the tanx = sinxcosx x

4 Why does the following limit NOT existlimx2 x3 ndash 1

x - 2

Drill solutionsBy substitution1 -92 33 Using the product rule

limx0 (sinxx) (1cosx) limx0 (sinxx) limx0 (1cosx) Look at graph 1(1cosx) 1 1 = 1

4 since you cannot factor the given rational function you must just substitute 2 however by doing so the denominator would be 0 which is not possible

One-sided and Two-sided LimitsSometimes the values of a function f tend

to different limits as x approaches a number c from the left and from the right

Right-hand limit The limit of f as x approaches c from the

rightLeft-hand limit

The limit of f as x approaches c from the leftExample Use the graph to the right

) ( limx fc x

)(lim xfcx

1)(lim2

xfx

0)(lim2

xfx

Definition of Step FunctionA step function is a special type of function

whose graph is a series of line segments The graph of a step function looks like a

series of small steps There are two types of step functions

rounding UP and rounding DOWNRounding UP

Greatest Integer Functionf(x) = -int (-x)Ceiling FunctionExample f(4) = 1

Rounding DownLeast Integer Functionf(x) = int(x)Floor FunctionExample f(21) = 2

One-Sided and Two-sided LimitsA function f(x) has a limit as x approaches c

if and only if (IFF) the right-hand and left-hand limits at c exist and are equal

LandLLxfcxcxcx

limlim)(lim

Exploring Right-handed and Left-handed LimitsUse the graph and function below to

determine the limits

435

321

22211

101

)(

xx

xx

xx

xx

xf

)(lim0

xfx

)(lim1

xfx

)(lim1

xfx 1 10Therefore no limit exists

as x1 since they are NOT the same

The rest of the limits regarding this problem can be found on page 64

Exploring Left-handed and Right-Handed Limits

)(lim1

xfx

)(lim1

xfx

1 2)(lim

2xf

x )(lim

2xf

x

2 3

Sandwich TheoremIf g(x) lt f(x) lt h(x) for all x ne c in some interval

about c and limxc g(x) = limxc h(x) = L then limxc f(x) = L

Using the sandwich theoremShow that the limx0 [x2 sin(1x)] = 0We know that all of the values of the sine function

lie between -1 and 1 (range)The greatest value that sin(1x) can have is 1 and

the lowest value is -1 sohellip-x2 lt x2 sin(1x)] lt x2

Because the limx0 -x2 = 0 and limx0 x2 = 0 then limx0 [x2 sin(1x)] = 0

Closure

• Drill Find f(2)
• Lesson 21 Rates of Changes and Limits
• Average and Instantaneous Speed
• Finding Instantaneous Speed
• Definition of Limit
• Determining the limit by substitution and confirm graphically (
• Determine the limit graphically Then prove algebraically(Alwa
• Determine the limit graphically Then prove algebraically(Alwa (2)
• Properties of Limits
• Properties of Limits (2)
• Properties of Limits (3)
• Using Properties of Limits
• Drill
• Drill solutions
• One-sided and Two-sided Limits
• Definition of Step Function
• Rounding Down
• One-Sided and Two-sided Limits
• Exploring Right-handed and Left-handed Limits
• Exploring Left-handed and Right-Handed Limits
• Sandwich Theorem
• Closure

day 1 homework p 66 1-6 8-14(even) 16-18 20-28 (even)

day 2 homework p 66-67 29-50

Lesson 21Rates of Changes and Limits

Average and Instantaneous Speed

A moving bodyrsquos average speed during an interval of time is found by dividing the distance covered by elapsed time

Example 1A rock breaks lose from the top of a tall cliff What

is its average speed during the first 2 seconds of fall

SolutionUse the equation y(t) = 16t2 where y = distance

and t = time in secondsFor the first two seconds of the fall starting time is

0 and ending time is 2Use the formula sec32

2

64

02

)0(16)2(16 22ft

timeoflength

travdist

t

y

Finding Instantaneous SpeedTo find instantaneous speed at a time t and some

time h seconds from t use the formula

Find the speed of the rock from example 1 at the instant t = 2 seconds

If we are looking for the speed to be instantaneous h = 0 but that can only be evaluate at the end so 64 + 16(0) = 64 ftsec

h

tyhty

tht

tyhty

t

y )()(

)(

)()(

hh

hh

h

hh

h

hh

h

h

h

tyhy

t

y

16641664

6416646464)44(16

)2(16)2(16

2)2(

)()2(

2

22

22

Definition of LimitThe limit is a method of evaluating an

expression as an argument approaches a value This value can be any point on the number line and often limits are evaluated as an argument approaches infinity or minus infinity The following expression states that as x approaches the value c the function approaches the value L

When looking at a graph the limit of a rational function (f(x)g(x)) is the HORIZONTAL asymptote

Lxfcx

)(lim

Determining the limit by substitution and confirm graphically (put in calc)

)12(3lim 2

21

xx

x)1)21(2()21(3 2

)11)(41(3 51)2(43

Determine the limit graphically Then prove algebraically(Always FACTOR with rational expressions)

1

1lim

21

x

xx

the limit appears to

be 12

2

1

1

1

)1)(1(

1

1

1lim

21

xxx

x

x

xx

Determine the limit graphically Then prove algebraically(Always FACTOR with rational expressions)

24

23

0 163

85lim

xx

xxx

the limit appears to

be -12

516)0(3

8)0(5

163

85

)163(

)85(

163

85lim

22

22

2

24

23

0

x

x

xx

xx

xx

xxx

Properties of LimitsKnown limits

limit of the function with the constant k

limit of the identity function of x = c

Properties of Limits If L M c and k are real number and

kkcx

)(lim

cxcx

)(lim

Lxfcx

)(lim Mxgcx

)(lim

Properties of LimitsSum Rule

The limit of the sum of 2 functions is the sum of their limits

Difference RuleThe limit of the difference of 2 functions is

the difference of their limits

Product RuleThe limit of a product of 2 functions is the

product of their limits

MLxgxfcx

))()((lim

MLxgxfcx

))()((lim

MLxgxfcx

))()((lim

Properties of LimitsConstant Multiple Rule

The limit of a constant times a function is the constant times the limit of the function

Quotient RuleThe limit of a quotient of two functions is the

quotient of their limits provided that the denominator is not zero

Power RuleIf r and s are integers and s ne 0 then

Lkxfkcx

))((lim

0)(

)(lim

MM

L

xg

xfcx

srsr

cxLxf ))((lim

Using Properties of LimitsUse the observations that and

and the properties of limits to evaluate the following

limxc (x4 + 4x2 -3)limxc x4 + limxc 4x2 ndash limxc 3c4 + 4c2 -3

limxc x4 + x2 -1 x2 + 5 limxc x4 + limxc x2 - limxc 1

limxc x2 + limxc 5

c4 + c2 -1 c2 + 5

kkcx

)(lim cxcx

)(lim

DrillEvaluate the following limits

1 limx3 x2 (2 ndash x)

2 limx2 x2 + 2x + 4 x + 2

3 limx0 tanx you will want to use the tanx = sinxcosx x

4 Why does the following limit NOT existlimx2 x3 ndash 1

x - 2

Drill solutionsBy substitution1 -92 33 Using the product rule

limx0 (sinxx) (1cosx) limx0 (sinxx) limx0 (1cosx) Look at graph 1(1cosx) 1 1 = 1

4 since you cannot factor the given rational function you must just substitute 2 however by doing so the denominator would be 0 which is not possible

One-sided and Two-sided LimitsSometimes the values of a function f tend

to different limits as x approaches a number c from the left and from the right

Right-hand limit The limit of f as x approaches c from the

rightLeft-hand limit

The limit of f as x approaches c from the leftExample Use the graph to the right

) ( limx fc x

)(lim xfcx

1)(lim2

xfx

0)(lim2

xfx

Definition of Step FunctionA step function is a special type of function

whose graph is a series of line segments The graph of a step function looks like a

series of small steps There are two types of step functions

rounding UP and rounding DOWNRounding UP

Greatest Integer Functionf(x) = -int (-x)Ceiling FunctionExample f(4) = 1

Rounding DownLeast Integer Functionf(x) = int(x)Floor FunctionExample f(21) = 2

One-Sided and Two-sided LimitsA function f(x) has a limit as x approaches c

if and only if (IFF) the right-hand and left-hand limits at c exist and are equal

LandLLxfcxcxcx

limlim)(lim

Exploring Right-handed and Left-handed LimitsUse the graph and function below to

determine the limits

435

321

22211

101

)(

xx

xx

xx

xx

xf

)(lim0

xfx

)(lim1

xfx

)(lim1

xfx 1 10Therefore no limit exists

as x1 since they are NOT the same

The rest of the limits regarding this problem can be found on page 64

Exploring Left-handed and Right-Handed Limits

)(lim1

xfx

)(lim1

xfx

1 2)(lim

2xf

x )(lim

2xf

x

2 3

Sandwich TheoremIf g(x) lt f(x) lt h(x) for all x ne c in some interval

about c and limxc g(x) = limxc h(x) = L then limxc f(x) = L

Using the sandwich theoremShow that the limx0 [x2 sin(1x)] = 0We know that all of the values of the sine function

lie between -1 and 1 (range)The greatest value that sin(1x) can have is 1 and

the lowest value is -1 sohellip-x2 lt x2 sin(1x)] lt x2

Because the limx0 -x2 = 0 and limx0 x2 = 0 then limx0 [x2 sin(1x)] = 0

Closure

• Drill Find f(2)
• Lesson 21 Rates of Changes and Limits
• Average and Instantaneous Speed
• Finding Instantaneous Speed
• Definition of Limit
• Determining the limit by substitution and confirm graphically (
• Determine the limit graphically Then prove algebraically(Alwa
• Determine the limit graphically Then prove algebraically(Alwa (2)
• Properties of Limits
• Properties of Limits (2)
• Properties of Limits (3)
• Using Properties of Limits
• Drill
• Drill solutions
• One-sided and Two-sided Limits
• Definition of Step Function
• Rounding Down
• One-Sided and Two-sided Limits
• Exploring Right-handed and Left-handed Limits
• Exploring Left-handed and Right-Handed Limits
• Sandwich Theorem
• Closure

Average and Instantaneous Speed

A moving bodyrsquos average speed during an interval of time is found by dividing the distance covered by elapsed time

