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 c
f 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 c
f 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 c
f 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
6
8
2
343
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
3lim 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 sin
1 coslim
0
sinlim 1
0 0
sinlim limsin 1 0(0)x x
xx
x
Right-Hand Limit: lim
Left-Hand Limit: lim
x 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
limg(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 sinx
x 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 0x
x x
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
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