2.1

Rates of Change and Limits

Quick ReviewIn Exercises 1 – 4, find f (2).

452 .1 23 xxxf 4

54 .2

3

2

x

xxf

2 sin .3x

xf

2 ,1

1

2 ,13 .4

2x

x

xxxf

0

12

11

0

3

1

Quick ReviewIn Exercises 5 – 8, write the inequality in the form a < x < b.

4 .5 x 2 .6 cx

32 .7 x 2 .8 dcx

44 x 22 cxc

51 x 22 dcxdc

Quick ReviewIn Exercises 9 and 10, write the fraction in reduced form.

3

183 .9

2

x

xx12

2 .10

2

2

xx

xx 6x 3x

6x

x 12 x

12 x 1x

1x

x

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

Essential QuestionHow can limits be used to describe continuity, the derivative

and the integral: the ideas giving the foundation of

Calculus?

Average and Instantaneous SpeedA body’s average speed during an interval of time is found by dividing the distance covered by the elapsed time.

Experiments show that a dense solid object dropped from rest to fall freely near the surface of the earth will fall y = 16t 2 feet in the first t seconds.

1. Wile E Coyote drops an anvil from the top of a cliff. What is its average rate of speed during the first 5 seconds of its fall?

traveleddistance timeelapsed t

y

22 016516

05

80 ft/sec

Average and Instantaneous SpeedA body’s average speed during an interval of time is found by dividing the distance covered by the elapsed time.

Experiments show that a dense solid object dropped from rest to fall freely near the surface of the earth will fall y = 16t 2 feet in the first t seconds.

2. What is the speed of the anvil at the instant t = 5?

t

y

22 516516

h

55 h

Since we cannot calculate the speed right at 5 sec. Calculate the average between 5 sec and slightly after 5 sec.

2516102516 2

hhh

h

hh 216160 h16160

The smaller h becomes the closer the average will get to 160.

Definition of LimitLet c and L be real numbers. The function f has limit L as x approaches c if, given any positive number , there is a positive number such that for all x,

We write

.0 Lxfcx

Lxfcx

lim

The sentence is read, “The limit of f of x as x approaches

c equals L.” The notation means that the values of x approach (but does

not equal) c.

Lxfcx

lim

The next figures illustrate the fact that the existence of a limit as x → c never depend on how the function may not be defined at c.

Definition of Limit continuedThe function f has a limit 2 as x → 1 even though f is not defined at 1.

The function g has a limit 2 as x → 1 even though g(1) ≠ 2.

The function h is the only one whose limit as x → 1 equals its value at x = 1.

Properties of Limits

If , , , and are real numbers and

lim and lim , then

1. : lim

The limit of the sum of two functions is the sum of their limits.

2. : lim

The limit

x c x c

x c

x c

L M c k

f x L g x M

Sum Rule f x g x L M

DifferenceRule f x g x L M

of the difference of two functions is the difference

of their limits.

Properties of Limits continued

3. lim

The limit of the product of two functions is the product of their limits.

4. lim

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

of the function.

5.

x c

x c

f x g x L M

k f x k L

Quot

: lim , 0

The limit of the quotient of two functions is the quotient

of their limits, provided the limit of the denominator is not zero.

x c

f x Lient Rule M

g x M

Product Rule:

Constant Multiple Rule:

Properties of Limits continued

6. : If and are integers, 0, then

lim

provided that is a real number.

The limit of a rational power of a function is that power of the

limit of the function, provided the latte

rrss

x c

r

s

Power Rule r s s

f x L

L

r is a real number.

Other properties of limits:

lim

limx c

x c

k k

x c

Example Properties of LimitsUse any properties of limits to find:

2135lim .3 4

xxcx

Sum and difference rules45lim xcx

xcx13lim

2lim

cx

Product and multiple rules45c 13c 2

8

4lim .4

2

3

x

xxcx

Sum and difference rules lim 3xcx

xcx4lim

2lim xcx

Product and multiple rules c3

4c

2c

Quotient rule 8lim

4lim2

3

x

xx

cx

cx

8limcx

8

Polynomial and Rational Functions

11 0

11 0

1. If ... is any polynomial function and

is any real number, then

lim ...