Example 1A rock breaks lose from the top of a tall cliff What

is its average speed during the first 2 seconds of fall

SolutionUse the equation y(t) = 16t2 where y = distance

and t = time in secondsFor the first two seconds of the fall starting time is

0 and ending time is 2Use the formula sec32

2

64

02

)0(16)2(16 22ft

timeoflength

travdist

t

y

Finding Instantaneous SpeedTo find instantaneous speed at a time t and some

time h seconds from t use the formula

Find the speed of the rock from example 1 at the instant t = 2 seconds

If we are looking for the speed to be instantaneous h = 0 but that can only be evaluate at the end so 64 + 16(0) = 64 ftsec

h

tyhty

tht

tyhty

t

y )()(

)(

)()(

hh

hh

h

hh

h

hh

h

h

h

tyhy

t

y

16641664

6416646464)44(16

)2(16)2(16

2)2(

)()2(

2

22

22

Definition of LimitThe limit is a method of evaluating an

expression as an argument approaches a value This value can be any point on the number line and often limits are evaluated as an argument approaches infinity or minus infinity The following expression states that as x approaches the value c the function approaches the value L

When looking at a graph the limit of a rational function (f(x)g(x)) is the HORIZONTAL asymptote

Lxfcx

)(lim

Determining the limit by substitution and confirm graphically (put in calc)

)12(3lim 2

21

xx

x)1)21(2()21(3 2

)11)(41(3 51)2(43

Determine the limit graphically Then prove algebraically(Always FACTOR with rational expressions)

1

1lim

21

x

xx

the limit appears to

be 12

2

1

1

1

)1)(1(

1

1

1lim

21

xxx

x

x

xx

Determine the limit graphically Then prove algebraically(Always FACTOR with rational expressions)

24

23

0 163

85lim

xx

xxx

the limit appears to

be -12

516)0(3

8)0(5

163

85

)163(

)85(

163

85lim

22

22

2

24

23

0

x

x

xx

xx

xx

xxx

Properties of LimitsKnown limits

limit of the function with the constant k

limit of the identity function of x = c

Properties of Limits If L M c and k are real number and

kkcx

)(lim

cxcx

)(lim

Lxfcx

)(lim Mxgcx

)(lim

Properties of LimitsSum Rule

The limit of the sum of 2 functions is the sum of their limits

Difference RuleThe limit of the difference of 2 functions is

the difference of their limits

Product RuleThe limit of a product of 2 functions is the

product of their limits

MLxgxfcx

))()((lim

MLxgxfcx

))()((lim

MLxgxfcx

))()((lim

Properties of LimitsConstant Multiple Rule

The limit of a constant times a function is the constant times the limit of the function

Quotient RuleThe limit of a quotient of two functions is the

quotient of their limits provided that the denominator is not zero

Power RuleIf r and s are integers and s ne 0 then

Lkxfkcx

))((lim

0)(

)(lim

MM

L

xg

xfcx

srsr

cxLxf ))((lim

Using Properties of LimitsUse the observations that and

and the properties of limits to evaluate the following

limxc (x4 + 4x2 -3)limxc x4 + limxc 4x2 ndash limxc 3c4 + 4c2 -3

limxc x4 + x2 -1 x2 + 5 limxc x4 + limxc x2 - limxc 1

limxc x2 + limxc 5

c4 + c2 -1 c2 + 5

kkcx

)(lim cxcx

)(lim

DrillEvaluate the following limits

1 limx3 x2 (2 ndash x)

2 limx2 x2 + 2x + 4 x + 2

3 limx0 tanx you will want to use the tanx = sinxcosx x

4 Why does the following limit NOT existlimx2 x3 ndash 1

x - 2

Drill solutionsBy substitution1 -92 33 Using the product rule

limx0 (sinxx) (1cosx) limx0 (sinxx) limx0 (1cosx) Look at graph 1(1cosx) 1 1 = 1

4 since you cannot factor the given rational function you must just substitute 2 however by doing so the denominator would be 0 which is not possible

One-sided and Two-sided LimitsSometimes the values of a function f tend

to different limits as x approaches a number c from the left and from the right

Right-hand limit The limit of f as x approaches c from the

rightLeft-hand limit

The limit of f as x approaches c from the leftExample Use the graph to the right

) ( limx fc x

)(lim xfcx

1)(lim2

xfx

0)(lim2

xfx

Definition of Step FunctionA step function is a special type of function

whose graph is a series of line segments The graph of a step function looks like a

series of small steps There are two types of step functions

rounding UP and rounding DOWNRounding UP

Greatest Integer Functionf(x) = -int (-x)Ceiling FunctionExample f(4) = 1

Rounding DownLeast Integer Functionf(x) = int(x)Floor FunctionExample f(21) = 2

One-Sided and Two-sided LimitsA function f(x) has a limit as x approaches c

if and only if (IFF) the right-hand and left-hand limits at c exist and are equal

LandLLxfcxcxcx

limlim)(lim

Exploring Right-handed and Left-handed LimitsUse the graph and function below to

determine the limits

435

321

22211

101

)(

xx

xx

xx

xx

xf

)(lim0

xfx

)(lim1

xfx

)(lim1

xfx 1 10Therefore no limit exists

as x1 since they are NOT the same

The rest of the limits regarding this problem can be found on page 64

Exploring Left-handed and Right-Handed Limits

)(lim1

xfx

)(lim1

xfx

1 2)(lim

2xf

x )(lim

2xf

x

2 3

Sandwich TheoremIf g(x) lt f(x) lt h(x) for all x ne c in some interval

about c and limxc g(x) = limxc h(x) = L then limxc f(x) = L

Using the sandwich theoremShow that the limx0 [x2 sin(1x)] = 0We know that all of the values of the sine function

lie between -1 and 1 (range)The greatest value that sin(1x) can have is 1 and

the lowest value is -1 sohellip-x2 lt x2 sin(1x)] lt x2

Because the limx0 -x2 = 0 and limx0 x2 = 0 then limx0 [x2 sin(1x)] = 0

Closure

• Drill Find f(2)
• Lesson 21 Rates of Changes and Limits
• Average and Instantaneous Speed
• Finding Instantaneous Speed
• Definition of Limit
• Determining the limit by substitution and confirm graphically (
• Determine the limit graphically Then prove algebraically(Alwa
• Determine the limit graphically Then prove algebraically(Alwa (2)
• Properties of Limits
• Properties of Limits (2)
• Properties of Limits (3)
• Using Properties of Limits
• Drill
• Drill solutions
• One-sided and Two-sided Limits
• Definition of Step Function
• Rounding Down
• One-Sided and Two-sided Limits
• Exploring Right-handed and Left-handed Limits
• Exploring Left-handed and Right-Handed Limits
• Sandwich Theorem
• Closure

Finding Instantaneous SpeedTo find instantaneous speed at a time t and some

time h seconds from t use the formula

Find the speed of the rock from example 1 at the instant t = 2 seconds

If we are looking for the speed to be instantaneous h = 0 but that can only be evaluate at the end so 64 + 16(0) = 64 ftsec

h

tyhty

tht

tyhty

t

y )()(

)(

)()(

hh

hh

h

hh

h

hh

h

h

h

tyhy

t

y

16641664

6416646464)44(16

)2(16)2(16

2)2(

)()2(

2

22

22

Definition of LimitThe limit is a method of evaluating an

expression as an argument approaches a value This value can be any point on the number line and often limits are evaluated as an argument approaches infinity or minus infinity The following expression states that as x approaches the value c the function approaches the value L

When looking at a graph the limit of a rational function (f(x)g(x)) is the HORIZONTAL asymptote

Lxfcx

)(lim

Determining the limit by substitution and confirm graphically (put in calc)

)12(3lim 2

21

xx

x)1)21(2()21(3 2

)11)(41(3 51)2(43

Determine the limit graphically Then prove algebraically(Always FACTOR with rational expressions)

1

1lim

21

x

xx

the limit appears to

be 12

2

1

1

1

)1)(1(

1

1

1lim

21

xxx

x

x

xx

Determine the limit graphically Then prove algebraically(Always FACTOR with rational expressions)

24

23

0 163

85lim

xx

xxx

the limit appears to

be -12

516)0(3

8)0(5

163

85

)163(

)85(

163

85lim

22

22

2

24

23

0

x

x

xx

xx

xx

xxx

Properties of LimitsKnown limits

limit of the function with the constant k

limit of the identity function of x = c

Properties of Limits If L M c and k are real number and

kkcx

)(lim

cxcx

)(lim

Lxfcx

)(lim Mxgcx

)(lim

Properties of LimitsSum Rule

The limit of the sum of 2 functions is the sum of their limits

Difference RuleThe limit of the difference of 2 functions is

the difference of their limits

Product RuleThe limit of a product of 2 functions is the

product of their limits

MLxgxfcx

))()((lim

MLxgxfcx

))()((lim

MLxgxfcx

))()((lim

Properties of LimitsConstant Multiple Rule

The limit of a constant times a function is the constant times the limit of the function

Quotient RuleThe limit of a quotient of two functions is the

quotient of their limits provided that the denominator is not zero

Power RuleIf r and s are integers and s ne 0 then

Lkxfkcx

))((lim

0)(

)(lim

MM

L

xg

xfcx

srsr

cxLxf ))((lim

Using Properties of LimitsUse the observations that and

and the properties of limits to evaluate the following

limxc (x4 + 4x2 -3)limxc x4 + limxc 4x2 ndash limxc 3c4 + 4c2 -3

limxc x4 + x2 -1 x2 + 5 limxc x4 + limxc x2 - limxc 1

limxc x2 + limxc 5

c4 + c2 -1 c2 + 5

kkcx

)(lim cxcx

)(lim

DrillEvaluate the following limits

1 limx3 x2 (2 ndash x)

2 limx2 x2 + 2x + 4 x + 2

3 limx0 tanx you will want to use the tanx = sinxcosx x

4 Why does the following limit NOT existlimx2 x3 ndash 1

x - 2

Drill solutionsBy substitution1 -92 33 Using the product rule

limx0 (sinxx) (1cosx) limx0 (sinxx) limx0 (1cosx) Look at graph 1(1cosx) 1 1 = 1

4 since you cannot factor the given rational function you must just substitute 2 however by doing so the denominator would be 0 which is not possible

