ESSENTIAL CALCULUS CH01 Functions

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ESSENTIAL CALCULUSCH01 Functions Limits
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In this Chapter

• 1.1 Functions and Their Representations
• 1.2 A Catalog of Essential Functions
• 1.3 The Limit of a Function
• 1.4 Calculating Limits
• 1.5 Continuity
• 1.6 Limits Involving Infinity
• Review

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Some Terminologiesdomainset Arangeindependen
t varibleA symbol representing
any number in the
domaindependent varible A symbol
representing any number in
the range
Chapter 1, 1.1, P2
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A function f is a rule that assigns to each
element x in a set A exactly one element, called
f(x) , in a set B.
Chapter 1, 1.1, P2
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Chapter 1, 1.1, P2
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Chapter 1, 1.1, P2
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If f is a function with domain A, then its graph
is the set of ordered pairs
(Notice that these are input-output pairs.) In
other words, the graph of f consists of all
Points(x,y) in the coordinate plane such that
yf(x) and x is in the domain of f.
Chapter 1, 1.1, P2
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Chapter 1, 1.1, P2
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Chapter 1, 1.1, P2
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Chapter 1, 1.1, P2
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• EXAMPLE 1 The graph of a function f is shown in
  Figure 6.
• Find the values of f(1) and f(5) .
• (b) What are the domain and range of f ?

Chapter 1, 1.1, P2
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EXAMPLE 3 Find the domain of each function.
Chapter 1, 1.1, P4
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THE VERTICAL LINE TEST A curve in the xy-plane is
the graph of a function of x if and only if no
vertical line intersects the curve more than once.
Chapter 1, 1.1, P4
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Chapter 1, 1.1, P5
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EXAMPLE 4 A function f is defined by
1-X if X1 X2 if Xgt1
f(x)
Evaluate f(0) ,f(1) , and f(2) and sketch the
graph.
Chapter 1, 1.1, P5
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Chapter 1, 1.1, P5
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EXAMPLE 5 Sketch the graph of the absolute value
function f(x)X.
Chapter 1, 1.1, P6
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EXAMPLE 6 In Example C at the beginning of this
section we considered the cost C(w) of mailing a
first-class letter with weight w. In effect, this
is a piecewise defined function because, from the
table of values, we have
0.39 if oltw1 0.63 if 1ltw2 0.87 if
2ltw3 1.11 if 3ltw4
C(w)
Chapter 1, 1.1, P6
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Chapter 1, 1.1, P6
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If a function f satisfies f(-x)f(x) for every
number x in its domain, then f is called an even
function.
Chapter 1, 1.1, P6
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Chapter 1, 1.1, P6
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Chapter 1, 1.1, P6
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If f satisfies f(-x)-f(x) for every number x in
its domain, then f is called an odd function.
Chapter 1, 1.1, 07
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EXAMPLE 7 Determine whether each of the following
functions is even, odd, or neither even nor odd.

