2.1: Rates of Change & Limits

Greg Kelly, Hanford High School, Richland, Washington

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

Then your average speed is:mi

200 mi 4 hr 50 hr

distanceaverage speed

elapsed time

x

t

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

A rock falls from a high cliff.

The position of the rock is given by:216y t

After 2 seconds:216 2 64y

average speed: av

64 ft ft32

2 sec secV

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

We can use the TI-89 to evaluate this expression for smaller and smaller values of h.

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

0

16 2 64limh

h

h

The limit as h approaches zero:

2

0

16 4 4 64limh

h h

h

2

0

64 64 16 64limh

h h

h

0lim 64 16h

h

0

64

Limit notation: limx c

f x L

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

LIMITS AT INFINITY (Page 608)

and*If x is negative, xn does not exist for certain values of n, so the second limit is undefined.

01

lim x

nx

.0

1 lim

x nx

1. If f ( x ) becomes infinitely large in magnitude (positive or negative) as x approaches the number a from either side, we write or In either case,

the limit does not exist.

.)( lim

xfax

)( lim

xfax

2. If f ( x ) becomes infinitely large in magnitude (positive) as x approaches a from one side and infinitely large in magnitude (negative) as x approaches a from the other side, then does not exist.)( lim

xf

ax

3. If and , and L ≠ M, then does not exist.

Lxfax

)( lim

Mxfax

)( lim

)( lim

xfax

EXISTENCE OF LIMITS: (see page 602)

Means approach from the right

Means approach from the left

The limit of a function refers to the 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 page 603 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.

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!

1 2 3 4

1

2

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

CONTINUITY AT x = c (see page 617)

A function f is continuous at x = c if the following three conditions are satisfied:

1. f (c) is defined

If f is not continuous at c, it is discontinuous there.

and exists, )( lim 2.

xfcx

).()( lim 3.

cfxfcx

1 2 3 4

1

2

4 is a point of discontinuity becausef (4) does not exist. (rule 1)

Examples:

1 is a point of discontinuity even though f (1) exists, thedoes not exist. (rule 2)

)( lim 1

xfx

2 is a point of discontinuity even though f (2) exists and

exists, f (2) ≠ (rule 3)

)( lim 2

xfx

).( lim 2

xfx

3 is a point of continuity because f (3) exists, exists, and

f (3) =

)( lim3 x

xf

).( lim3 x

xf

Interval notation Graph

( #, means to start just to the right of the number

[ #, means to start exactly at the number

# ) means to stop just to the left of the number (exclude the #)

# ] means to stop exactly at the number (include the #)

( #1, #2 ) is an open interval

( #1, #2 ] is a half-open interval

[ #1, #2 ) is a half-open interval

[ #1, #2 ] is a closed interval

#

#

#

#

#1 #2

# ∞ – ∞ #

– ∞ # # ∞ #1 #2

#1 #2

#1 #2

Finding Limits at Infinity (page 610)

,01

lim x

nx0

1 lim

x

nx

2. Use the rules for limits, including the rules for limits at infinity,

and

to find the limit of the result from step 1.

1. Divide p(x) and q(x) by the highest power of x in q(x).

If f (x) = p(x)/q(x), for polynomials p(x) and q(x), q(x) ≠ 0,

and can be found as follows.)( lim x

xf

)( lim x

xf

EXAMPLE # 1

33

2

3

3

333

3

752

246

x lim

xxx

xx

xxx

xx

2

6

3

32

75

24

x lim

xx

xx

752

246

23

3

x lim

xx

xx

0

0 0

0

3 2

6

EXAMPLE # 2

356

49

24

2

x lim

xx

xx

43

4

254

46

44

4

29

x lim

xx

x

x

x

x

x

x

x

6

43

25

34

29

x lim

xx

xx

0 6

0

00

00

EXAMPLE # 3

54

1112

24

5

x lim

xx

x

4

254

44

411

4

512

x lim

x

x

x

x

xx

x

4

12

25

411

x lim

x

xx

4

12 lim

x

x

∞

0

0

IN SUMMARY

If the degree of the numerator is larger than the degree of the denominator the limit is infinity.

If the degree of the denominator is larger than the degree of the numerator the limit is zero.

If the degree of the numerator is the same as the degree of the denominator the limit is the quotient formed by their coefficients a/b.

However on a test, I want to see the steps involved. I do not want you to use this summary. You may use it only to verify your answers.