One-sided and Two-sided LimitsSometimes the values of a function f tend

to different limits as x approaches a number c from the left and from the right

Right-hand limit The limit of f as x approaches c from the

rightLeft-hand limit

The limit of f as x approaches c from the leftExample Use the graph to the right

) ( limx fc x

)(lim xfcx

1)(lim2

xfx

0)(lim2

xfx

Definition of Step FunctionA step function is a special type of function

whose graph is a series of line segments The graph of a step function looks like a

series of small steps There are two types of step functions

rounding UP and rounding DOWNRounding UP

Greatest Integer Functionf(x) = -int (-x)Ceiling FunctionExample f(4) = 1

Rounding DownLeast Integer Functionf(x) = int(x)Floor FunctionExample f(21) = 2

One-Sided and Two-sided LimitsA function f(x) has a limit as x approaches c

if and only if (IFF) the right-hand and left-hand limits at c exist and are equal

LandLLxfcxcxcx

limlim)(lim

Exploring Right-handed and Left-handed LimitsUse the graph and function below to

determine the limits

435

321

22211

101

)(

xx

xx

xx

xx

xf

)(lim0

xfx

)(lim1

xfx

)(lim1

xfx 1 10Therefore no limit exists

as x1 since they are NOT the same

The rest of the limits regarding this problem can be found on page 64

Exploring Left-handed and Right-Handed Limits

)(lim1

xfx

)(lim1

xfx

1 2)(lim

2xf

x )(lim

2xf

x

2 3

Sandwich TheoremIf g(x) lt f(x) lt h(x) for all x ne c in some interval

about c and limxc g(x) = limxc h(x) = L then limxc f(x) = L

Using the sandwich theoremShow that the limx0 [x2 sin(1x)] = 0We know that all of the values of the sine function

lie between -1 and 1 (range)The greatest value that sin(1x) can have is 1 and

the lowest value is -1 sohellip-x2 lt x2 sin(1x)] lt x2

Because the limx0 -x2 = 0 and limx0 x2 = 0 then limx0 [x2 sin(1x)] = 0

Closure

• Drill Find f(2)
• Lesson 21 Rates of Changes and Limits
• Average and Instantaneous Speed
• Finding Instantaneous Speed
• Definition of Limit
• Determining the limit by substitution and confirm graphically (
• Determine the limit graphically Then prove algebraically(Alwa
• Determine the limit graphically Then prove algebraically(Alwa (2)
• Properties of Limits
• Properties of Limits (2)
• Properties of Limits (3)
• Using Properties of Limits
• Drill
• Drill solutions
• One-sided and Two-sided Limits
• Definition of Step Function
• Rounding Down
• One-Sided and Two-sided Limits
• Exploring Right-handed and Left-handed Limits
• Exploring Left-handed and Right-Handed Limits
• Sandwich Theorem
• Closure

Definition of LimitThe limit is a method of evaluating an

expression as an argument approaches a value This value can be any point on the number line and often limits are evaluated as an argument approaches infinity or minus infinity The following expression states that as x approaches the value c the function approaches the value L

When looking at a graph the limit of a rational function (f(x)g(x)) is the HORIZONTAL asymptote

Lxfcx

)(lim

Determining the limit by substitution and confirm graphically (put in calc)

)12(3lim 2

21

xx

x)1)21(2()21(3 2

)11)(41(3 51)2(43

Determine the limit graphically Then prove algebraically(Always FACTOR with rational expressions)

1

1lim

21

x

xx

the limit appears to

be 12

2

1

1

1

)1)(1(

1

1

1lim

21

xxx

x

x

xx

Determine the limit graphically Then prove algebraically(Always FACTOR with rational expressions)

24

23

0 163

85lim

xx

xxx

the limit appears to

be -12

516)0(3

8)0(5

163

85

)163(

)85(

163

85lim

22

22

2

24

23

0

x

x

xx

xx

xx

xxx

Properties of LimitsKnown limits

limit of the function with the constant k

limit of the identity function of x = c

Properties of Limits If L M c and k are real number and

kkcx

)(lim

cxcx

)(lim

Lxfcx

)(lim Mxgcx

)(lim

Properties of LimitsSum Rule

The limit of the sum of 2 functions is the sum of their limits

Difference RuleThe limit of the difference of 2 functions is

the difference of their limits

Product RuleThe limit of a product of 2 functions is the

product of their limits

MLxgxfcx

))()((lim

MLxgxfcx

))()((lim

MLxgxfcx

))()((lim

Properties of LimitsConstant Multiple Rule

The limit of a constant times a function is the constant times the limit of the function

Quotient RuleThe limit of a quotient of two functions is the

quotient of their limits provided that the denominator is not zero

Power RuleIf r and s are integers and s ne 0 then

Lkxfkcx

))((lim

0)(

)(lim

MM

L

xg

xfcx

srsr

cxLxf ))((lim

Using Properties of LimitsUse the observations that and

and the properties of limits to evaluate the following

limxc (x4 + 4x2 -3)limxc x4 + limxc 4x2 ndash limxc 3c4 + 4c2 -3

limxc x4 + x2 -1 x2 + 5 limxc x4 + limxc x2 - limxc 1

limxc x2 + limxc 5

c4 + c2 -1 c2 + 5

kkcx

)(lim cxcx

)(lim

DrillEvaluate the following limits

1 limx3 x2 (2 ndash x)

2 limx2 x2 + 2x + 4 x + 2

3 limx0 tanx you will want to use the tanx = sinxcosx x

4 Why does the following limit NOT existlimx2 x3 ndash 1

x - 2

Drill solutionsBy substitution1 -92 33 Using the product rule

limx0 (sinxx) (1cosx) limx0 (sinxx) limx0 (1cosx) Look at graph 1(1cosx) 1 1 = 1

4 since you cannot factor the given rational function you must just substitute 2 however by doing so the denominator would be 0 which is not possible

One-sided and Two-sided LimitsSometimes the values of a function f tend

to different limits as x approaches a number c from the left and from the right

Right-hand limit The limit of f as x approaches c from the

rightLeft-hand limit

The limit of f as x approaches c from the leftExample Use the graph to the right

) ( limx fc x

)(lim xfcx

1)(lim2

xfx

0)(lim2

xfx

Definition of Step FunctionA step function is a special type of function

whose graph is a series of line segments The graph of a step function looks like a

series of small steps There are two types of step functions

rounding UP and rounding DOWNRounding UP

Greatest Integer Functionf(x) = -int (-x)Ceiling FunctionExample f(4) = 1

Rounding DownLeast Integer Functionf(x) = int(x)Floor FunctionExample f(21) = 2

One-Sided and Two-sided LimitsA function f(x) has a limit as x approaches c

if and only if (IFF) the right-hand and left-hand limits at c exist and are equal

LandLLxfcxcxcx

limlim)(lim

Exploring Right-handed and Left-handed LimitsUse the graph and function below to

determine the limits

435

321

22211

101

)(

xx

xx

xx

xx

xf

)(lim0

xfx

)(lim1

xfx

)(lim1

xfx 1 10Therefore no limit exists

as x1 since they are NOT the same

The rest of the limits regarding this problem can be found on page 64

Exploring Left-handed and Right-Handed Limits

)(lim1

xfx

)(lim1

xfx

1 2)(lim

2xf

x )(lim

2xf

x

2 3

Sandwich TheoremIf g(x) lt f(x) lt h(x) for all x ne c in some interval

about c and limxc g(x) = limxc h(x) = L then limxc f(x) = L

Using the sandwich theoremShow that the limx0 [x2 sin(1x)] = 0We know that all of the values of the sine function

lie between -1 and 1 (range)The greatest value that sin(1x) can have is 1 and

the lowest value is -1 sohellip-x2 lt x2 sin(1x)] lt x2

Because the limx0 -x2 = 0 and limx0 x2 = 0 then limx0 [x2 sin(1x)] = 0

Closure

• Drill Find f(2)
• Lesson 21 Rates of Changes and Limits
• Average and Instantaneous Speed
• Finding Instantaneous Speed
• Definition of Limit
• Determining the limit by substitution and confirm graphically (
• Determine the limit graphically Then prove algebraically(Alwa
• Determine the limit graphically Then prove algebraically(Alwa (2)
• Properties of Limits
• Properties of Limits (2)
• Properties of Limits (3)
• Using Properties of Limits
• Drill
• Drill solutions
• One-sided and Two-sided Limits
• Definition of Step Function
• Rounding Down
• One-Sided and Two-sided Limits
• Exploring Right-handed and Left-handed Limits
• Exploring Left-handed and Right-Handed Limits
• Sandwich Theorem
• Closure

Determining the limit by substitution and confirm graphically (put in calc)

)12(3lim 2

21

xx

x)1)21(2()21(3 2

)11)(41(3 51)2(43

Determine the limit graphically Then prove algebraically(Always FACTOR with rational expressions)

1

1lim

21

x

xx

the limit appears to

be 12

2

1

1

1

)1)(1(

1

1

1lim

21

xxx

x

x

xx

Determine the limit graphically Then prove algebraically(Always FACTOR with rational expressions)

24

23

0 163

85lim

xx

xxx

the limit appears to

be -12

516)0(3

8)0(5

163

85

)163(

)85(

163

85lim

22

22

2

24

23

0

x

x

xx

xx

xx

xxx

Properties of LimitsKnown limits

limit of the function with the constant k

limit of the identity function of x = c

Properties of Limits If L M c and k are real number and

kkcx

)(lim

cxcx

)(lim

Lxfcx

)(lim Mxgcx

)(lim

Properties of LimitsSum Rule

The limit of the sum of 2 functions is the sum of their limits

Difference RuleThe limit of the difference of 2 functions is

the difference of their limits

Product RuleThe limit of a product of 2 functions is the

product of their limits

MLxgxfcx

))()((lim

MLxgxfcx

))()((lim

MLxgxfcx

))()((lim

Properties of LimitsConstant Multiple Rule

The limit of a constant times a function is the constant times the limit of the function

Quotient RuleThe limit of a quotient of two functions is the

quotient of their limits provided that the denominator is not zero

Power RuleIf r and s are integers and s ne 0 then

Lkxfkcx

))((lim

0)(

)(lim

MM

L

xg

xfcx

srsr

cxLxf ))((lim

Using Properties of LimitsUse the observations that and

and the properties of limits to evaluate the following

limxc (x4 + 4x2 -3)limxc x4 + limxc 4x2 ndash limxc 3c4 + 4c2 -3

limxc x4 + x2 -1 x2 + 5 limxc x4 + limxc x2 - limxc 1

limxc x2 + limxc 5

c4 + c2 -1 c2 + 5

kkcx

)(lim cxcx

)(lim

DrillEvaluate the following limits

1 limx3 x2 (2 ndash x)

