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  • 7/26/2019 Chapter 1_Function and Limit

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    CALCULUSCALCULUS

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    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

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    %'%

    Functions and

    (heir )epresentations

    FU*C("O*S A*+ L","(S

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    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' %.

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    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

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    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

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    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*

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    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

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    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.

    :

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    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

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    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(

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    The reason for the truth of the %ertical *ine Test can be

    seen in the figure.

    (1: $:)("CAL L"*: (:S(

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    +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.html
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    +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.html
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    *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

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    :;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.

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    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

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    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

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    %'% >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

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    %'.,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

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    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

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    "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

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    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,

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    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

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    ()"/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 + = + =

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    ()"/O*O,:()"CFU*C("O*S

    sin

    tan cos

    x

    x x=

    , ,2 2x

    ( , )R =

    tan( ) tan ;x k x k Z+ =

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    ()"/O*O,:()"CFU*C("O*S

    The reciprocals of the sine, cosine, and tangentfunctions are

    1cos

    sin1

    seccos

    1cottan

    ecx

    x

    xx

    anxx

    =

    =

    =

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    A function of the formf(x) =x a , where a is constant,

    is called a power function.

    0O@:) FU*C("O*S

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    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, )

    :

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    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.

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    *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

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    =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

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    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("*/

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    *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

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    *:@ 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=

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    =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#

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    +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

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    +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

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    *:@ 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=

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    *:@ 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=

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    = The figure illustrates these stretching

    = transformations when applied to the

    cosine

    = function with c 1.

    ()A*SFO),A("O*S

    :;ample

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    :;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

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    = 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

    = =

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    *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

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    %'. >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 " $

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    Answer: f(x), f( x), f(x)

    %'. >U"? >U:S("O*S

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    %'-

    (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.

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    +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*

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    "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 .

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    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

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    %'9

    Calculating Limits

    Using the Limit Laws

    L","(S

    +n this section, we will:

    Cse the *imit *aws to calculate limits.

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    [ ]".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

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    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 =

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    "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:)(=

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    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 %

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    /rove that does not exist.

    US"*/ (1: L","( LA@S :;ample E

    0limx

    x

    x

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    +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 .

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    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 -

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    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

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    '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

    =

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    %'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

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    %'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

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    %'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"(=

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    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"(=

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    +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"(=

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    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"(=

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    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"(=

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    +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"(=

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    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"(=

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    +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

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    +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

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    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

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    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

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    %'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

    =

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    %'G

    Limits involving infinity

    @e will study&

    "nfinity Limits

    Limits at "nfinity

    "*F"*"(: L","(S + fi iti 9

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    *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

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    *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

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    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

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    = 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"*"(=

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    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*

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    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

    = =

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    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

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    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

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    %'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

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    %'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

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    :;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

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    (hanIs