1. f(x)x5x
2. g(x)1-x4
3. h(x)2xx2

Chapter 1, 1.1, 07
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Chapter 1, 1.1, 07
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A function f is called increasing on an interval
if f (x1)lt f (x2) whenever x1lt x2 in I It is
called decreasing on I if f (x1)gt f (x2)
whenever x1 lt x2 in I
Chapter 1, 1.1, 07
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1. The graph of a function f is given. (a) State
the value of f(-1). (b) Estimate the value of
f(2). (c) For what values of x is f(x)2? (d)
Estimate the values of x such that f(x)0 . (e)
State the domain and range of f . (f ) On what
interval is f increasing?
Chapter 1, 1.1, 08
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Chapter 1, 1.1, 08
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2. The graphs of f and g are given. (a) State the
values of f(-4)and g(3). (b) For what values of x
is f(x)g(x)? (c) Estimate the solution of the
equation f(x)-1. (d) On what interval is f
decreasing? (e) State the domain and range of
f. (f ) State the domain and range of g.
Chapter 1, 1.1, 08
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Chapter 1, 1.1, 08
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36 Determine whether the curve is the graph of
a function of x. If it is, state the domain and
range of the function.
Chapter 1, 1.1, 08
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5354 Graphs of f and g are shown. Decide
whether each function is even, odd, or neither.
Explain your reasoning.
Chapter 1, 1.1, 10
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A function P is called a polynomial if
P(x)anxnan-1xn-1???a2x2a1xa0
where n is a nonnegative integer and the numbers
a0,a1,a2,..an are constants called the
coefficients of the polynomial. The domain of any
polynomial is R(-8,8) If the leading coefficient
an?0, then the degree of the polynomial is n.
Chapter 1, 1.2, 13
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Chapter 1, 1.2, 14
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Chapter 1, 1.2, 14
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Chapter 1, 1.2, 14
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Chapter 1, 1.2, 14
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Chapter 1, 1.2, 14
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Chapter 1, 1.2, 14
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Chapter 1, 1.2, 14
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Chapter 1, 1.2, 14
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Chapter 1, 1.2, 14
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Chapter 1, 1.2, 15
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A rational function f is a ratio of two
polynomials
Where P and Q are polynomials. The domain
consists of all values of x such that Q(x)?0.
Chapter 1, 1.2, 15
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Chapter 1, 1.2, 15
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Chapter 1, 1.2, 15
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-1 son x1 -1 cos x1
Chapter 1, 1.2, 15
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Chapter 1, 1.2, 16
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Chapter 1, 1.2, 16
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Chapter 1, 1.2, 16
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Sin(x2p)sin x cos(x2p)cos x
Chapter 1, 1.2, 16
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The exponential functions are the functions of
the form f(x)ax , where the base is a positive
constant.
Chapter 1, 1.2, 16
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The logarithmic functions f(x)logax , where the
base a is a positive constant, are the inverse
functions of the exponential functions.
Chapter 1, 1.2, 16
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Figure 15 illustrates these shifts by showing
how the graph of y(x3)21 is obtained from the
graph of the parabola yx2 Shift 3 units to the
left and 1 unit upward.
Y(x3)21
Chapter 1, 1.2, 17
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VERTICAL AND HORIZONTAL SHIFTS Suppose cgt0. To
obtain the graph of Y f(x)c, shift the graph
of yf(x) a distance c units c units upward Y
f(x)- c, shift the graph of yf(x) a distance c
units c units downward Y f(x- c), shift the
graph of yf(x) a distance c units c units to the
right Yf(x c), shift the graph of yf(x) a
distance c units c units to the left
Chapter 1, 1.2, 17
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VERTICAL AND HORIZONTAL STRETCHING AND
REFLECTING Suppose cgt1. To obtain the graph
of ycf(x), stretch the graph of yf(x)
vertically by a factor of c y(1/c)f(x),
compress the graph of yf(x) vertically by a
factor of c Yf(cx), compress the graph of
yf(x) horizontally by a factor of c Yf(x/c),
stretch the graph of yf(x) horizontally by a
factor of c Y-f(x), reflect the graph of yf(x)
about the x-axis Yf(-x), reflect the graph of
yf(x) about they-axis
Chapter 1, 1.2, 17
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Chapter 1, 1.2, 17
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Chapter 1, 1.2, 17
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EXAMPLE 2 Given the graph of y , use
transformations to graph y -2 , y ,
y- , y2 , and y
Chapter 1, 1.2, 18
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Chapter 1, 1.2, 18
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Chapter 1, 1.2, 18
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Chapter 1, 1.2, 18
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Chapter 1, 1.2, 18
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Chapter 1, 1.2, 18
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Chapter 1, 1.2, 18
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EXAMPLE 3 Sketch the graph of the function
y1-sin x.
Chapter 1, 1.2, 18
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Chapter 1, 1.2, 18
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Chapter 1, 1.2, 18
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(fg)(x)f(x)g(x) (f-g)(x)f(x)-g(x)
If the domain of f is A and the domain of g is B,
then the domain of f g is the intersection A n B
Chapter 1, 1.2, 18
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(fg)(x)f(x)g(x)
The domain of fg is A nB, but we cant divide by
0 and so the domain of f/g is
Chapter 1, 1.2, 18
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DEFINITION Given two functions f and g , the
composite function f?g (also called the
composition of f and g ) is defined by
(f?g)(x)f(g(x))
Chapter 1, 1.2, 19
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Chapter 1, 1.2, 19
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EXAMPLE 5 If f(x) and g(x) , find
each function and its domain.
(a) f?g (b) g?f (c) f?f (d)g?g
Chapter 1, 1.2, 20
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EXAMPLE 6 Given F(x)cos2(x9) , find functions f
,g ,and h such that Ff?g?H.
Chapter 1, 1.2, 20
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• 17. The graph of yf(x) is given. Match each
  equation with its graph and give reasons for your
  choices.
• yf(x-4)
• yf(x)3
• y f(x)
• y-f(x4)
• y2f(x6)

