Contributed by:

TOPICS DISCUSSED:

1. rates of change of limits

2. the sandwich theorem

1. rates of change of limits

2. the sandwich theorem

1.
2.1

Rates of Change

and Limits

Grand Teton National Park, Wyoming

Photo by Vickie Kelly, 2007 Greg Kelly, Hanford High School, Richland, Washington

Rates of Change

and Limits

Grand Teton National Park, Wyoming

Photo by Vickie Kelly, 2007 Greg Kelly, Hanford High School, Richland, Washington

2.
Suppose you drive 200 miles, and it takes you 4 hours.

mi

Then your average speed is: 200 mi 4 hr 50

hr

distance x

average speed

elapsed time t

If you look at your speedometer during this trip, it might

read 65 mph. This is your instantaneous speed.

mi

Then your average speed is: 200 mi 4 hr 50

hr

distance x

average speed

elapsed time t

If you look at your speedometer during this trip, it might

read 65 mph. This is your instantaneous speed.

3.
A rock falls from a high cliff.

2

The position of the rock is given by: y 16t

After 2 seconds: y 16 22 64

64 ft ft

average speed: Vav 32

2 sec sec

What is the instantaneous speed at 2 seconds?

2

The position of the rock is given by: y 16t

After 2 seconds: y 16 22 64

64 ft ft

average speed: Vav 32

2 sec sec

What is the instantaneous speed at 2 seconds?

4.
2 2

y 16 2 h 16 2

Vinstantaneous

t h

for some very small where h = some very

change in t small change in t

We can use the TI-89 to evaluate this expression for

smaller and smaller values of h.

y 16 2 h 16 2

Vinstantaneous

t h

for some very small where h = some very

change in t small change in t

We can use the TI-89 to evaluate this expression for

smaller and smaller values of h.

5.
2 2

y 16 2 h 16 2

Vinstantaneous

t h

16 2 h ^ 2 64 h h 1,.1,.01,.001,.0001,.00001

y

We can see that the velocity

approaches 64 ft/sec as h becomes

h t

very small.

1 80

We say that the velocity has a limiting

0.1 65.6

value of 64 as h approaches zero.

.01 64.16

(Note that h never actually becomes .001 64.016

zero.) 64.0016

.0001

.00001 64.0002

y 16 2 h 16 2

Vinstantaneous

t h

16 2 h ^ 2 64 h h 1,.1,.01,.001,.0001,.00001

y

We can see that the velocity

approaches 64 ft/sec as h becomes

h t

very small.

1 80

We say that the velocity has a limiting

0.1 65.6

value of 64 as h approaches zero.

.01 64.16

(Note that h never actually becomes .001 64.016

zero.) 64.0016

.0001

.00001 64.0002

6.
2

The limit as h 16 2 h 16 22

approaches zero:

lim

h 0 h

16 lim

4 4 h h 2

4

Since the 16 is

unchanged as h

h 0 h

approaches zero,

we can factor 16 4 4h h 2 4

out. 16 lim

h 0 h

0

16 lim 4 h 64

h 0

The limit as h 16 2 h 16 22

approaches zero:

lim

h 0 h

16 lim

4 4 h h 2

4

Since the 16 is

unchanged as h

h 0 h

approaches zero,

we can factor 16 4 4h h 2 4

out. 16 lim

h 0 h

0

16 lim 4 h 64

h 0

7.
sin x

Consider: y

x

What happens as x approaches zero?

Y= y sin x / x

2

2

/2

WINDOW

GRAPH

Consider: y

x

What happens as x approaches zero?

Y= y sin x / x

2

2

/2

WINDOW

GRAPH

8.
y sin x / x

Looks like y=1

Looks like y=1

9.
y sin x / x

Numerically:

TblSet

TABLE

You can scroll

down to see

more values.

Numerically:

TblSet

TABLE

You can scroll

down to see

more values.

10.
y sin x / x

sin x

It appears that the limit of as x approaches zero is 1

x

TABLE

You can scroll

down to see

more values.

sin x

It appears that the limit of as x approaches zero is 1

x

TABLE

You can scroll

down to see

more values.

11.
Limit notation: lim f x L

x c

“The limit of f of x as x approaches c is L.”

sin x

So: lim 1

x 0 x

x c

“The limit of f of x as x approaches c is L.”

sin x

So: lim 1

x 0 x

12.
The limit of a function refers to the value that the

function approaches, not the actual value (if any).

lim f x 2

x 2

not 1

function approaches, not the actual value (if any).

lim f x 2

x 2

not 1

13.
Properties of Limits:

Limits can be added, subtracted, multiplied, multiplied

by a constant, divided, and raised to a power.

(See your book for details.)

For a limit to exist, the function must approach the

same value from both sides.

One-sided limits approach from either the left or right side only.

Limits can be added, subtracted, multiplied, multiplied

by a constant, divided, and raised to a power.

(See your book for details.)

For a limit to exist, the function must approach the

same value from both sides.

One-sided limits approach from either the left or right side only.

14.
lim f x

2

x 1 does not exist

because the left and

1

right hand limits do not

match!

1 2 3 4

At x=1: lim f x 0 left hand limit

x 1

lim f x 1 right hand limit

x 1

f 1 1 value of the function

2

x 1 does not exist

because the left and

1

right hand limits do not

match!

1 2 3 4

At x=1: lim f x 0 left hand limit

x 1

lim f x 1 right hand limit

x 1

f 1 1 value of the function

15.
2 lim f x 1

x 2

1

because the left and

right hand limits match.

1 2 3 4

At x=2: lim f x 1 left hand limit

x 2

lim f x 1 right hand limit

x 2

f 2 2 value of the function

x 2

1

because the left and

right hand limits match.

1 2 3 4

At x=2: lim f x 1 left hand limit

x 2

lim f x 1 right hand limit

x 2

f 2 2 value of the function

16.
2 lim f x 2

x 3

1

because the left and

right hand limits match.

1 2 3 4

At x=3: lim f x 2 left hand limit

x 3

lim f x 2 right hand limit

x 3

f 3 2 value of the function

x 3

1

because the left and

right hand limits match.

1 2 3 4

At x=3: lim f x 2 left hand limit

x 3

lim f x 2 right hand limit

x 3

f 3 2 value of the function

17.
The Sandwich Theorem:

If g x f x h x for all x c in some interval about c

and lim g x lim h x L, then lim f x L.

x c x c x c

2 1

Show that: lim x sin 0

x 0

x

1

The maximum value of sine is 1, so x sin x 2

2

x

1

The minimum value of sine is -1, so x sin x 2

2

x

1

So: x x sin x 2

2 2

x

If g x f x h x for all x c in some interval about c

and lim g x lim h x L, then lim f x L.

x c x c x c

2 1

Show that: lim x sin 0

x 0

x

1

The maximum value of sine is 1, so x sin x 2

2

x

1

The minimum value of sine is -1, so x sin x 2

2

x

1

So: x x sin x 2

2 2

x

18.
1

lim x lim x sin lim x 2

2 2

x 0 x 0

x x 0

2 1

0 lim x sin 0

x 0

x

2 1

By the sandwich theorem: lim x sin 0

x 0

x

Y= WINDOW

lim x lim x sin lim x 2

2 2

x 0 x 0

x x 0

2 1

0 lim x sin 0

x 0

x

2 1

By the sandwich theorem: lim x sin 0

x 0

x

Y= WINDOW

19.