2.1

Rates of Change and Limits

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

…and why

Limits can be used to describe continuity, the derivative

and the integral: the ideas giving the foundation of

calculus.

Average and Instantaneous Speed

A 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 sur2

face of the earth will fall

16 feet in the first seconds. y t t

Definition of Limit

Let and be real numbers. The function has limit as

approaches if, given any positive number , there is a positive

number such that for all ,

0 .

We write

limx c

c L f L x

c

x

x c f x L

f x L

Definition of Limit continued

The sentence lim is read, "The limit of of as

approaches equals ". the notation means that the values

of the function approach or equal as the values of approach

(but do not equ

x cf x L f x x

c L f x

f L x

al) .

Figure 2.2 illustrates the fact that the existence of a limit as never

depends on how the function may or may not be defined at .

c

x c

c

Definition of Limit continued

The function has limit 2 as 1 even though is not defined at 1.

The function has limit 2 as 1 even though 1 2.

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

f x f

g x g

h x x

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 Limits

3

Use any of the properties of limits to find

lim 3 2 9x c

x x

3 3

3

sum and difference rules

product and multiple rules

lim 3 2 9 lim3 lim2 lim9

3 2 9

x c x c x c x cx x x x

c c

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 Limits 2

5Use Theorem 2 to find lim 4 2 6

xx x

22

5lim 4 -2 6 4 2 6 4 25 15 0 6 100 0 95 1 6 6x

x x

Evaluating LimitsAs with polynomials, limits of many familiar

functions can be found by substitution at points where they are defined. This includes trigonometric functions, exponential and logarithmic functions, and composites of these functions.

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

Example Limits0

5Find lim

x x

[-6,6] by [-10,10]

5Solve graphically: The graph of suggests that

the limit does not exist.

f xx

Confirm Analytically :

We can't use substitution in this example because when is relaced by 0,

the denominator becomes 0 and the function is undefined.

This would suggest that we rely on the graph to s

x

ee that the

limit does not exist.

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

One-Sided and Two-Sided Limits (continued)

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 Limits

o

1 2 3

4

Find the following limits from the given graph.

0

2

2

2

3

a. lim

b. lim

c. lim

d. lim

e. lim

x

x

x

x

x

f x

f x

f x

f x

f x

0

Does Not Exist

4

Does Not Exist

0

Sandwich Theorem

If 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

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