Upload
nam-hai
View
232
Download
0
Embed Size (px)
Citation preview
7/26/2019 Chapter 1_Function and Limit
1/87
CALCULUSCALCULUS
7/26/2019 Chapter 1_Function and Limit
2/87
Objectives
Four ways to represent a function
Basis functions and the transformations of functions
Calculating limits of functions
limits at a point! limits involving "nfinity#
$erifying continuity of a function
Chapter %& Function and limit
7/26/2019 Chapter 1_Function and Limit
3/87
%'%
Functions and
(heir )epresentations
FU*C("O*S A*+ L","(S
7/26/2019 Chapter 1_Function and Limit
4/87
A function f is a rule that assigns to each element x in a
set Dexactly one element, called f(x), in a set E.
The set Dis called the domainof the function f.
The range of fis the set
of all possible values
of f(x) asxvaries
throughout the domain.
FU*C("O*
Fig' %'%'-! p' %.
7/26/2019 Chapter 1_Function and Limit
5/87
The graph of f is the set of all points (x, y) in the
coordinate plane such that y = f(x) and x is in the domain
of f.
The graph of falso allows us to picture:
The domain of fon thex-axis
"ts rangeon the y-axis
/)A01
7/26/2019 Chapter 1_Function and Limit
6/87
The graph of a function fis shown.a. Find the values of f() and f(!).
b. "hat is the domain and range of f #
f%# 2 -
f3# 4 5'6
+ 2 75! 68
"mf# 2 74.! 98
:;ample %/)A01
7/26/2019 Chapter 1_Function and Limit
7/87
Find the domain and region of thefunctions ( if it is a function).
a.
b.
( ) for all natural numbers n.f n n=
):0):S:*(A("O*S
( ) is any real number such that larger than xg x
+"SCUSS"O*
7/26/2019 Chapter 1_Function and Limit
8/87
There are four possible wa$s to represent a function:
Algebraicall$ (b$ an explicit formula)
%isuall$ (b$ a graph)
&umericall$ (b$ a table of values)
%erball$ (b$ a description in words)
):0):S:*(A("O*S OF FU*C("O*S
7/26/2019 Chapter 1_Function and Limit
9/87
The human population of the world Pdepends on
the time t.
The table gives estimates of the
world population P(t) at time t,
for certain $ears.
'owever, for each value of the
time t,there is a corresponding
value of P, and we sa$ that
Pis a function of t.
:
7/26/2019 Chapter 1_Function and Limit
10/87
When you turn on a hot-water faucet, the temperature T of
the water depends on how long the water has been
running.
raw a rough graph of Tas a function of the time t that has
elapsed since the faucet was turned on.
:;ample 9):0):S:*(A("O*S
7/26/2019 Chapter 1_Function and Limit
11/87
A curve in thexy-plane is the graph of a function of
x if and onl$ if no vertical line intersects thecurve more than once.
(1: $:)("CAL L"*: (:S(
7/26/2019 Chapter 1_Function and Limit
12/87
The reason for the truth of the %ertical *ine Test can be
seen in the figure.
(1: $:)("CAL L"*: (:S(
7/26/2019 Chapter 1_Function and Limit
13/87
+f a function fsatisfies:
f(-x) =f(x), x D
then f is called an even function'
The geometric significance of an even function is that itsgraph is symmetric with respect to the y4a;is.
S=,,:()=& :$:* FU*C("O*
y = x4
movie
http://../phuoc%20vinh/Truong%20FPT/math%201/Calculus/Animation_Chap01/01_02_12a.htmlhttp://../phuoc%20vinh/Truong%20FPT/math%201/Calculus/Animation_Chap01/01_02_12a.html7/26/2019 Chapter 1_Function and Limit
14/87
+f f satisfies:
f(-x) = -f(x), x
D
then fis called an odd function.
The graph of an odd function is symmetric about the origin.