2 limx2 x2 + 2x + 4 x + 2

3 limx0 tanx you will want to use the tanx = sinxcosx x

4 Why does the following limit NOT existlimx2 x3 ndash 1

x - 2

Drill solutionsBy substitution1 -92 33 Using the product rule

limx0 (sinxx) (1cosx) limx0 (sinxx) limx0 (1cosx) Look at graph 1(1cosx) 1 1 = 1

4 since you cannot factor the given rational function you must just substitute 2 however by doing so the denominator would be 0 which is not possible

One-sided and Two-sided LimitsSometimes the values of a function f tend

to different limits as x approaches a number c from the left and from the right

Right-hand limit The limit of f as x approaches c from the

rightLeft-hand limit

The limit of f as x approaches c from the leftExample Use the graph to the right

) ( limx fc x

)(lim xfcx

1)(lim2

xfx

0)(lim2

xfx

Definition of Step FunctionA step function is a special type of function

whose graph is a series of line segments The graph of a step function looks like a

series of small steps There are two types of step functions

rounding UP and rounding DOWNRounding UP

Greatest Integer Functionf(x) = -int (-x)Ceiling FunctionExample f(4) = 1

Rounding DownLeast Integer Functionf(x) = int(x)Floor FunctionExample f(21) = 2

One-Sided and Two-sided LimitsA function f(x) has a limit as x approaches c

if and only if (IFF) the right-hand and left-hand limits at c exist and are equal

LandLLxfcxcxcx

limlim)(lim

Exploring Right-handed and Left-handed LimitsUse the graph and function below to

determine the limits

435

321

22211

101

)(

xx

xx

xx

xx

xf

)(lim0

xfx

)(lim1

xfx

)(lim1

xfx 1 10Therefore no limit exists

as x1 since they are NOT the same

The rest of the limits regarding this problem can be found on page 64

Exploring Left-handed and Right-Handed Limits

)(lim1

xfx

)(lim1

xfx

1 2)(lim

2xf

x )(lim

2xf

x

2 3

Sandwich TheoremIf g(x) lt f(x) lt h(x) for all x ne c in some interval

about c and limxc g(x) = limxc h(x) = L then limxc f(x) = L

Using the sandwich theoremShow that the limx0 [x2 sin(1x)] = 0We know that all of the values of the sine function

lie between -1 and 1 (range)The greatest value that sin(1x) can have is 1 and

the lowest value is -1 sohellip-x2 lt x2 sin(1x)] lt x2

Because the limx0 -x2 = 0 and limx0 x2 = 0 then limx0 [x2 sin(1x)] = 0

Closure

• Drill Find f(2)
• Lesson 21 Rates of Changes and Limits
• Average and Instantaneous Speed
• Finding Instantaneous Speed
• Definition of Limit
• Determining the limit by substitution and confirm graphically (
• Determine the limit graphically Then prove algebraically(Alwa
• Determine the limit graphically Then prove algebraically(Alwa (2)
• Properties of Limits
• Properties of Limits (2)
• Properties of Limits (3)
• Using Properties of Limits
• Drill
• Drill solutions
• One-sided and Two-sided Limits
• Definition of Step Function
• Rounding Down
• One-Sided and Two-sided Limits
• Exploring Right-handed and Left-handed Limits
• Exploring Left-handed and Right-Handed Limits
• Sandwich Theorem
• Closure

Determine the limit graphically Then prove algebraically(Always FACTOR with rational expressions)

1

1lim

21

x

xx

the limit appears to

be 12

2

1

1

1

)1)(1(

1

1

1lim

21

xxx

x

x

xx

Determine the limit graphically Then prove algebraically(Always FACTOR with rational expressions)

24

23

0 163

85lim

xx

xxx

the limit appears to

be -12

516)0(3

8)0(5

163

85

)163(

)85(

163

85lim

22

22

2

24

23

0

x

x

xx

xx

xx

xxx

Properties of LimitsKnown limits

limit of the function with the constant k

limit of the identity function of x = c

Properties of Limits If L M c and k are real number and

kkcx

)(lim

cxcx

)(lim

Lxfcx

)(lim Mxgcx

)(lim

Properties of LimitsSum Rule

The limit of the sum of 2 functions is the sum of their limits

Difference RuleThe limit of the difference of 2 functions is

the difference of their limits

Product RuleThe limit of a product of 2 functions is the

product of their limits

MLxgxfcx

))()((lim

MLxgxfcx

))()((lim

MLxgxfcx

))()((lim

Properties of LimitsConstant Multiple Rule

The limit of a constant times a function is the constant times the limit of the function

Quotient RuleThe limit of a quotient of two functions is the

quotient of their limits provided that the denominator is not zero

Power RuleIf r and s are integers and s ne 0 then

Lkxfkcx

))((lim

0)(

)(lim

MM

L

xg

xfcx

srsr

cxLxf ))((lim

Using Properties of LimitsUse the observations that and

and the properties of limits to evaluate the following

limxc (x4 + 4x2 -3)limxc x4 + limxc 4x2 ndash limxc 3c4 + 4c2 -3

limxc x4 + x2 -1 x2 + 5 limxc x4 + limxc x2 - limxc 1

limxc x2 + limxc 5

c4 + c2 -1 c2 + 5

kkcx

)(lim cxcx

)(lim

DrillEvaluate the following limits

1 limx3 x2 (2 ndash x)

2 limx2 x2 + 2x + 4 x + 2

3 limx0 tanx you will want to use the tanx = sinxcosx x

4 Why does the following limit NOT existlimx2 x3 ndash 1

x - 2

Drill solutionsBy substitution1 -92 33 Using the product rule

limx0 (sinxx) (1cosx) limx0 (sinxx) limx0 (1cosx) Look at graph 1(1cosx) 1 1 = 1

4 since you cannot factor the given rational function you must just substitute 2 however by doing so the denominator would be 0 which is not possible

One-sided and Two-sided LimitsSometimes the values of a function f tend

to different limits as x approaches a number c from the left and from the right

Right-hand limit The limit of f as x approaches c from the

rightLeft-hand limit

The limit of f as x approaches c from the leftExample Use the graph to the right

) ( limx fc x

)(lim xfcx

1)(lim2

xfx

0)(lim2

xfx

Definition of Step FunctionA step function is a special type of function

whose graph is a series of line segments The graph of a step function looks like a

series of small steps There are two types of step functions

rounding UP and rounding DOWNRounding UP

Greatest Integer Functionf(x) = -int (-x)Ceiling FunctionExample f(4) = 1

Rounding DownLeast Integer Functionf(x) = int(x)Floor FunctionExample f(21) = 2

One-Sided and Two-sided LimitsA function f(x) has a limit as x approaches c

if and only if (IFF) the right-hand and left-hand limits at c exist and are equal

LandLLxfcxcxcx

limlim)(lim

Exploring Right-handed and Left-handed LimitsUse the graph and function below to

determine the limits

435

321

22211

101

)(

xx

xx

xx

xx

xf

)(lim0

xfx

)(lim1

xfx

)(lim1

xfx 1 10Therefore no limit exists

as x1 since they are NOT the same

The rest of the limits regarding this problem can be found on page 64

Exploring Left-handed and Right-Handed Limits

)(lim1

xfx

)(lim1

xfx

1 2)(lim

2xf

x )(lim

2xf

x

2 3

Sandwich TheoremIf g(x) lt f(x) lt h(x) for all x ne c in some interval

about c and limxc g(x) = limxc h(x) = L then limxc f(x) = L

Using the sandwich theoremShow that the limx0 [x2 sin(1x)] = 0We know that all of the values of the sine function

lie between -1 and 1 (range)The greatest value that sin(1x) can have is 1 and

the lowest value is -1 sohellip-x2 lt x2 sin(1x)] lt x2

Because the limx0 -x2 = 0 and limx0 x2 = 0 then limx0 [x2 sin(1x)] = 0

Closure

• Drill Find f(2)
• Lesson 21 Rates of Changes and Limits
• Average and Instantaneous Speed
• Finding Instantaneous Speed
• Definition of Limit
• Determining the limit by substitution and confirm graphically (
• Determine the limit graphically Then prove algebraically(Alwa
• Determine the limit graphically Then prove algebraically(Alwa (2)
• Properties of Limits
• Properties of Limits (2)
• Properties of Limits (3)
• Using Properties of Limits
• Drill
• Drill solutions
• One-sided and Two-sided Limits
• Definition of Step Function
• Rounding Down
• One-Sided and Two-sided Limits
• Exploring Right-handed and Left-handed Limits
• Exploring Left-handed and Right-Handed Limits
• Sandwich Theorem
• Closure

Determine the limit graphically Then prove algebraically(Always FACTOR with rational expressions)

24

23

0 163

85lim

xx

xxx

the limit appears to

be -12

516)0(3

8)0(5

163

85

)163(

)85(

163

85lim

22

22

2

24

23

0

x

x

xx

xx

xx

xxx

Properties of LimitsKnown limits

limit of the function with the constant k

limit of the identity function of x = c

Properties of Limits If L M c and k are real number and

kkcx

)(lim

cxcx

)(lim

Lxfcx

)(lim Mxgcx

)(lim

Properties of LimitsSum Rule

The limit of the sum of 2 functions is the sum of their limits

Difference RuleThe limit of the difference of 2 functions is

the difference of their limits

Product RuleThe limit of a product of 2 functions is the

product of their limits

MLxgxfcx

))()((lim

MLxgxfcx

))()((lim

MLxgxfcx

))()((lim

Properties of LimitsConstant Multiple Rule

The limit of a constant times a function is the constant times the limit of the function

Quotient RuleThe limit of a quotient of two functions is the

quotient of their limits provided that the denominator is not zero

Power RuleIf r and s are integers and s ne 0 then

Lkxfkcx

))((lim

0)(

)(lim

MM

L

xg

xfcx

srsr

cxLxf ))((lim

Using Properties of LimitsUse the observations that and

and the properties of limits to evaluate the following

limxc (x4 + 4x2 -3)limxc x4 + limxc 4x2 ndash limxc 3c4 + 4c2 -3

limxc x4 + x2 -1 x2 + 5 limxc x4 + limxc x2 - limxc 1

limxc x2 + limxc 5

c4 + c2 -1 c2 + 5

kkcx

)(lim cxcx

)(lim

DrillEvaluate the following limits

1 limx3 x2 (2 ndash x)