Chapter 1, 1.2, 22
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18. The graph of f is given. Draw the graphs of
the following functions.
(a)yf(x4) (b) yf(x)4 (c) y2f(x)
(d) y- f(x)3
Chapter 1, 1.2, 22
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19 The graph of f is given. Use it to graph
the following functions. (a) yf(2x) (b)
yf( x) (c) yf(-x) (d)y-f(-x)
Chapter 1, 1.2, 22
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• 53 Use the given graphs of f and g to evaluate
  each expression, or explain why it is undefined.
• f(g(2)) (b) g(f(0)) (c) (f?g)(0)
• (g?F)(6) (e) (g?g)(-2) (f) (f?f)(4)

Chapter 1, 1.2, 22
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Chapter 1, 1.3, 25
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Chapter 1, 1.3, 25
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1 DEFINITION We write
limf(x)L
X?a
and say the limit of f(X), as x approaches
, equals L if we can make the values of f(x)
arbitrarily close to L (as close to L as we like)
by taking x to be sufficiently close to a (on
either side of ) but not equal to a.
Chapter 1, 1.3, 25
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limf(x)L
X?a
is f(x)?L as x?a which is
usually read f(x) approaches L as x approaches
a.
Chapter 1, 1.3, 25
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Chapter 1, 1.3, 26
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Chapter 1, 1.3, 26
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Chapter 1, 1.3, 26
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Chapter 1, 1.3, 26
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Chapter 1, 1.3, 26
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Chapter 1, 1.3, 28
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Chapter 1, 1.3, 28
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2. DEFINITION We write
limf(x)L
X?a-
and say the left-hand limit of f(x) as X
approaches a or the limit of f(x) as
X approaches a from the left is equal to L if we
can make the values of f(X) arbitrarily close to
L by taking x to L be sufficiently close to a and
x less than a.
Chapter 1, 1.3, 29
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Chapter 1, 1.3, 30
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Chapter 1, 1.3, 30
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3 limf(x)L if and only if limf(x)L and
limf(x)L
X?a
X?a- X?a
Chapter 1, 1.3, 30
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EXAMPLE 7 The graph of a function g is shown is
Figure 10. Use it to state the values(if they
exist) of the following

• lim g(x) (b) lim g(x) (c)lim g(x)
• (d) lim g(x) (e) lim g(x) (f)lim g(x)

X?2- X?2
X?2
X?5- X?5
x?5
Chapter 1, 1.3, 30
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Chapter 1, 1.3, 30
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Chapter 1, 1.3, 31
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FINITION Let f be a function defined on some open
interval that contains the number a , except
possibly at a itself. Then we say that the limit
of as approaches is , and we write
lim g(x)L
X?a
if for every number egt0 there is a corresponding
number dgt0 such that if 0ltx-altd then
f(x)-Llte
Chapter 1, 1.3, 31
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Chapter 1, 1.3, 32
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Chapter 1, 1.31, 32
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Chapter 1, 1.3, 32
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Chapter 1, 1.3, 32
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Chapter 1, 1.3, 32
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Chapter 1, 1.3, 33
105