S=,,:()=& O++ FU*C("O*
y = x3 y = x5 y = x7
movie
http://../phuoc%20vinh/Truong%20FPT/math%201/Calculus/Animation_Chap01/01_02_12b.htmlhttp://../phuoc%20vinh/Truong%20FPT/math%201/Calculus/Animation_Chap01/01_02_12b.html7/26/2019 Chapter 1_Function and Limit
15/87
*et f is an odd function. +f (-,!) is in the graph of f then
which point is also in the graph of f#
a. (,!) b. (-,-!) c. (,-!) d. All of the others
Answer: c
:;ample
7/26/2019 Chapter 1_Function and Limit
16/87
:;ample %%
uppose f is an odd function and g is an even function.
"hat can we sa$ about the function f.g defined b$ (f.g)(x)f(x)g(x)#
/rove $our result.
7/26/2019 Chapter 1_Function and Limit
17/87
A function fis called increasing on an intervalIif:
f(x) 0 f(x1) wheneverx 0x1 inI
+t is called decreasing onIif:
f(x) 2 f(x1) wheneverx 0x1 inI
"*C):AS"*/ A*+ +:C):AS"*/ FU*C("O*S
7/26/2019 Chapter 1_Function and Limit
18/87
The function f is said to be increasing on theinterval 7a! b8, decreasing on 3b, c4, and
increasing again on 3c, d4.
"*C):AS"*/ A*+ +:C):AS"*/ FU*C("O*S
7/26/2019 Chapter 1_Function and Limit
19/87
%'% >U"? >U:S("O*S
) +f f is a function then f(x51)f(x)5f(1)
a. True b. False
1) +f f(s)f(t) then s t
a. True b. False
) *et f be a function.
"e can find s and t such that st and f(s) is note6ual to f(t)
a. True b. False
7/26/2019 Chapter 1_Function and Limit
20/87
%'.,A(1:,A("CAL ,O+:LS&
A CA(ALO/ OF:SS:*("AL FU*C("O*S
+n this section, we will learn about:
The purpose of mathematical models.
FU*C("O*S A*+ ,O+:LS
7/26/2019 Chapter 1_Function and Limit
21/87
A mathematical model is a mathematical
description7often b$ means of a function or ane6uation7of a real-world phenomenon such as:
i8e of a population
emand for a product
peed of a falling ob9ect
*ife expectanc$ of a person at birth
ost of emission reductions
,A(1:,A("CAL ,O+:LS
7/26/2019 Chapter 1_Function and Limit
22/87
"hen we sa$ that y is a linear function of x, wemean that the graph of the function is a line.
o, we can use the slope-intercept form of the e6uation
of a line to write a formula for the function as
where m is the slope of the line and b is the y-intercept.
( )y f x mx b= = +
L"*:A) ,O+:LS
7/26/2019 Chapter 1_Function and Limit
23/87
A function Pis called a pol$nomialif
P(x) = anxn+ an-1xn-1+ + a2x2+ a1x+ a0
where n is a nonnegative integer and the numbers
a;, a, a1,
7/26/2019 Chapter 1_Function and Limit
24/87
A rational function fis a ratio of two pol$nomials
where Pand are pol$nomials.
The domain consists of all values ofx
such that .
( )( )
( )
P xf x
Q x=
( ) 0Q x
)A("O*ALFU*C("O*S
7/26/2019 Chapter 1_Function and Limit
25/87
()"/O*O,:()"CFU*C("O*S
f(x) sinx
f(x) = cos x
( , )D =
! 3-, 4
sin( 2 ) sin cos( 2 ) cos ;x k x x k x k Z + = + =
7/26/2019 Chapter 1_Function and Limit
26/87
()"/O*O,:()"CFU*C("O*S
sin
tan cos
x
x x=
, ,2 2x
( , )R =
tan( ) tan ;x k x k Z+ =
7/26/2019 Chapter 1_Function and Limit
27/87
()"/O*O,:()"CFU*C("O*S
The reciprocals of the sine, cosine, and tangentfunctions are
1cos
sin1
seccos
1cottan
ecx
x
xx
anxx
=
=
=
7/26/2019 Chapter 1_Function and Limit
28/87
A function of the formf(x) =x a , where a is constant,
is called a power function.