2 limx2 x2 + 2x + 4 x + 2

3 limx0 tanx you will want to use the tanx = sinxcosx x

4 Why does the following limit NOT existlimx2 x3 ndash 1

x - 2

Drill solutionsBy substitution1 -92 33 Using the product rule

limx0 (sinxx) (1cosx) limx0 (sinxx) limx0 (1cosx) Look at graph 1(1cosx) 1 1 = 1

4 since you cannot factor the given rational function you must just substitute 2 however by doing so the denominator would be 0 which is not possible

One-sided and Two-sided LimitsSometimes the values of a function f tend

to different limits as x approaches a number c from the left and from the right

Right-hand limit The limit of f as x approaches c from the

rightLeft-hand limit

The limit of f as x approaches c from the leftExample Use the graph to the right

) ( limx fc x

)(lim xfcx

1)(lim2

xfx

0)(lim2

xfx

Definition of Step FunctionA step function is a special type of function

whose graph is a series of line segments The graph of a step function looks like a

series of small steps There are two types of step functions

rounding UP and rounding DOWNRounding UP

Greatest Integer Functionf(x) = -int (-x)Ceiling FunctionExample f(4) = 1

Rounding DownLeast Integer Functionf(x) = int(x)Floor FunctionExample f(21) = 2

One-Sided and Two-sided LimitsA function f(x) has a limit as x approaches c

if and only if (IFF) the right-hand and left-hand limits at c exist and are equal

LandLLxfcxcxcx

limlim)(lim

Exploring Right-handed and Left-handed LimitsUse the graph and function below to

determine the limits

435

321

22211

101

)(

xx

xx

xx

xx

xf

)(lim0

xfx

)(lim1

xfx

)(lim1

xfx 1 10Therefore no limit exists

as x1 since they are NOT the same

The rest of the limits regarding this problem can be found on page 64

Exploring Left-handed and Right-Handed Limits

)(lim1

xfx

)(lim1

xfx

1 2)(lim

2xf

x )(lim

2xf

x

2 3

Sandwich TheoremIf g(x) lt f(x) lt h(x) for all x ne c in some interval

about c and limxc g(x) = limxc h(x) = L then limxc f(x) = L

Using the sandwich theoremShow that the limx0 [x2 sin(1x)] = 0We know that all of the values of the sine function

lie between -1 and 1 (range)The greatest value that sin(1x) can have is 1 and

the lowest value is -1 sohellip-x2 lt x2 sin(1x)] lt x2

Because the limx0 -x2 = 0 and limx0 x2 = 0 then limx0 [x2 sin(1x)] = 0

Closure

• Drill Find f(2)
• Lesson 21 Rates of Changes and Limits
• Average and Instantaneous Speed
• Finding Instantaneous Speed
• Definition of Limit
• Determining the limit by substitution and confirm graphically (
• Determine the limit graphically Then prove algebraically(Alwa
• Determine the limit graphically Then prove algebraically(Alwa (2)
• Properties of Limits
• Properties of Limits (2)
• Properties of Limits (3)
• Using Properties of Limits
• Drill
• Drill solutions
• One-sided and Two-sided Limits
• Definition of Step Function
• Rounding Down
• One-Sided and Two-sided Limits
• Exploring Right-handed and Left-handed Limits
• Exploring Left-handed and Right-Handed Limits
• Sandwich Theorem
• Closure

Properties of LimitsKnown limits

limit of the function with the constant k

limit of the identity function of x = c

Properties of Limits If L M c and k are real number and

kkcx

)(lim

cxcx

)(lim

Lxfcx

)(lim Mxgcx

)(lim

Properties of LimitsSum Rule

The limit of the sum of 2 functions is the sum of their limits

Difference RuleThe limit of the difference of 2 functions is

the difference of their limits

Product RuleThe limit of a product of 2 functions is the

product of their limits

MLxgxfcx

))()((lim

MLxgxfcx

))()((lim

MLxgxfcx

))()((lim

Properties of LimitsConstant Multiple Rule

The limit of a constant times a function is the constant times the limit of the function

Quotient RuleThe limit of a quotient of two functions is the

quotient of their limits provided that the denominator is not zero

Power RuleIf r and s are integers and s ne 0 then

Lkxfkcx

))((lim

0)(

)(lim

MM

L

xg

xfcx

srsr

cxLxf ))((lim

Using Properties of LimitsUse the observations that and

and the properties of limits to evaluate the following

limxc (x4 + 4x2 -3)limxc x4 + limxc 4x2 ndash limxc 3c4 + 4c2 -3

limxc x4 + x2 -1 x2 + 5 limxc x4 + limxc x2 - limxc 1

limxc x2 + limxc 5

c4 + c2 -1 c2 + 5

kkcx

)(lim cxcx

)(lim

DrillEvaluate the following limits

1 limx3 x2 (2 ndash x)

2 limx2 x2 + 2x + 4 x + 2

3 limx0 tanx you will want to use the tanx = sinxcosx x

4 Why does the following limit NOT existlimx2 x3 ndash 1

x - 2

Drill solutionsBy substitution1 -92 33 Using the product rule

limx0 (sinxx) (1cosx) limx0 (sinxx) limx0 (1cosx) Look at graph 1(1cosx) 1 1 = 1

4 since you cannot factor the given rational function you must just substitute 2 however by doing so the denominator would be 0 which is not possible

One-sided and Two-sided LimitsSometimes the values of a function f tend

to different limits as x approaches a number c from the left and from the right

Right-hand limit The limit of f as x approaches c from the

rightLeft-hand limit

The limit of f as x approaches c from the leftExample Use the graph to the right

) ( limx fc x

)(lim xfcx

1)(lim2

xfx

0)(lim2

xfx

Definition of Step FunctionA step function is a special type of function

whose graph is a series of line segments The graph of a step function looks like a

series of small steps There are two types of step functions

rounding UP and rounding DOWNRounding UP

Greatest Integer Functionf(x) = -int (-x)Ceiling FunctionExample f(4) = 1

Rounding DownLeast Integer Functionf(x) = int(x)Floor FunctionExample f(21) = 2

One-Sided and Two-sided LimitsA function f(x) has a limit as x approaches c

if and only if (IFF) the right-hand and left-hand limits at c exist and are equal

LandLLxfcxcxcx

limlim)(lim

Exploring Right-handed and Left-handed LimitsUse the graph and function below to

determine the limits

435

321

22211

101

)(

xx

xx

xx

xx

xf

)(lim0

xfx

)(lim1

xfx

)(lim1

xfx 1 10Therefore no limit exists

as x1 since they are NOT the same

The rest of the limits regarding this problem can be found on page 64

Exploring Left-handed and Right-Handed Limits

)(lim1

xfx

)(lim1

xfx

1 2)(lim

2xf

x )(lim

2xf

x

2 3

Sandwich TheoremIf g(x) lt f(x) lt h(x) for all x ne c in some interval

about c and limxc g(x) = limxc h(x) = L then limxc f(x) = L

Using the sandwich theoremShow that the limx0 [x2 sin(1x)] = 0We know that all of the values of the sine function

lie between -1 and 1 (range)The greatest value that sin(1x) can have is 1 and

the lowest value is -1 sohellip-x2 lt x2 sin(1x)] lt x2

Because the limx0 -x2 = 0 and limx0 x2 = 0 then limx0 [x2 sin(1x)] = 0

Closure

• Drill Find f(2)
• Lesson 21 Rates of Changes and Limits
• Average and Instantaneous Speed
• Finding Instantaneous Speed
• Definition of Limit
• Determining the limit by substitution and confirm graphically (
• Determine the limit graphically Then prove algebraically(Alwa
• Determine the limit graphically Then prove algebraically(Alwa (2)
• Properties of Limits
• Properties of Limits (2)
• Properties of Limits (3)
• Using Properties of Limits
• Drill
• Drill solutions
• One-sided and Two-sided Limits
• Definition of Step Function
• Rounding Down
• One-Sided and Two-sided Limits
• Exploring Right-handed and Left-handed Limits
• Exploring Left-handed and Right-Handed Limits
• Sandwich Theorem
• Closure

Properties of LimitsSum Rule

The limit of the sum of 2 functions is the sum of their limits

Difference RuleThe limit of the difference of 2 functions is

the difference of their limits

Product RuleThe limit of a product of 2 functions is the

product of their limits

MLxgxfcx

))()((lim

MLxgxfcx

))()((lim

MLxgxfcx

))()((lim

Properties of LimitsConstant Multiple Rule

The limit of a constant times a function is the constant times the limit of the function

Quotient RuleThe limit of a quotient of two functions is the

quotient of their limits provided that the denominator is not zero

Power RuleIf r and s are integers and s ne 0 then

Lkxfkcx

))((lim

0)(

)(lim

MM

L

xg

xfcx

srsr

cxLxf ))((lim

Using Properties of LimitsUse the observations that and

and the properties of limits to evaluate the following

limxc (x4 + 4x2 -3)limxc x4 + limxc 4x2 ndash limxc 3c4 + 4c2 -3

limxc x4 + x2 -1 x2 + 5 limxc x4 + limxc x2 - limxc 1

limxc x2 + limxc 5

c4 + c2 -1 c2 + 5

kkcx

)(lim cxcx

)(lim

DrillEvaluate the following limits

1 limx3 x2 (2 ndash x)

2 limx2 x2 + 2x + 4 x + 2

3 limx0 tanx you will want to use the tanx = sinxcosx x

4 Why does the following limit NOT existlimx2 x3 ndash 1

x - 2

Drill solutionsBy substitution1 -92 33 Using the product rule

limx0 (sinxx) (1cosx) limx0 (sinxx) limx0 (1cosx) Look at graph 1(1cosx) 1 1 = 1

4 since you cannot factor the given rational function you must just substitute 2 however by doing so the denominator would be 0 which is not possible

One-sided and Two-sided LimitsSometimes the values of a function f tend

to different limits as x approaches a number c from the left and from the right

Right-hand limit The limit of f as x approaches c from the

rightLeft-hand limit

The limit of f as x approaches c from the leftExample Use the graph to the right

) ( limx fc x

)(lim xfcx

1)(lim2

xfx

0)(lim2

xfx

Definition of Step FunctionA step function is a special type of function

whose graph is a series of line segments The graph of a step function looks like a

series of small steps There are two types of step functions

rounding UP and rounding DOWNRounding UP

Greatest Integer Functionf(x) = -int (-x)Ceiling FunctionExample f(4) = 1

Rounding DownLeast Integer Functionf(x) = int(x)Floor FunctionExample f(21) = 2

One-Sided and Two-sided LimitsA function f(x) has a limit as x approaches c

if and only if (IFF) the right-hand and left-hand limits at c exist and are equal

