Objective: Be able to understand the definition of a limit, the properties of limits, and one-sided and two-sided limits.

( ) ( )( )

( ) ( )

f c x f cm PQ

c x cf c x f c

x

As Q gets closer and closer to P,

the slope of the secant line gets closer

to the slope of the tangent line.

( , ( ))P c f c

( , ( ))Q c x f c x

x

The slope of the tangent line is the limit of the slope of the secant line.

lim ( ) Lx cf x

If f(x) becomes arbitrarily close to a single number L as our x approaches c from either side, then the limit of f(x), as x approaches c, is L.

c

(c,L)

3

1

Estimate the limit graphically and numerically.

11) lim

1x

x

x

( )f x

x 0.8

2.44

0.9 0.990.999 1 1.0011.01 1.1 1.2

2.7102.970 2.997 3 3.003 3.030 3.310 3.64

3

1

1lim 3

1x

x

x

The existence or nonexistence of ( ) at =c

has no bearing on the existence of the limit of

( ) as approaches c.

f x x

f x x

2) Find the limit of ( ) as approaches

4 where is defined by

f x x

f

( )f x 1, 4

0, =4

x

x

4

lim ( ) 1x

f x

If

as from either side, then the limit of

( ) as approach

( ) becomes arbitrarily clos

es c, is L, writt

e to a sing

en as

ap

proac

le

num

ber

hes c

L

lim ( ) Lx c

x

f

x

x x

f x

f

L L

L

c c

c

L L

L

c c

c

( ) becomes arbitrarily close to a single number Lf x

( ) lies in the interval (L ,L ) or

( )-L

f x

f x

approaches cxThere exists a positive

number, , such that

lies in either the interval

(c- ,c) or (c, c+ ) or

0< -c

x

x

Let c and L be real numbers. The function

if, given any positive

, there is a positive number such that for all ,

0

has limit L as approaches

-c (

c

)-L

We write

x

x f x

f x

lim ( ) Lx cf x

If L, M, , and are real numbers and

lim ( ) L and lim ( ) M, then

1) Sum Rule: lim( ( ) ( )) L M

2) Difference Rule: lim( ( ) ( )) L-M

3) Product Rule: lim( ( ) ( )) L M

.

x c x c

x c

x c

x c

c k

f x g x

f x g x

f x g x

f x g x

( ) L4) Quotient Rule: lim , M 0

( ) M

5) Constant Multiple Rule: lim( ( )) L

6) Power Rule: If and are integers, 0, then

lim( ( )) L provided that L is a real number.

x c

x c

r r rs s s

x c

f x

g x

kf x k

r s s

f x

8

2

0

0

2

1

3

4

2

3

Find the limits.

3) lim3

4) lim(2 4) ( 4)

5) lim( 3)( 2)

16) lim

7) lim(2 1)

8) lim 2 10

x

x

x

x

x

x

x

x x

x x

x

x

x

x

24

68

2343

2 2

Direct Substitution:

does not work

if you get the

0indeterminate form !!!

0

Let be a real number and let ( ) ( ) for all

in an open interval containing . If the limit

of ( ) as approaches exists, then the limit of

( ) also exists and

lim ( ) ( )x c

c f x g x

x c c

g x x c

f x

f x g x

3

2

19) Show that the functions ( )

1

and ( ) 1 have the same values

for all other than 1.

xf x

x

g x x x

x x

3 1( )

1

xf x

x

21 1

1

x x x

x

2 1x x ( )g x

110) Find lim ( ).

xf x

3

2

3

611) Find lim .

3x

x x

x

3

3 2lim

3x

x x

x

3

lim 2x

x

5

0

1 112) Find lim .

x

x

x

0

1 1 1 1lim

1 1x

x x

x x

0

1 1lim

1 1x

x

x x

0lim

1 1x

x

x x

0

1lim

1 1x x

1

2

0

sin1) lim 1

x

x

x

0

1 cos2) lim 0

x

x

x

0

2

0

Find the limit of the trigonometric function.

cos tan13) lim

sin14) lim

x

x

x

0

cos sin1 coslim

0

sinlim 1

0 0

sinlim limsin 1 0(0)x x

xx

x

Right-Hand Limit: lim

Left-Hand Limit: limx c

x c

Graph : int or

(Greatest Integer Funtion)

y x y x

Observe the behavior of the graph as you

approach 1 from the left and from the right.

What do you notice?

limit of as c

from the right

f x

limit of as c

from the left

f x

A function ( ) has a limit as approaches c

if and only if the right-hand and left-hand limits

at c exist and are equal. In symbols,

lim ( ) L lim ( ) L and lim ( ) L.x c x c x c

f x x

f x f x f x

15) Find the limit of ( ) as

approaches 0 from the left

and from the right.

f x x

x

lim 1x c

x

lim 0x c

x

limx c

x DNE

(Used when a limit cannot

be found directly)

If ( ) ( ) ( ) for all c

in some interval about c, and

lim g(x)= lim ( ) L, then

lim ( ) L.x c x c

x c

g x f x h x x

h x

f x

( )h x

( )f x( )g x

c

0

Use the Sandwich Theorem to find the limit.

16) lim sinxx x

RECALL: sin oscillates between -1 and 1x

0 0 0lim lim sin limx x x

x x x x

00 lim sin 0

xx x

0lim sin 0xx x

Objective: Be able to

calculate the average and

instantaneous speed.

Change in Distance

Change in Time

s

t

position

time

s

t

2

1) A rock breaks loose from the top of a cliff.

What is its average speed during the first 2

seconds of fall? (Use 16 for the distance

equation, in terms of feet, of a free-falling object.)

y t

2 216(2) 16(0)

2 0

s

t

32 fe

64 0et s

2/ ec

(Speed over an interval of time)

(Speed at a

specific time)

**Review Tangent Line Problem

0

( ) ( )limt

s t t s t

t

2) Find the speed of the rock in

Example 1 at the instant 2.t 2 2

0

16(2 ) 16(2)limt

t

t

2

0

64 64 16( ) 64limt

t t

t

2

0

64 16( )limt

t t

t

0lim 64 16t

t

64 feet/sec

Homework:Page 66 #1,2,3,4,8,9,12,16,19,23,25,26,27,28,35,36,38,41,44,45,46,47,48,50,51,54,61,64