0O@:) FU*C("O*S
7/26/2019 Chapter 1_Function and Limit
29/87
The e;ponential functions are the functions of the
form , where the base a is a positive
constant.
The graphs of y 1x and y (;.!)xare shown.
+n both cases, the domain is and the range
is .
( ) xf x a=
( , )
(0, )
:
7/26/2019 Chapter 1_Function and Limit
30/87
The logarithmic functions ,
where the base ais a positive constant, are the
inverse functions of the exponential functions.
( ) loga
f x x=
LO/A)"(1,"CFU*C("O*S
The figure shows the graphs of
four logarithmic functions with
various bases.
7/26/2019 Chapter 1_Function and Limit
31/87
*abel the following graph from the graph of
the function $f(x) shown in the part (a)
$f(x)-1, $f(x-1), $-f(x), $1f(x), $f(-x)#
()A*SFO),A("O*S
7/26/2019 Chapter 1_Function and Limit
32/87
=uppose c2 ;.
To obtain the graph of
y =f(x)+c, shiftthe
graph of y =f(x)
a distance c unitsupward.
To obtain the graph
ofy =f(x) -c, shiftthe graph of y =f(x)
a distance c units
downward.
S1"F("*/
@hy dont we consider the case c5
7/26/2019 Chapter 1_Function and Limit
33/87
To obtain the graph ofy =f(x - c),shift the graph of
y =f(x) a distance c units to the right.
To obtain the graph
ofy =f(x " c),shiftthe graph of y =f(x)
a distance c units to
the left.
S1"F("*/
7/26/2019 Chapter 1_Function and Limit
34/87
*abel the following graph from the graph of
the function $f(x) shown in the part (a)
$f(x)-1, $f(x-1), $-f(x), $1f(x), $f(-x)#
*:@ FU*C("O*S F)O, OL+ FU*C("O*S
7/26/2019 Chapter 1_Function and Limit
35/87
*:@ FU*C("O*S F)O, OL+ FU*C("O*S
b$ shifting 1 units downward. b$ shifting 1 units to the right.
2y x= 2y x=
*abel the following graph from the graph of
the function shown in the part (a):
$f(x)-1, $f(x-1),$-f(x), $1f(x), $f(-x)#
y x=
7/26/2019 Chapter 1_Function and Limit
36/87
=uppose c2 .
To obtain the graph
ofy= cf(x), stretch
the graph of y = f(x)
verticall$b$ a factorof c#
To obtain the graph
ofy= (1!c)f(x),
compressthe graph
of y = f(x) verticall$ b$
a factor of c#
()A*SFO),A("O*S
'ow about the case c0#
7/26/2019 Chapter 1_Function and Limit
37/87
+n order to obtain the graph of y f(cx),
compress the graph of y = f(x) hori8ontall$b$ a factor of c#
To obtain the graphof y f(x>c), stretch
the graph of y = f(x)
hori8ontall$ b$ a factor
of c.
()A*SFO),A("O*S
7/26/2019 Chapter 1_Function and Limit
38/87
+n order to obtain the graph of y -f(x),
reflect the graph of y = f(x) about thex-axis.
To obtain the graphof y f(-x), reflectthe graph of y = f(x)about the y-axis.