LandLLxfcxcxcx

limlim)(lim

Exploring Right-handed and Left-handed LimitsUse the graph and function below to

determine the limits

435

321

22211

101

)(

xx

xx

xx

xx

xf

)(lim0

xfx

)(lim1

xfx

)(lim1

xfx 1 10Therefore no limit exists

as x1 since they are NOT the same

The rest of the limits regarding this problem can be found on page 64

Exploring Left-handed and Right-Handed Limits

)(lim1

xfx

)(lim1

xfx

1 2)(lim

2xf

x )(lim

2xf

x

2 3

Sandwich TheoremIf g(x) lt f(x) lt h(x) for all x ne c in some interval

about c and limxc g(x) = limxc h(x) = L then limxc f(x) = L

Using the sandwich theoremShow that the limx0 [x2 sin(1x)] = 0We know that all of the values of the sine function

lie between -1 and 1 (range)The greatest value that sin(1x) can have is 1 and

the lowest value is -1 sohellip-x2 lt x2 sin(1x)] lt x2

Because the limx0 -x2 = 0 and limx0 x2 = 0 then limx0 [x2 sin(1x)] = 0

Closure

• Drill Find f(2)
• Lesson 21 Rates of Changes and Limits
• Average and Instantaneous Speed
• Finding Instantaneous Speed
• Definition of Limit
• Determining the limit by substitution and confirm graphically (
• Determine the limit graphically Then prove algebraically(Alwa
• Determine the limit graphically Then prove algebraically(Alwa (2)
• Properties of Limits
• Properties of Limits (2)
• Properties of Limits (3)
• Using Properties of Limits
• Drill
• Drill solutions
• One-sided and Two-sided Limits
• Definition of Step Function
• Rounding Down
• One-Sided and Two-sided Limits
• Exploring Right-handed and Left-handed Limits
• Exploring Left-handed and Right-Handed Limits
• Sandwich Theorem
• Closure

Properties of LimitsConstant Multiple Rule

The limit of a constant times a function is the constant times the limit of the function

Quotient RuleThe limit of a quotient of two functions is the

quotient of their limits provided that the denominator is not zero

Power RuleIf r and s are integers and s ne 0 then

Lkxfkcx

))((lim

0)(

)(lim

MM

L

xg

xfcx

srsr

cxLxf ))((lim

Using Properties of LimitsUse the observations that and

and the properties of limits to evaluate the following

limxc (x4 + 4x2 -3)limxc x4 + limxc 4x2 ndash limxc 3c4 + 4c2 -3

limxc x4 + x2 -1 x2 + 5 limxc x4 + limxc x2 - limxc 1

limxc x2 + limxc 5

c4 + c2 -1 c2 + 5

kkcx

)(lim cxcx

)(lim

DrillEvaluate the following limits

1 limx3 x2 (2 ndash x)

2 limx2 x2 + 2x + 4 x + 2

3 limx0 tanx you will want to use the tanx = sinxcosx x

4 Why does the following limit NOT existlimx2 x3 ndash 1

x - 2

Drill solutionsBy substitution1 -92 33 Using the product rule

limx0 (sinxx) (1cosx) limx0 (sinxx) limx0 (1cosx) Look at graph 1(1cosx) 1 1 = 1

4 since you cannot factor the given rational function you must just substitute 2 however by doing so the denominator would be 0 which is not possible

One-sided and Two-sided LimitsSometimes the values of a function f tend

to different limits as x approaches a number c from the left and from the right

Right-hand limit The limit of f as x approaches c from the

rightLeft-hand limit

The limit of f as x approaches c from the leftExample Use the graph to the right

) ( limx fc x

)(lim xfcx

1)(lim2

xfx

0)(lim2

xfx

Definition of Step FunctionA step function is a special type of function

whose graph is a series of line segments The graph of a step function looks like a

series of small steps There are two types of step functions

rounding UP and rounding DOWNRounding UP

Greatest Integer Functionf(x) = -int (-x)Ceiling FunctionExample f(4) = 1

Rounding DownLeast Integer Functionf(x) = int(x)Floor FunctionExample f(21) = 2

One-Sided and Two-sided LimitsA function f(x) has a limit as x approaches c

if and only if (IFF) the right-hand and left-hand limits at c exist and are equal

LandLLxfcxcxcx

limlim)(lim

Exploring Right-handed and Left-handed LimitsUse the graph and function below to

determine the limits

435

321

22211

101

)(

xx

xx

xx

xx

xf

)(lim0

xfx

)(lim1

xfx

)(lim1

xfx 1 10Therefore no limit exists

as x1 since they are NOT the same

The rest of the limits regarding this problem can be found on page 64

Exploring Left-handed and Right-Handed Limits

)(lim1

xfx

)(lim1

xfx

1 2)(lim

2xf

x )(lim

2xf

x

2 3

Sandwich TheoremIf g(x) lt f(x) lt h(x) for all x ne c in some interval

about c and limxc g(x) = limxc h(x) = L then limxc f(x) = L

Using the sandwich theoremShow that the limx0 [x2 sin(1x)] = 0We know that all of the values of the sine function

lie between -1 and 1 (range)The greatest value that sin(1x) can have is 1 and

the lowest value is -1 sohellip-x2 lt x2 sin(1x)] lt x2

Because the limx0 -x2 = 0 and limx0 x2 = 0 then limx0 [x2 sin(1x)] = 0

Closure

• Drill Find f(2)
• Lesson 21 Rates of Changes and Limits
• Average and Instantaneous Speed
• Finding Instantaneous Speed
• Definition of Limit
• Determining the limit by substitution and confirm graphically (
• Determine the limit graphically Then prove algebraically(Alwa
• Determine the limit graphically Then prove algebraically(Alwa (2)
• Properties of Limits
• Properties of Limits (2)
• Properties of Limits (3)
• Using Properties of Limits
• Drill
• Drill solutions
• One-sided and Two-sided Limits
• Definition of Step Function
• Rounding Down
• One-Sided and Two-sided Limits
• Exploring Right-handed and Left-handed Limits
• Exploring Left-handed and Right-Handed Limits
• Sandwich Theorem
• Closure

Using Properties of LimitsUse the observations that and

and the properties of limits to evaluate the following

limxc (x4 + 4x2 -3)limxc x4 + limxc 4x2 ndash limxc 3c4 + 4c2 -3

limxc x4 + x2 -1 x2 + 5 limxc x4 + limxc x2 - limxc 1

limxc x2 + limxc 5

c4 + c2 -1 c2 + 5

kkcx

)(lim cxcx

)(lim

DrillEvaluate the following limits

1 limx3 x2 (2 ndash x)

2 limx2 x2 + 2x + 4 x + 2

3 limx0 tanx you will want to use the tanx = sinxcosx x

4 Why does the following limit NOT existlimx2 x3 ndash 1

x - 2

Drill solutionsBy substitution1 -92 33 Using the product rule

limx0 (sinxx) (1cosx) limx0 (sinxx) limx0 (1cosx) Look at graph 1(1cosx) 1 1 = 1

4 since you cannot factor the given rational function you must just substitute 2 however by doing so the denominator would be 0 which is not possible

One-sided and Two-sided LimitsSometimes the values of a function f tend

to different limits as x approaches a number c from the left and from the right

Right-hand limit The limit of f as x approaches c from the

rightLeft-hand limit

The limit of f as x approaches c from the leftExample Use the graph to the right

) ( limx fc x

)(lim xfcx

1)(lim2

xfx

0)(lim2

xfx

Definition of Step FunctionA step function is a special type of function

whose graph is a series of line segments The graph of a step function looks like a

series of small steps There are two types of step functions

rounding UP and rounding DOWNRounding UP

Greatest Integer Functionf(x) = -int (-x)Ceiling FunctionExample f(4) = 1

Rounding DownLeast Integer Functionf(x) = int(x)Floor FunctionExample f(21) = 2

One-Sided and Two-sided LimitsA function f(x) has a limit as x approaches c

if and only if (IFF) the right-hand and left-hand limits at c exist and are equal

LandLLxfcxcxcx

limlim)(lim

Exploring Right-handed and Left-handed LimitsUse the graph and function below to

determine the limits

435

321

22211

101

)(

xx

xx

xx

xx

xf

)(lim0

xfx

)(lim1

xfx

)(lim1

xfx 1 10Therefore no limit exists

as x1 since they are NOT the same

The rest of the limits regarding this problem can be found on page 64

Exploring Left-handed and Right-Handed Limits

)(lim1

xfx

)(lim1

xfx

1 2)(lim

2xf

x )(lim

2xf

x

2 3

Sandwich TheoremIf g(x) lt f(x) lt h(x) for all x ne c in some interval

about c and limxc g(x) = limxc h(x) = L then limxc f(x) = L

Using the sandwich theoremShow that the limx0 [x2 sin(1x)] = 0We know that all of the values of the sine function

lie between -1 and 1 (range)The greatest value that sin(1x) can have is 1 and

the lowest value is -1 sohellip-x2 lt x2 sin(1x)] lt x2

Because the limx0 -x2 = 0 and limx0 x2 = 0 then limx0 [x2 sin(1x)] = 0

Closure

• Drill Find f(2)
• Lesson 21 Rates of Changes and Limits
• Average and Instantaneous Speed
• Finding Instantaneous Speed
• Definition of Limit
• Determining the limit by substitution and confirm graphically (
• Determine the limit graphically Then prove algebraically(Alwa
• Determine the limit graphically Then prove algebraically(Alwa (2)
• Properties of Limits
• Properties of Limits (2)
• Properties of Limits (3)
• Using Properties of Limits
• Drill
• Drill solutions
• One-sided and Two-sided Limits
• Definition of Step Function
• Rounding Down
• One-Sided and Two-sided Limits
• Exploring Right-handed and Left-handed Limits
• Exploring Left-handed and Right-Handed Limits
• Sandwich Theorem
• Closure

DrillEvaluate the following limits

1 limx3 x2 (2 ndash x)

2 limx2 x2 + 2x + 4 x + 2

3 limx0 tanx you will want to use the tanx = sinxcosx x

4 Why does the following limit NOT existlimx2 x3 ndash 1

x - 2

Drill solutionsBy substitution1 -92 33 Using the product rule

limx0 (sinxx) (1cosx) limx0 (sinxx) limx0 (1cosx) Look at graph 1(1cosx) 1 1 = 1

4 since you cannot factor the given rational function you must just substitute 2 however by doing so the denominator would be 0 which is not possible

