What you ’ ll learn about Average and Instantaneous Speed Definition of Limit Properties of Limits One-Sided and Two-Sided Limits Sandwich Theorem The.

What you’ll learn about• Average and Instantaneous Speed• Definition of Limit• Properties of Limits• One-Sided and Two-Sided Limits• Sandwich TheoremThe Limit concept is a calculus concept. Limits can be

used to describe continuity, the Derivative and the Integral: the ideas giving the foundation of calculus.

In Ch2 we will define and calculate Limits, by substitution, numerically, analytically, graphical investigation – or a combination of these.

A. Average Rate of Change vs Instantaneous Rate of Change

1. Let’s begin our discussion of the difference between the Average rate of change (ROC) and the Instantaneous ROC by first reviewing the slope of the graph of a linear function.

2. The slope of a line is constant! That is, Δx and Δy don’t change as you move on the graph.

3. Let’s do a real world example.

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

Then your average speed is:mi

200 mi 4 hr 50 hr

But, how do we set this up using points on a graph?If

4. In this application the slope is constant, but, how do you find the slope of a curve, where the slope varies?

4. If you look at your speedometer during this trip at a particular time, it might read 65 mph. This is your instantaneous speed.

A real world example: A rock falls from a high cliff. Find the instantaneous speed after 2 seconds.

The position of the rock at any time t is given by the function: 216y t

At t = 2 sec.:

average speed: av

64 ft ft32

2 sec secV

32 is the average speed, but what is the instantaneous speed at 2 seconds?

instantaneous

yV

t

for some very small change in t

2 216 2 16 2h

h

where h = some very small change in t, or

We can use a graphing calculator to evaluate this expression for smaller and smaller values of h. You can see this in the book on page 60.

Why can you NOT use the average speed method to find the instantaneous speed?

instantaneous

yV

t

2 2

16 2 16 2h

h

hy

t

1 80

0.1 65.6

.01 64.16

.001 64.016

.0001 64.0016

.00001 64.0002

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

We can see that the velocity approaches 64 ft/sec as h becomes very small.

We say that the velocity has a limiting value of 64 as h approaches zero.

(Note that h never actually becomes zero.)

2 2

0

16 2 16 2limh

h

h

The limit as h approaches zero:

2

0

4 4 416 lim

h

h h

h

2

0

4 4 416 lim

h

h h

h

0

Since the 16 is unchanged as h approaches zero, we can factor 16 out.

Now, let’s find the limit Algebraically (Analytically)

Okay now, Let’s find the limit of:

First, we cannot use substitution; can you see why?So, we do this graphicaally

sin xy

x

What happens as x approaches zero?

sin /y x x

22

/ 2

WINDOW

Y=

GRAPH

sin /y x x

Looks like y=1

sin /y x x

Numerically:

TblSet

You can scroll down to see more values.

TABLE

sin /y x x

You can scroll down to see more values.

TABLE

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

x

Limit notation: limx c

f x L

“The limit of f of x as x approaches c is L.”The notation means that the values f (x) of the function f approach or equal L as the valuesof x approach (but do not equal) c.

So: 0

sinlim 1x

x

x

Okay what is the the formal and informal definitions of a Limit. For the formal definition see page 60. The informal definition is as follows:

Slide 2- 13

Example Limits

Solve graphically:

1 sinThe graph of suggests that the limit exists and is 1.

cos

xf x

x

0

1 sinFind lim

cosx

x

x

0

00

Confirm Analytically:

lim 1 sin 1 sin 01 sinFind lim

cos lim cos cos0

1 0 1

1

x

xx

xx

x x

The limit of a function refers to the “y” value that the function approaches, not the actual value (if any).

2

lim 2x

f x

not 1

Properties of Limits:

Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power. See book for details. Let’s do some examples:

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.

Slide 2- 16

One-Sided and Two-Sided LimitsSometimes the values of a function tend to different limits as approaches a

number from opposite sides. When this happens, we call the limit of as

approaches from the right the right-ha

f x

c f x

c

nd limit of at and the limit as

approaches from the left the left-hand limit.

right-hand: lim The limit of as approaches from the right.

left-hand: lim The limit of as apx c

x c

f c x

c

f x f x c

f x f x

proaches from the left.c

1 2 3 4

1

2

At x=1: 1

lim 0x

f x

1

lim 1x

f x

1 1f

left hand limit

right hand limit

value of the function

1

limx

f x does not exist

because the left and right hand limits do not match!

At x=2: 2

lim 1x

f x

2

lim 1x

f x

2 2f

left hand limit

right hand limit

value of the function

2

lim 1x

f x

because the left and right hand limits match.

1 2 3 4

1

2

At x=3: 3

lim 2x

f x

3

lim 2x

f x

3 2f

left hand limit

right hand limit

value of the function

3

lim 2x

f x

because the left and right hand limits match.

1 2 3 4

1

2

The Sandwich Theorem:

If for all in some interval about

and lim lim , then lim .x c x c x c

g x f x h x x c c

g x h x L f x L

Show that: 2

0

1lim sin 0x

xx

The maximum value of sine is 1, so 2 21sinx x

x

The minimum value of sine is -1, so 2 21sinx x

x

So: 2 2 21sinx x x

x

2 2 2

0 0 0

1lim lim sin limx x x

x x xx

2

0

10 lim sin 0

xx

x

2

0

1lim sin 0x

xx

By the sandwich theorem:

Y= WINDOW

p