()A*SFO),A("O*S
7/26/2019 Chapter 1_Function and Limit
39/87
*:@ FU*C("O*S F)O, OL+ FU*C("O*S
*abel the following graph from the graph of
the function shown in the part (a):
$f(x)-1, $f(x-1),$-f(x), $1f(x), $f(-x)#
y x=
7/26/2019 Chapter 1_Function and Limit
40/87
*:@ FU*C("O*S F)O, OL+ FU*C("O*S
b$ reflecting about thex-axis. b$ stretching verticall$ b$ a factor of 1. b$ reflecting about the y-axis
y x=
2y x=
y x=
*abel the following graph from the graph of
the function shown in the part (a):
$f(x)-1, $f(x-1), $-f(x), $1f(x), $f(-x)#
y x=
7/26/2019 Chapter 1_Function and Limit
41/87
= The figure illustrates these stretching
= transformations when applied to the
cosine
= function with c 1.
()A*SFO),A("O*S
:;ample
7/26/2019 Chapter 1_Function and Limit
42/87
:;ample
uppose that the graph of f is given.
escribe how the graph of the function f(x-1)51
can be obtained from the graph of f.
elect the correctanswer.
a. hift the graph 1 units to the left and 1 units down.
b. hift the graph 1 units to the right and 1 units down.
c. hift the graph 1 units to the right and 1 units up.
d. hift the graph 1 units to the left and 1 units up.
e. none of these
Answer: c
7/26/2019 Chapter 1_Function and Limit
43/87
= Two functions fand gcan be combined to formnew functions:
(f5 g)x f(x) 5 g(x)
(f? g)x f(x) ? g(x)
CO,B"*A("O*S OF FU*C("O*S
( )( )( ) ( ) ( ) ( )
( )
f f xfg x f x g x x
g g x
= =
7/26/2019 Chapter 1_Function and Limit
44/87
*et h(x)f(g(x)).
) +f g(x)x- and h(x)x51 then f(x) is:
a. x5 b. x5@ c. x5 d. &one of them
1) +f h(x)x51 and f(x)x- then g(x) is:
a. x5 b. x5@ c.x5 d. &one of them
Answer: ) d 1) a
7/26/2019 Chapter 1_Function and Limit
45/87
%'. >U"? >U:S("O*S
) +f f and g are functions, then
a. True b. False
1)
is
a. ! b. c. 1 d. &one of the others
f g g f=o o
( )(2)f go
)(xf
)(xg
.
x 1 2 " # $
2 1 0 1 2
$ # 2 " $
7/26/2019 Chapter 1_Function and Limit
46/87
Answer: f(x), f( x), f(x)
%'. >U"? >U:S("O*S
7/26/2019 Chapter 1_Function and Limit
47/87
%'-
(he Limit of a Function
L","(S
+n this section, we will learn:
About limits in general and about numericaland graphical methods for computing them.
7/26/2019 Chapter 1_Function and Limit
48/87
+n general, we write
if we can maBe the values
of f(x) arbitraril$ close to $b$ taBingxto be
sufficientl$ close to a
but not eDual to a.
( )limx a
f x L
=
(1: L","( OF A FU*C("O*
7/26/2019 Chapter 1_Function and Limit
49/87
"e write
if we can maBe the values of f(x) arbitraril$ close to $b$
taBingxto be sufficientl$ close to aandxless than a.
( )limx a f x L
=
O*:4S"+:+ L","(S +efinition .
7/26/2019 Chapter 1_Function and Limit
50/87
imilarl$, %the right-hand limit of f(x) as x approaches a is
e&ual to $'and we write
( )limx a
f x L+
=
O*:4S"+:+ L","(S
( )2
limx
g x
( )2
limx
g x+
( )2
limx
g x
( )#
limx
g x
( )#
limx
g x+
( )
#limx
g x
7/26/2019 Chapter 1_Function and Limit
51/87
%'9
Calculating Limits
Using the Limit Laws
L","(S
+n this section, we will:
Cse the *imit *aws to calculate limits.