2. If and are polynomials and is any real number, then

lim , prov

n nn n

n nn nx c

x c

f x a x a x a

c

f x f c a c a c a

f x g x c

f x f c

g x g c

ided that 0.g c

Example Limits5. Determine the limit by substitution. Support graphically.

143lim 2

2

xx

x 23 124 2

90 6 15

Evaluating LimitsAs with polynomials, limits of many familiar functionscan be found by substitution at points where they aredefined. This includes trigonometric functions, exponential and logarithmic functions, and compositesof these functions.

You can evaluate limits either graphically, numerically, or algebraically.

Example Limits6. Determine the limit by substitution. Support graphically.

5

12lim

2

3

x

xxx

1332 2 53

8

16

2

Example Limits

x

xx cos

sin1lim Find .7

0

Solve graphically:

1. is and existslimit that thesuggests cos

sin1 ofgraph The

x

xxf

Confirm algebraically:

x

xx cos

sin1lim

0

xx

sin1lim0

xx

coslim0

0sin1

01

0cos 11

Determine the limit graphically. Support algebraically.

Example Limits

xx

5lim Find .8

0

Solve graphically:

exist.not doeslimit that thesuggests 5

ofgraph Thex

xf

Confirm algebraically: undefined. is 0

5 because possibleNot

Confirm numerically:

x

y

1.0 01.0 001.0 1.0 01.0 001.0

50 500 5000 50 500 5000

Limit does not exist.

Determine the limit graphically. Support algebraically.

Example Limits

25

158lim Find .9

2

2

5

x

xxx

Solve graphically:

.5

1 is and existslimit that thesuggests

25

158 ofgraph The

2

2

x

xxxf

Confirm algebraically:

25

158lim

2

2

5

x

xxx

35lim

5

xxx 55 xx

3lim

5

xx

2 5x 10 5

1

Determine the limit graphically. Support algebraically.

Example Limits

xxx

xx 2

3

0 cos

sinlim Find .10

Solve graphically:

1. is and existslimit that thesuggests cos

sin ofgraph The

2

3

xxx

xxf

Confirm algebraically:

xxx

xx 2

3

0 cos

sinlim

sinlim

3

0

xx

xx 2cos1 xx

xx 2

3

5 sin

sinlim

x

xx

sinlim

0

xsin

1

Determine the limit graphically. Support algebraically.

On page 60, the graph and table shows that this limit approaches 1.

One-Sided and Two-Sided Limits

Sometimes 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

We sometimes call lim the two-sided limit of at to distinguish it from

the one-sided right-hand and left-hand limits of at .

A function has a limit as approaches if and only if the r

x cf x f c

f c

f x x c

ight-hand

and left-hand limits at exist and are equal. In symbols,

lim lim and lim .x c x c x c

c

f x L f x L f x L

Example One-Sided and Two-Sided Limitsx

xintlim 11.

2

Solve graphically:

2

xx

intlim 12.1

Solve graphically:

0

Example One-Sided and Two-Sided LimitsGiven the following graph, compute the limits.

4 13. f

DNE

xfx 4lim 14.

2

xfx 4lim 15.

2

xfx 4lim 16.

2

Example One-Sided and Two-Sided LimitsGiven the following graph, compute the limits.

1 17. f

4

xfx 1lim 18.

4

xfx 1lim 19.

2

xfx 1lim 20.

DNE

Example One-Sided and Two-Sided LimitsGiven the following graph, compute the limits.

6 21. f

2

xfx 6lim 22.

5

xfx 6lim 23.

5

xfx 6lim 24.

5

Sandwich TheoremIf we cannot find a limit directly, we may be able to find

it indirectly with the Sandwich Theorem. The theorem

refers to a function whose values are sandwiched between

the values of two other func

f

tions, and .

If and have the same limit as then has that limit too.

g h

g h x c f

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

Pg. 66, 2.1 #1-47 odd