• 3. Use the given graph of f to state the value of
  each quantity, if it exists. If it does not
  exist, explain why.
• Lim f(X) (b) lim f(X) (C)lim f(X)
• (d) Lim f(X) (e)F(5)

X?1- X?1
X?1
X?5
Chapter 1, 1.3, 33
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4. For the function f whose graph is given, state
the value of each quantity, if it exists. If it
does not exist, explain why.
(a_Lim f(X) (b) lim f(X) (C)lim
f(X) (d) Lim f(X) (e)F(5)
X?0 X?3-
X?3
X?3
Chapter 1, 1.3, 33
107

• 5. For the function g whose graph is given, state
  the value of each quantity, if it exists. If it
  does not exist, explain why.
• lim g(t) (b) lim g(t) (c) lim g(t)
• (d)lim g(t) (e) lim g(t) (f) lim g(t)
• (g)g(2) (h)lim g(t)

X?0- X?0
X?0
X?2- X?2
X?2
X?4
Chapter 1, 1.3, 33
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• LIMIT LAWS Suppose that c is a constant and the
  limits
• lim f(X) and lim g(x)
• Exist Then
• lim?f(x)g(x)?lim f(x)lim g(x)
• lim?f(x)-g(x)?limf(x)-lim g(x)
• lim ?cf(x)?c lim f(x)
• lim ?f(x)g(x)?lim f(x)?lim g(x)
• lim if lim g(x)?0

X?a X?a
X?a
X?a X?a
X?a X?a
X?a
X?a X?a

X?a
X?a X?a
X?a
X?a
Chapter 1, 1.4, 35
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Sum Law Difference Law Constant Multiple
Law Product Law Quotient Law
Chapter 1, 1.4, 36
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1. The limit of a sum is the sum of the
limits. 2. The limit of a difference is the
difference of the limits. 3. The limit of
a constant times a function is the constant
times the limit of the function. 4. The limit of
a product is the product of the limits. 5.
The limit of a quotient is the quotient of the
limits (provided that the limit of the
denominator is not 0).
Chapter 1, 1.4, 36
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6. limf(x)nlimf(x)n where n is a positive
integer
X?a X?a
Chapter 1, 1.4, 36
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7. lim cc 8. lim xa
X?a
X?a
Chapter 1, 1.4, 36
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9. lim xnan where n is a positive integer
X?a
Chapter 1, 1.4, 36
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10. lim where n is a positive
integer (If n is even, we assume that agt0.)
X?a
Chapter 1, 1.4, 36
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11.Lim where n is a
positive integer If n is even, we assume that
lim f(X)gt0.
X?a
X?a
X?a
Chapter 1, 1.4, 36
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DIRECT SUBSTITUTION PROPERTY If f is a polynomial
or a rational function and is in the domain of f,
then
lim f(X)gtf(a)
X?a
Chapter 1, 1.4, 37
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If f(x)g(x) when x ? a, then lim f(x)lim g(x),
provided the limits exist.
X?a
X?a
Chapter 1, 1.4, 38
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FIGURE 2 The graphs of the functions f (from
Example 2) and g (from Example 3)
Chapter 1, 1.4, 39
119
2 THEOREM lim f(x)L if and only if lim
f(x)Llim f(x)
X?a
X?a-
X?a
Chapter 1, 1.4, 39
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Chapter 1, 1.4, 40
121
Chapter 1, 1.4, 40
122
Chapter 1, 1.4, 40
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3. THEOREM If f(x)g(x) when x is near a (except
possibly at a) and the limits of f and g both
exist as x approaches a, then
lim f(x) lim g(x)
X?a
X?a
Chapter 1, 1.4, 41
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4. THE SQUEEZE THEOREM If f(x) g(x) h(x) when x
is near a (except possibly at a) and
limf(x)lim h(X) L Then lim
g(X)L
X?a
X?a
X?a
Chapter 1, 1.4, 41
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Chapter 1, 1.4, 41
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Chapter 1, 1.4, 41
127
2. The graphs of f and g are given. Use them to
evaluate each limit, if it exists. If the limit
does not exist, explain why.
(a)limf(x)g(x) (b) lim f(x)g(x) (c)lim
f(x)g(x) (d) lim (e)Limx3f(x) (f)
lim
X?2
X?1
X?0
X? -1
X?2
X?1
Chapter 1, 1.4, 43
128
As illustrated in Figure 1, if f is
continuous, then the points (x, f(x)) on the
graph of f approach the point (a, f(a)) on the
graph. So there is no gap in the curev.
Chapter 1, 1.5, 46
129
Chapter 1, 1.5, 46
130