7/26/2019 Chapter 1_Function and Limit
52/87
[ ]".lim ( ) ( ) lim ( ) lim ( )x a x a x a
f x g x f x g x
=
[ ]1.lim ( ) ( ) lim ( ) lim ( )x a x a x a
f x g x f x g x
=
(1: L","( LA@S
[ ].lim ( ) lim ( )x a x a
cf x c f x
=
lim ( )( )#.lim if lim ( ) 0
( ) lim ( )
= x a
x a x a
x a
f xf xg x
g x g x
uppose that cis a constant and the limits
and exist. Then
lim ( )x a
f x
lim ( )x ag x
7/26/2019 Chapter 1_Function and Limit
53/87
where nis a positive integer.
10.lim n nx a
x a
=
US"*/ (1: L","( LA@S
%.lim n nx a
x a
=
&.lim
xac = c
'.limx a
x a
=
[ ]$.lim ( ) lim ( )
nn
x a x af x f x =
11.lim ( ) lim ( )n nx a x a
f x f x =
7/26/2019 Chapter 1_Function and Limit
54/87
"e state this fact as follows. +f fis a polynomialor a
rational function and ais in the domain of f, then
lim ( ) ( )x a
f x f a
=
+"):C( SUBS("(U("O* 0)O0:)(=
7/26/2019 Chapter 1_Function and Limit
55/87
if and onl$ if
US"*/ (1: L","( LA@S
lim ( )x a f x L = lim ( ) lim ( )x a x af x L f x + = =
(heorem %
7/26/2019 Chapter 1_Function and Limit
56/87
/rove that does not exist.
US"*/ (1: L","( LA@S :;ample E
0limx
x
x
7/26/2019 Chapter 1_Function and Limit
57/87
+f when x is near a (except possibl$ at a)
and the limits of fand g both exist asxapproaches a,
then
( ) ( )f x g x
lim ( ) lim ( )x a x a
f x g x
0)O0:)(":S OF L","(S (heorem .
7/26/2019 Chapter 1_Function and Limit
58/87
The 6uee8e Theorem (the andwich Theorem or the /inching Theorem)
states that,if
whenxis near a (except possibl$ at a)
and .Then
( ) ( ) ( )f x g x h x
lim ( ) lim ( )x a x a
f x h x L
= = lim ( )x a
g x L
=
S>U::?: (1:O):, (heorem -
7/26/2019 Chapter 1_Function and Limit
59/87
how that
&ote that we cannot use
This is because does not exist.
2
0
1lim sin 0.x
xx
=
US"*/ (1: L","( LA@S :;ample %%
2 2
0 0 0
1 1lim sin lim limsinx x x
x xx x
=
0
limsin(1! )x
x
7/26/2019 Chapter 1_Function and Limit
60/87
'owever, since ,
we have:
11 sin 1
x
2 2 21sinx x x
x
US"*/ (1: L","( LA@S :;ample %%
TaBing f(x) -x1, and h(x) x1
in the 6uee8e Theorem,
we obtain:
2
0
1lim sin 0x
xx
=
7/26/2019 Chapter 1_Function and Limit
61/87
%'9 >U"? >U:S("O*S
) +f
then
a. True b. False
lim ( ) 0, lim ( ) 0x x
f x g x
= =
( )lim oes not exist
( )x
f x
g x
7/26/2019 Chapter 1_Function and Limit
62/87
%'9 >U"? >U:S("O*S
+f
a. True b. False
lim ( ) ( ) exists, then the limit must be () ()x f x g x f g
L","(S
7/26/2019 Chapter 1_Function and Limit
63/87
%'3
Continuity
L","(S
+n this section, we will:
ee that the mathematical definition of continuit$
corresponds closel$ with the meaning of the word
continuit$ in ever$da$ language.
CO*("*U"(=
7/26/2019 Chapter 1_Function and Limit
64/87
A function fis continuous at a number a if:
lim ( ) ( )x a
f x f a
=
CO*("*U"(= %' +efinition
&otice that :
f(a) is defined - that is,
a is in the domain of f exists.
lim ( )x a
f x
lim ( ) ( )x a
f x f a
=
CO*("*U"(=
7/26/2019 Chapter 1_Function and Limit
65/87
+f fis defined near a - that is, fis defined on an open intervalcontaining a, except perhaps at a - wesa$ that fis
discontinuousat a if fis not continuous at a.
