AP CALCULUS AB

Chapter 2:Limits and Continuity

Section 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

Average speed y

t

Section 2.1 – Rates of Change and Limits Instantaneous Speed:

To find the instantaneous speed, we start by calculating the average speed over an interval from time t1 to any slight time later t2=t1+h

To find the instantaneous speed, we take increasingly smaller values of h --- in other words, we let h approach 0.Then

h

tyhty

tht

tyhty

t

y 11

11

11

h

tfhtfh

0lim Speed ousInstantane

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

(If the horizontal distance between x and c is less than , then the vertical distance between and L is less than ).

or as x gets increasingly closer to c, then gets increasingly closer to L.

xf

xf

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 ...

n nn n

n nn nx c

f x a x a x a

c

f x f c a c a c a

2

4

2

Ex: 3 2 1 for

lim 3 2 1

3 16 2 4 1

48 8

4

1

5

4 4 4

5

x

f x x x c

f x f

Polynomial and Rational Functions

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

lim , provided that 0.x c

f x g x c

f x f cg c

g x g c

2

3

Ex: 2 3 1 and for

2 3 1

3

3 9 3

3

10lim 10

123

2

x

g x x

g

f x x x c

f

g

x f

x

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

Section 2.1 – Rates of Change and Limits Techniques for Finding Limits

1. Numerically – plug in values that approach c from both the right and left.

2. Algebraically – factor and cancel/simplify first. Then plug in c for x.

3. Graphically.

In “well-behaved” functions we can find the by direct substitution of c for x:

xfcx

lim

3

4 3 4 7Ex: lim 7

2 3 2 1x

x

x

xfcx

lim

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.

Section 2.1 – Rates of Change and Limits Limits of Trigonometric Functions:1. 2. 3. 4. 5. 6.

cx

cx

cx

cx

cx

cx

cx

cx

cx

cx

cx

cx

cotcotlim

secseclim

csccsclim

tantanlim

coscoslim

sinsinlim

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.

Section 2.1 – Rates of Change and Limits Functions that agree in all but 1 point

.limlim

and exists also

lim then exists, lim and , containing

intervalopen an in allfor If

xgxf

xfxgc

cxxgxf

cxcx

cxcx

Section 2.1 – Rates of Change and Limits Cancellation Techniques for finding Limits

If you have a rational function

5

312

32lim

1

132lim

1

32lim :Ex

on.substituti use then , offactor

a with cancel to then try ,0

0 so

0 and 0 and such that

1

1

2

1

x

x

xx

x

xx

xp

xqcq

cpcr

cqcpxq

xpxrxr

x

xx

Section 2.1 – Rates of Change and Limits Rationalization Techniques for Finding Limits

22

1202

122

1lim

22lim

22

22lim

22

2222lim

22lim

0

0

0

0

0

x

xx

x

xx

xx

x

x

x

x

x

x

x

x

x

x

Section 2.1 – Rates of Change and Limits Special Limits from Trigonometry1.

2.

1sin

lim0

x

xx

0cos1

lim0

x

xx

Section 2.1 – Rates of Change and Limits Remember:

2233

2233

babababa

babababa

Section 2.1 – Rates of Change and Limits

As you approach 2 from the left, you get closer and closer to 12.

As you approach 2 from the right, you get closer and closer to 12.

So,

2

422

2

8)(

2

3

x

xxx

x

xxf

122

8lim

3

2

x

xx

Section 2.1 – Rates of Change and Limits Limit of a Composite Function:

If and

then

Lxgcx

)(lim LfxfLx

)(lim

Lfxgfcx

)(lim

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 ap

x 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

Section 2.1 – Rates of Change and LimitsIf

As x approaches 2 from the left, f(x) approaches 3.

As x approaches 2 from the right, f(x) approaches 3.

So,

2 ,1

2 ,3)2(2)(

x

xxxf

3)(lim2

xfx

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

Section 2.1 – Rates of Change and Limits Limits that do not exist:

As x approaches 0 from the left, f(x) approaches 0.

As x approaches 0 from the right, f(x) approaches 1.

So,

does not exist

0 ,

0 ,1)(

3

xx

xxxf

)(lim0

xfx

Section 2.1 – Rates of Change and Limits More Limits that do not

exist:

As x approaches , from the left, f(x) goes to

As x approaches , from the right, f(x) goes to

So

does not exist.

xxf tan)(

2

2

)(lim2

xfx

Section 2.1 – Rates of Change and Limits More Limits that do not exist:

Oscillating behavior

Graph on calculator and zoom in about 4 times around x=0.

xxf

1sin)(

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

Section 2.1 – Rates of Change and Limits The Sandwich Theorem:If for all In some interval about c, and

Then

)()()( xhxfxg cx

Lxhxgcxcx

)(lim)(lim

Lxfcx

)(lim

g(x)

h(x)

f(x)