• DEFINITION A function f is continuous at a
• number a if
• lim f(X)f(a)

X?a
Chapter 1, 1.5, 46
131

• Notice that Definition I implicitly requires
  three things if f is continuous at a
• f(a)is defined (that is, a is in the domain of f
  )
• lim f(x) exists
• lim f(x) f(a)

X?a
X?a
Chapter 1, 1.5, 46
132
If f is defined near a(in other words, f is
defined on an open interval containing a, except
perhaps at a), we say that f is discontinuous at
a (or f has a discontinuity at a) if f is not
continuous at a.
Chapter 1, 1.5, 46
133
2. DEFINITION A function f is continuous from the
right t a number a if lim
f(x)f(a) And f is continuous from the left at a
if lim f(x)f(a)
X?a
X?a-
Chapter 1, 1.5, 47
134
3. DEFINITION A function f is continuous on an
interval if it is continuous at every number in
the interval. (If f is defined only on one side
of an endpoint of the interval, we understand
continuous at the endpoint to mean
continuous from the right or continuous from the
left.)
Chapter 1, 1.5, 48
135

• 4. THEOREM If f and g are continuous at a and c
  is a constant, then the following functions are
  also continuous at a
• fg 2 f-g 3 cf
• 4. fg 5. if g(a)?0

Chapter 1, 1.5, 48
136

• 5. THEOREM
• Any polynomial is continuous everywhere that is,
  it is continuous on R(-8,8).
• (b) Any rational function is continuous wherever
  it is defined that is, it is
• continuous on its domain.