CO*("*U"(=
The figure shows the graph of a
function f.
At which numbers is fdiscontinuous#
"h$#
+efinition
CO*("*U"(=
7/26/2019 Chapter 1_Function and Limit
66/87
A function fis continuous from the right
at a number a if
and fis continuous from the left at a if
lim ( ) ( )x a
f x f a+
=
lim ( ) ( )x a
f x f a
=
CO*("*U"(= .' +efinition
CO*("*U"(=
7/26/2019 Chapter 1_Function and Limit
67/87
A function fis continuous on an interval if it is
continuous at ever$ number in the interval.
+f fis defined onl$ on one side of an endpoint of the interval,
we understand Dcontinuous at the endpointE to mean
Dcontinuous from the rightE or Dcontinuous from the left.E
CO*("*U"(= -' +efinition
CO*("*U"(=
7/26/2019 Chapter 1_Function and Limit
68/87
+f fand gare continuous at aand cis a constant, then the
following functions are also continuous at a:
.f +g
2.f -g
. cf
".fg
#. ( ) 0f
i f g ag
CO*("*U"(= 9' (heorem
CO*("*U"(=
7/26/2019 Chapter 1_Function and Limit
69/87
The following t$pes of functions are continuous atever$ number in their domains:
/ol$nomials
ational functions
oot functions
Trigonometric functions
CO*("*U"(= 6' (heorem
CO*("*U"(=
7/26/2019 Chapter 1_Function and Limit
70/87
+f fis continuous at band then
+n other words,
+fxis close to a, then g(x) is close to bG and, since fis continuous at b, if g(x) is close to b, then f(g(x))
is close to f(b).
lim ( )x a
g x b
=
lim ( ( )) ( )x a
f g x f b
=
( )lim ( ( )) lim ( )x a x a
f g x f g x
=
CO*("*U"(= E' (heorem
CO*("*U"(= (h
7/26/2019 Chapter 1_Function and Limit
71/87
+f gis continuous at aand fis continuous at g(a),then the composite function
is continuous at a.
This theorem is often expressed informall$ b$ sa$ing
Ha continuous function of a continuous function is
a continuous function.I
( ) ( ) ( ( ))f g x f g x=o
CO*("*U"(= ' (heorem
"*(:),:+"A(: $ALU: (1:O):, %5 (h
7/26/2019 Chapter 1_Function and Limit
72/87
uppose that fis continuous on the closed interval
7a! b8and let be an$ number between f(a) and f(b),
where
Then, there exists a number cin (a, b)such that f(c) .
( ) ( )f a f b
"*(:),:+"A(: $ALU: (1:O):, %5' (heorem
"*(:),:+"A(: $ALU: (1:O):, : l
7/26/2019 Chapter 1_Function and Limit
73/87
how that there is a root of the e6uation
between and 1.
*et .
"e are looBing for a solution of the given e6uation7
that is, a number cbetween and 1 such that f(c) ;.
Therefore, we taBe a , b 1, and ; in the theorem.
"e have
and
2" $ 2 0x x x + =
2( ) " $ 2f x x x x= +
"*(:),:+"A(: $ALU: (1:O):, :;ample
(1) " $ 2 1 0f = + = U:S("O*S
7/26/2019 Chapter 1_Function and Limit
74/87
%'3 >U"? >U:S("O*S
) +f f()2; and f()0; then there exists a number c between and such that f(c);
a. True b. False
1) "hich is the e6uation expressing the fact that H f is continuous at 1I#
a. b.
c. d.
e.