One-sided and Two-sided LimitsSometimes the values of a function f tend

to different limits as x approaches a number c from the left and from the right

Right-hand limit The limit of f as x approaches c from the

rightLeft-hand limit

The limit of f as x approaches c from the leftExample Use the graph to the right

) ( limx fc x

)(lim xfcx

1)(lim2

xfx

0)(lim2

xfx

Definition of Step FunctionA step function is a special type of function

whose graph is a series of line segments The graph of a step function looks like a

series of small steps There are two types of step functions

rounding UP and rounding DOWNRounding UP

Greatest Integer Functionf(x) = -int (-x)Ceiling FunctionExample f(4) = 1

Rounding DownLeast Integer Functionf(x) = int(x)Floor FunctionExample f(21) = 2

One-Sided and Two-sided LimitsA function f(x) has a limit as x approaches c

if and only if (IFF) the right-hand and left-hand limits at c exist and are equal

LandLLxfcxcxcx

limlim)(lim

Exploring Right-handed and Left-handed LimitsUse the graph and function below to

determine the limits

435

321

22211

101

)(

xx

xx

xx

xx

xf

)(lim0

xfx

)(lim1

xfx

)(lim1

xfx 1 10Therefore no limit exists

as x1 since they are NOT the same

The rest of the limits regarding this problem can be found on page 64

Exploring Left-handed and Right-Handed Limits

)(lim1

xfx

)(lim1

xfx

1 2)(lim

2xf

x )(lim

2xf

x

2 3

Sandwich TheoremIf g(x) lt f(x) lt h(x) for all x ne c in some interval

about c and limxc g(x) = limxc h(x) = L then limxc f(x) = L

Using the sandwich theoremShow that the limx0 [x2 sin(1x)] = 0We know that all of the values of the sine function

lie between -1 and 1 (range)The greatest value that sin(1x) can have is 1 and

the lowest value is -1 sohellip-x2 lt x2 sin(1x)] lt x2

Because the limx0 -x2 = 0 and limx0 x2 = 0 then limx0 [x2 sin(1x)] = 0

Closure

• Drill Find f(2)
• Lesson 21 Rates of Changes and Limits
• Average and Instantaneous Speed
• Finding Instantaneous Speed
• Definition of Limit
• Determining the limit by substitution and confirm graphically (
• Determine the limit graphically Then prove algebraically(Alwa
• Determine the limit graphically Then prove algebraically(Alwa (2)
• Properties of Limits
• Properties of Limits (2)
• Properties of Limits (3)
• Using Properties of Limits
• Drill
• Drill solutions
• One-sided and Two-sided Limits
• Definition of Step Function
• Rounding Down
• One-Sided and Two-sided Limits
• Exploring Right-handed and Left-handed Limits
• Exploring Left-handed and Right-Handed Limits
• Sandwich Theorem
• Closure

Drill solutionsBy substitution1 -92 33 Using the product rule

limx0 (sinxx) (1cosx) limx0 (sinxx) limx0 (1cosx) Look at graph 1(1cosx) 1 1 = 1

4 since you cannot factor the given rational function you must just substitute 2 however by doing so the denominator would be 0 which is not possible

One-sided and Two-sided LimitsSometimes the values of a function f tend

to different limits as x approaches a number c from the left and from the right

Right-hand limit The limit of f as x approaches c from the

rightLeft-hand limit

The limit of f as x approaches c from the leftExample Use the graph to the right

) ( limx fc x

)(lim xfcx

1)(lim2

xfx

0)(lim2

xfx

Definition of Step FunctionA step function is a special type of function

whose graph is a series of line segments The graph of a step function looks like a

series of small steps There are two types of step functions

rounding UP and rounding DOWNRounding UP

Greatest Integer Functionf(x) = -int (-x)Ceiling FunctionExample f(4) = 1

Rounding DownLeast Integer Functionf(x) = int(x)Floor FunctionExample f(21) = 2

One-Sided and Two-sided LimitsA function f(x) has a limit as x approaches c

if and only if (IFF) the right-hand and left-hand limits at c exist and are equal

LandLLxfcxcxcx

limlim)(lim

Exploring Right-handed and Left-handed LimitsUse the graph and function below to

determine the limits

435

321

22211

101

)(

xx

xx

xx

xx

xf

)(lim0

xfx

)(lim1

xfx

)(lim1

xfx 1 10Therefore no limit exists

as x1 since they are NOT the same

The rest of the limits regarding this problem can be found on page 64

Exploring Left-handed and Right-Handed Limits

)(lim1

xfx

)(lim1

xfx

1 2)(lim

2xf

x )(lim

2xf

x

2 3

Sandwich TheoremIf g(x) lt f(x) lt h(x) for all x ne c in some interval

about c and limxc g(x) = limxc h(x) = L then limxc f(x) = L

Using the sandwich theoremShow that the limx0 [x2 sin(1x)] = 0We know that all of the values of the sine function

lie between -1 and 1 (range)The greatest value that sin(1x) can have is 1 and

the lowest value is -1 sohellip-x2 lt x2 sin(1x)] lt x2

Because the limx0 -x2 = 0 and limx0 x2 = 0 then limx0 [x2 sin(1x)] = 0

Closure

• Drill Find f(2)
• Lesson 21 Rates of Changes and Limits
• Average and Instantaneous Speed
• Finding Instantaneous Speed
• Definition of Limit
• Determining the limit by substitution and confirm graphically (
• Determine the limit graphically Then prove algebraically(Alwa
• Determine the limit graphically Then prove algebraically(Alwa (2)
• Properties of Limits
• Properties of Limits (2)
• Properties of Limits (3)
• Using Properties of Limits
• Drill
• Drill solutions
• One-sided and Two-sided Limits
• Definition of Step Function
• Rounding Down
• One-Sided and Two-sided Limits
• Exploring Right-handed and Left-handed Limits
• Exploring Left-handed and Right-Handed Limits
• Sandwich Theorem
• Closure

One-sided and Two-sided LimitsSometimes the values of a function f tend

to different limits as x approaches a number c from the left and from the right

Right-hand limit The limit of f as x approaches c from the

rightLeft-hand limit

The limit of f as x approaches c from the leftExample Use the graph to the right

) ( limx fc x

)(lim xfcx

1)(lim2

xfx

0)(lim2

xfx

Definition of Step FunctionA step function is a special type of function

whose graph is a series of line segments The graph of a step function looks like a

series of small steps There are two types of step functions

rounding UP and rounding DOWNRounding UP

Greatest Integer Functionf(x) = -int (-x)Ceiling FunctionExample f(4) = 1

Rounding DownLeast Integer Functionf(x) = int(x)Floor FunctionExample f(21) = 2

One-Sided and Two-sided LimitsA function f(x) has a limit as x approaches c

if and only if (IFF) the right-hand and left-hand limits at c exist and are equal

LandLLxfcxcxcx

limlim)(lim

Exploring Right-handed and Left-handed LimitsUse the graph and function below to

determine the limits

435

321

22211

101

)(

xx

xx

xx

xx

xf

)(lim0

xfx

)(lim1

xfx

)(lim1

xfx 1 10Therefore no limit exists

as x1 since they are NOT the same

The rest of the limits regarding this problem can be found on page 64

Exploring Left-handed and Right-Handed Limits

)(lim1

xfx

)(lim1

xfx

1 2)(lim

2xf

x )(lim

2xf

x

2 3

Sandwich TheoremIf g(x) lt f(x) lt h(x) for all x ne c in some interval

about c and limxc g(x) = limxc h(x) = L then limxc f(x) = L

Using the sandwich theoremShow that the limx0 [x2 sin(1x)] = 0We know that all of the values of the sine function

lie between -1 and 1 (range)The greatest value that sin(1x) can have is 1 and

the lowest value is -1 sohellip-x2 lt x2 sin(1x)] lt x2

Because the limx0 -x2 = 0 and limx0 x2 = 0 then limx0 [x2 sin(1x)] = 0

Closure

• Drill Find f(2)
• Lesson 21 Rates of Changes and Limits
• Average and Instantaneous Speed
• Finding Instantaneous Speed
• Definition of Limit
• Determining the limit by substitution and confirm graphically (
• Determine the limit graphically Then prove algebraically(Alwa
• Determine the limit graphically Then prove algebraically(Alwa (2)
• Properties of Limits
• Properties of Limits (2)
• Properties of Limits (3)
• Using Properties of Limits
• Drill
• Drill solutions
• One-sided and Two-sided Limits
• Definition of Step Function
• Rounding Down
• One-Sided and Two-sided Limits
• Exploring Right-handed and Left-handed Limits
• Exploring Left-handed and Right-Handed Limits
• Sandwich Theorem
• Closure

Definition of Step FunctionA step function is a special type of function

whose graph is a series of line segments The graph of a step function looks like a

series of small steps There are two types of step functions

rounding UP and rounding DOWNRounding UP

Greatest Integer Functionf(x) = -int (-x)Ceiling FunctionExample f(4) = 1

Rounding DownLeast Integer Functionf(x) = int(x)Floor FunctionExample f(21) = 2

One-Sided and Two-sided LimitsA function f(x) has a limit as x approaches c

if and only if (IFF) the right-hand and left-hand limits at c exist and are equal

LandLLxfcxcxcx

limlim)(lim

Exploring Right-handed and Left-handed LimitsUse the graph and function below to

determine the limits

435

321

22211

101

)(

xx

xx

xx

xx

xf

)(lim0

xfx

)(lim1

xfx

)(lim1

xfx 1 10Therefore no limit exists

as x1 since they are NOT the same

The rest of the limits regarding this problem can be found on page 64

Exploring Left-handed and Right-Handed Limits

)(lim1

xfx

)(lim1

xfx

1 2)(lim

2xf

x )(lim

2xf

x

2 3

Sandwich TheoremIf g(x) lt f(x) lt h(x) for all x ne c in some interval

about c and limxc g(x) = limxc h(x) = L then limxc f(x) = L

Using the sandwich theoremShow that the limx0 [x2 sin(1x)] = 0We know that all of the values of the sine function

lie between -1 and 1 (range)The greatest value that sin(1x) can have is 1 and

the lowest value is -1 sohellip-x2 lt x2 sin(1x)] lt x2

Because the limx0 -x2 = 0 and limx0 x2 = 0 then limx0 [x2 sin(1x)] = 0

Closure

• Drill Find f(2)
• Lesson 21 Rates of Changes and Limits
• Average and Instantaneous Speed
• Finding Instantaneous Speed
• Definition of Limit
• Determining the limit by substitution and confirm graphically (
• Determine the limit graphically Then prove algebraically(Alwa
• Determine the limit graphically Then prove algebraically(Alwa (2)
• Properties of Limits
• Properties of Limits (2)
• Properties of Limits (3)
• Using Properties of Limits
• Drill
• Drill solutions
• One-sided and Two-sided Limits
• Definition of Step Function
• Rounding Down
• One-Sided and Two-sided Limits
• Exploring Right-handed and Left-handed Limits
• Exploring Left-handed and Right-Handed Limits
• Sandwich Theorem
• Closure