Chapter 1, 1.5, 49
137
6. THEOREM The following types of functions are
continuous at every number in their domains
polynomials, rational functions, root functions,
trigonometric functions
Chapter 1, 1.5, 50
138
7. THEOREM If f is continuous at b and
lim g(x)b, then lim f(g(X))f(b). in the
words lim f(g(X))f(lim g(X))
X?a
X?a
X?a
X?a
Chapter 1, 1.5, 51
139
8. THEOREM If g is continuous at a and f is
continuous at g(a), then the composite function
f?g given by(f?g)(x)f(g(x)) is continuous at a.
Chapter 1, 1.5, 51
140
9. INTERMEDIATE VALUE THEOREM Suppose that f is
continuous on the closed interval a,b and let N
be any number between f(a) and f(b) , where
f(a)?f(b). Then there exists a number c in(a,b)
such that f(c)N.
Chapter 1, 1.5, 52
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Chapter 1, 1.5, 52
142
Chapter 1, 1.5, 52
143
Chapter 1, 1.5, 53
144
3 (a) From the graph of f, state the numbers at
which f is discontinuous and explain
why. (b) For each of the numbers stated in
part (a), determine whether f is
continuous from the right, or from the
left, or neither.
Chapter 1, 1.5, 54
145
4. From the graph of g , state the intervals on
which g is continuous.
Chapter 1, 1.5, 54
146
1 DEFINITION The notation
lim f(x)8 means that the values
of f(x) can be made arbitrarily large (as large
as we please) by taking x sufficiently close to a
(on either side of a) but not equal to a.
X?a
Chapter 1, 1.6, 56
147
Chapter 1, 1.6, 57
148
Chapter 1, 1.6, 57
149
Chapter 1, 1.6, 57
150
Chapter 1, 1.6, 57
151
Chapter 1, 1.6, 57
152
Chapter 1, 1.6, 57
153
2. DEFINITION The line x a is called a vertical
asymptote of the curve yf(x) if at least one of
the following statements is true
lim f(x)8
lim f(x)8
lim f(x)8
X?a
X?a
X?a-
lim f(x)-8
lim f(x)-8
lim f(x)-8
X?a
X?a-
X?a
Chapter 1, 1.6, 57
154
Chapter 1, 1.6, 58
155
3. DEFINITION Let f be a function defined on some
interval(a, 8) . Then lim
f(x)L means that the values of f(x) can be made
as close to L as we like by taking x sufficiently
large.
X?a
Chapter 1, 1.6, 59
156
Chapter 1, 1.6, 59
157
Chapter 1, 1.6, 59
158
Chapter 1, 1.6, 59
159
Chapter 1, 1.6, 60
160
Chapter 1, 1.6, 60
161
Chapter 1, 1.6, 60
162
4. DEFINITION The line yL is called a horizontal
asymptote of the curve yf(x) if either
lim f(x)L or lim f(x)L
X?8
X?8
Chapter 1, 1.6, 60
163
EXAMPLE 3 Find the infinite limits, limits at
infinity, and asymptotes for the function f whose
graph is shown in Figure 11.
Chapter 1, 1.6, 60
164
5. If n is a positive integer, then
lim 0 lin 0
X?8
X?8
Chapter 1, 1.6, 61
165
6. DEFINITION Let f be a function defined on some
open interval that contains the number a, except
possibly at a itself. Then lim
f(x)8 means that for every positive number M
there is a positive number dsuch that
if 0ltx-altd then f(x)gtM
X? a
Chapter 1, 1.6, 64
166
7. DEFINITION Let f be a function defined on
some interval(a, 8) . Then
lim f(x)L
X?8
means that for every egt0 there is a
corresponding number N such that
if xgtN then f(x)-Llte
Chapter 1, 1.6, 65
167
Chapter 1, 1.6, 65
168
Chapter 1, 1.6, 65
169
8. DEFINITION Let f be a function defined on some
interval(a, 8) . Then lim
f(x)8 means that for every positive number M
there is a corresponding positive number N such
that if xgtN then f(x)gtM
X?8
Chapter 1, 1.6, 66
170

• 1.For the function f whose graph is given, state
  the following.
• lim f(x) (b) lim f(x)
• (c) lim f(x) (d) lim f(x)
• (e) lim f(x)
• (f) The equations of the asymptotes

X? 2
X? -1-
X?8
X? -1
X? -8
Chapter 1, 1.6, 66
171
Chapter 1, 1.6, 66
172

• 2. For the function g whose graph is given, state
  the following.
• lim g(x) (b) lim g(x)
• (c) lim g(x) (d) lim g(x)
• (e) lim g(x) (f) The equations of the
  asymptotes

X? -8
X?8
X?3
X? 0
X? -2
Chapter 1, 1.6, 67
173
1. Let f be the function whose graph is
given. (a) Estimate the value of f(2). (b)
Estimate the values of x such that f(x)3. (c)
State the domain of f. (d) State the range of
f. (e) On what interval is increasing? (f ) Is f
even, odd, or neither even nor odd? Explain.
Chapter 1, Review, 70
174
Chapter 1, Review, 70
175
2. Determine whether each curve is the graph of a
function of x. If it is, state the domain and
range of the function.
Chapter 1, Review, 71
176
8. The graph of f is given. Draw the graphs of
the following functions. (a)yf(x-8)
(b)y-f(x) (c)y2-f(x) (d)y
f(x)-1
Chapter 1, Review, 71
177

• 21. The graph of f is given.
• Fine each limit, or explain why it doex not
  exist.
• (i) lim f(x) (ii) lim f(x)
• (iii) lim f(x) (iv) lim f(x)
• (v) lim f(x) (vi) lim f(x)
• (vii) lim f(x) (viii) lim f(x)
• (b)State the equations of the horizontal
  asymptotes.
• (c)State the equations of the vertical
  asymptotes.
• (d)At what number is f discontinuous? Explain.

X? 2
X? -3
X? 4
X? -3
X?0
X?2-
X?8
X? -8
Chapter 1, Review, 71
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Chapter 1, Review, 71