2lim ( ) 0x
f x
=2
lim ( )x
f x
=
2lim ( ) 2x
f x
= lim ( ) (2)x
f x f
=
2lim ( ) (2)x
f x f
=
7/26/2019 Chapter 1_Function and Limit
75/87
%'G
Limits involving infinity
@e will study&
"nfinity Limits
Limits at "nfinity
"*F"*"(: L","(S + fi iti 9
7/26/2019 Chapter 1_Function and Limit
76/87
*et fbe a function defined on both sides of a, except
possibl$ at aitself. Then,
means that the values of f(x)
can be made arbitraril$ large
b$ taBingxsufficientl$
close to a, butnot e6ual to a.
( )limx a
f x
=
"*F"*"(: L","(S +efinition 9
"*F"*"(: L","(S + fi iti 3
7/26/2019 Chapter 1_Function and Limit
77/87
*et fbe defined on both sides of a, except possibl$ at a
itself. Then,
means that the values of f(x)
can be made arbitraril$
large negative b$ taBingx
sufficientl$ close to a,
but not e6ual to a.
( )limx a
f x
=
"*F"*"(: L","(S +efinition 3
"*F"*"(: L","(S
7/26/2019 Chapter 1_Function and Limit
78/87
imilar definitions can be given for the one-sided limits:
( )limx a
f x
= ( )limx a
f x+
=
( )limx a f x = ( )limx a f x+=
"*F"*"(: L","(S
+:F"*"("O*S
7/26/2019 Chapter 1_Function and Limit
79/87
= xais called the vertical as$mptoteof f(x)
if we have one of the following:
+:F"*"("O*S
( )limx a
f x
= ( )limx a
f x+
=
( )limx a
f x
= ( )limx a
f x+
=
L","(S A( "*F"*"(=
7/26/2019 Chapter 1_Function and Limit
80/87
L","(S A( "*F"*"(=
lim ( )x
f x L
=
if then ( )x M f x L > >
*et f be a function defined for ever$ x2a. Then
means that
+:F"*"("O*
7/26/2019 Chapter 1_Function and Limit
81/87
The line $*is called the horiHontal asymptote
of f(x) if we have oneof the following:
+:F"*"("O*
( ) ( )lim limx x
f x L f x L
= =
7/26/2019 Chapter 1_Function and Limit
82/87
2
1( )
2
xf x
x x
=
+
2
111
lim lim 11 22
1x x
x x
x x
x x
= =+
+
Find the as$mptotes of the function
( )( )2 2 2
21
1 11
2 ( 1)( 2 2)
1 lim
2 #x
x x xx
x x x x x
x
x x
+ +=
+ + +
+=
+
olution
$ is hori8ontal as$mptote
7/26/2019 Chapter 1_Function and Limit
83/87
ompute
a.
b.
c.
d.
2
1limsin
lim( 1 )
limsin
lim( )
x
x
x
x
x
x x
x
x x
+
;
;
oes not exist
%'G >U"? >U:S("O*S
7/26/2019 Chapter 1_Function and Limit
84/87
%'G >U"? >U:S("O*S
) Find
a. ; b. infinit$ c. d. oes not exist
limcosx
x
1) Find
a. ; b. infinit$ c. d. ose not exist
1lim cosx
xx
%'G >U"? >U:S("O*S
7/26/2019 Chapter 1_Function and Limit
85/87
%'G >U"? >U:S("O*S
) +f
Then
a. True b. False
0 0lim ( ) , lim ( )x x
f x g x = =
0lim ( ) ( )* 0x
f x g x
=
@) A function can have twodifferent hori8ontal as$mptotes
a. True b. False
:;ercises
7/26/2019 Chapter 1_Function and Limit
86/87
:;ercises
ection +. 1;,!,@,@J (p J,;)
ection +.1 : @J,!1,!@,K (p 1,1@)
+.@: L, 1J (p@@,@!)
+.K: 11,1@,1K (pKM)
@M page M1
7/26/2019 Chapter 1_Function and Limit
87/87
(hanIs