Rounding DownLeast Integer Functionf(x) = int(x)Floor FunctionExample f(21) = 2

One-Sided and Two-sided LimitsA function f(x) has a limit as x approaches c

if and only if (IFF) the right-hand and left-hand limits at c exist and are equal

LandLLxfcxcxcx

limlim)(lim

Exploring Right-handed and Left-handed LimitsUse the graph and function below to

determine the limits

435

321

22211

101

)(

xx

xx

xx

xx

xf

)(lim0

xfx

)(lim1

xfx

)(lim1

xfx 1 10Therefore no limit exists

as x1 since they are NOT the same

The rest of the limits regarding this problem can be found on page 64

Exploring Left-handed and Right-Handed Limits

)(lim1

xfx

)(lim1

xfx

1 2)(lim

2xf

x )(lim

2xf

x

2 3

Sandwich TheoremIf g(x) lt f(x) lt h(x) for all x ne c in some interval

about c and limxc g(x) = limxc h(x) = L then limxc f(x) = L

Using the sandwich theoremShow that the limx0 [x2 sin(1x)] = 0We know that all of the values of the sine function

lie between -1 and 1 (range)The greatest value that sin(1x) can have is 1 and

the lowest value is -1 sohellip-x2 lt x2 sin(1x)] lt x2

Because the limx0 -x2 = 0 and limx0 x2 = 0 then limx0 [x2 sin(1x)] = 0

Closure

• Drill Find f(2)
• Lesson 21 Rates of Changes and Limits
• Average and Instantaneous Speed
• Finding Instantaneous Speed
• Definition of Limit
• Determining the limit by substitution and confirm graphically (
• Determine the limit graphically Then prove algebraically(Alwa
• Determine the limit graphically Then prove algebraically(Alwa (2)
• Properties of Limits
• Properties of Limits (2)
• Properties of Limits (3)
• Using Properties of Limits
• Drill
• Drill solutions
• One-sided and Two-sided Limits
• Definition of Step Function
• Rounding Down
• One-Sided and Two-sided Limits
• Exploring Right-handed and Left-handed Limits
• Exploring Left-handed and Right-Handed Limits
• Sandwich Theorem
• Closure

One-Sided and Two-sided LimitsA function f(x) has a limit as x approaches c

if and only if (IFF) the right-hand and left-hand limits at c exist and are equal

LandLLxfcxcxcx

limlim)(lim

Exploring Right-handed and Left-handed LimitsUse the graph and function below to

determine the limits

435

321

22211

101

)(

xx

xx

xx

xx

xf

)(lim0

xfx

)(lim1

xfx

)(lim1

xfx 1 10Therefore no limit exists

as x1 since they are NOT the same

The rest of the limits regarding this problem can be found on page 64

Exploring Left-handed and Right-Handed Limits

)(lim1

xfx

)(lim1

xfx

1 2)(lim

2xf

x )(lim

2xf

x

2 3

Sandwich TheoremIf g(x) lt f(x) lt h(x) for all x ne c in some interval

about c and limxc g(x) = limxc h(x) = L then limxc f(x) = L

Using the sandwich theoremShow that the limx0 [x2 sin(1x)] = 0We know that all of the values of the sine function

lie between -1 and 1 (range)The greatest value that sin(1x) can have is 1 and

the lowest value is -1 sohellip-x2 lt x2 sin(1x)] lt x2

Because the limx0 -x2 = 0 and limx0 x2 = 0 then limx0 [x2 sin(1x)] = 0

Closure

• Drill Find f(2)
• Lesson 21 Rates of Changes and Limits
• Average and Instantaneous Speed
• Finding Instantaneous Speed
• Definition of Limit
• Determining the limit by substitution and confirm graphically (
• Determine the limit graphically Then prove algebraically(Alwa
• Determine the limit graphically Then prove algebraically(Alwa (2)
• Properties of Limits
• Properties of Limits (2)
• Properties of Limits (3)
• Using Properties of Limits
• Drill
• Drill solutions
• One-sided and Two-sided Limits
• Definition of Step Function
• Rounding Down
• One-Sided and Two-sided Limits
• Exploring Right-handed and Left-handed Limits
• Exploring Left-handed and Right-Handed Limits
• Sandwich Theorem
• Closure

Exploring Right-handed and Left-handed LimitsUse the graph and function below to

determine the limits

435

321

22211

101

)(

xx

xx

xx

xx

xf

)(lim0

xfx

)(lim1

xfx

)(lim1

xfx 1 10Therefore no limit exists

as x1 since they are NOT the same

The rest of the limits regarding this problem can be found on page 64

Exploring Left-handed and Right-Handed Limits

)(lim1

xfx

)(lim1

xfx

1 2)(lim

2xf

x )(lim

2xf

x

2 3

Sandwich TheoremIf g(x) lt f(x) lt h(x) for all x ne c in some interval

about c and limxc g(x) = limxc h(x) = L then limxc f(x) = L

Using the sandwich theoremShow that the limx0 [x2 sin(1x)] = 0We know that all of the values of the sine function

lie between -1 and 1 (range)The greatest value that sin(1x) can have is 1 and

the lowest value is -1 sohellip-x2 lt x2 sin(1x)] lt x2

Because the limx0 -x2 = 0 and limx0 x2 = 0 then limx0 [x2 sin(1x)] = 0

Closure

• Drill Find f(2)
• Lesson 21 Rates of Changes and Limits
• Average and Instantaneous Speed
• Finding Instantaneous Speed
• Definition of Limit
• Determining the limit by substitution and confirm graphically (
• Determine the limit graphically Then prove algebraically(Alwa
• Determine the limit graphically Then prove algebraically(Alwa (2)
• Properties of Limits
• Properties of Limits (2)
• Properties of Limits (3)
• Using Properties of Limits
• Drill
• Drill solutions
• One-sided and Two-sided Limits
• Definition of Step Function
• Rounding Down
• One-Sided and Two-sided Limits
• Exploring Right-handed and Left-handed Limits
• Exploring Left-handed and Right-Handed Limits
• Sandwich Theorem
• Closure

Exploring Left-handed and Right-Handed Limits

)(lim1

xfx

)(lim1

xfx

1 2)(lim

2xf

x )(lim

2xf

x

2 3

Sandwich TheoremIf g(x) lt f(x) lt h(x) for all x ne c in some interval

about c and limxc g(x) = limxc h(x) = L then limxc f(x) = L

Using the sandwich theoremShow that the limx0 [x2 sin(1x)] = 0We know that all of the values of the sine function

lie between -1 and 1 (range)The greatest value that sin(1x) can have is 1 and

the lowest value is -1 sohellip-x2 lt x2 sin(1x)] lt x2

Because the limx0 -x2 = 0 and limx0 x2 = 0 then limx0 [x2 sin(1x)] = 0

Closure

• Drill Find f(2)
• Lesson 21 Rates of Changes and Limits
• Average and Instantaneous Speed
• Finding Instantaneous Speed
• Definition of Limit
• Determining the limit by substitution and confirm graphically (
• Determine the limit graphically Then prove algebraically(Alwa
• Determine the limit graphically Then prove algebraically(Alwa (2)
• Properties of Limits
• Properties of Limits (2)
• Properties of Limits (3)
• Using Properties of Limits
• Drill
• Drill solutions
• One-sided and Two-sided Limits
• Definition of Step Function
• Rounding Down
• One-Sided and Two-sided Limits
• Exploring Right-handed and Left-handed Limits
• Exploring Left-handed and Right-Handed Limits
• Sandwich Theorem
• Closure

Sandwich TheoremIf g(x) lt f(x) lt h(x) for all x ne c in some interval

about c and limxc g(x) = limxc h(x) = L then limxc f(x) = L

Using the sandwich theoremShow that the limx0 [x2 sin(1x)] = 0We know that all of the values of the sine function

lie between -1 and 1 (range)The greatest value that sin(1x) can have is 1 and

the lowest value is -1 sohellip-x2 lt x2 sin(1x)] lt x2

Because the limx0 -x2 = 0 and limx0 x2 = 0 then limx0 [x2 sin(1x)] = 0

Closure

• Drill Find f(2)
• Lesson 21 Rates of Changes and Limits
• Average and Instantaneous Speed
• Finding Instantaneous Speed
• Definition of Limit
• Determining the limit by substitution and confirm graphically (
• Determine the limit graphically Then prove algebraically(Alwa
• Determine the limit graphically Then prove algebraically(Alwa (2)
• Properties of Limits
• Properties of Limits (2)
• Properties of Limits (3)
• Using Properties of Limits
• Drill
• Drill solutions
• One-sided and Two-sided Limits
• Definition of Step Function
• Rounding Down
• One-Sided and Two-sided Limits
• Exploring Right-handed and Left-handed Limits
• Exploring Left-handed and Right-Handed Limits
• Sandwich Theorem
• Closure

Closure

• Drill Find f(2)
• Lesson 21 Rates of Changes and Limits
• Average and Instantaneous Speed
• Finding Instantaneous Speed
• Definition of Limit
• Determining the limit by substitution and confirm graphically (
• Determine the limit graphically Then prove algebraically(Alwa
• Determine the limit graphically Then prove algebraically(Alwa (2)
• Properties of Limits
• Properties of Limits (2)
• Properties of Limits (3)
• Using Properties of Limits
• Drill
• Drill solutions
• One-sided and Two-sided Limits
• Definition of Step Function
• Rounding Down
• One-Sided and Two-sided Limits
• Exploring Right-handed and Left-handed Limits
• Exploring Left-handed and Right-Handed Limits
• Sandwich Theorem
• Closure