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ESSENTIAL CALCULUS CH01 Functions & Limits. 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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Essential calculus ch01 functions limits

ESSENTIAL CALCULUSCH01 Functions & Limits


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


Some Terminologies:domain:set Arange:independent varible:A symbol representing any number in the domaindependent varible: A symbol representing any number in the range

Chapter 1, 1.1, P2


A function fis 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




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 y=f(x) and x is in the domain of f.

Chapter 1, 1.1, P2





  • 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


EXAMPLE 3 Find the domain of each function.

Chapter 1, 1.1, P4


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



EXAMPLE 4 A function f is defined by

1-X if X≤1

X2 if X>1

f(x)=

Evaluate f(0) ,f(1) , and f(2) and sketch the graph.

Chapter 1, 1.1, P5



EXAMPLE 5 Sketch the graph of the

absolute value function f(x)=│X│.

Chapter 1, 1.1, P6


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 o<w≤1

0.63 if 1<w≤2

0.87 if 2<w≤3

1.11 if 3<w≤4

C(w)=

Chapter 1, 1.1, P6



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


Chapter 1, 1.1, P6 its domain, then f is called an


Chapter 1, 1.1, P6 its domain, then f is called an


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


EXAMPLE 7 domain, then f is called an Determine whether each of the following functions is even, odd, or neither even nor odd.

  • f(x)=x5+x

  • g(x)=1-x4

  • h(x)=2x=x2

Chapter 1, 1.1, 07


Chapter 1, 1.1, 07 domain, then f is called an


A function f is called domain, then f is called an increasing on an interval if

f (x1)< f (x2) whenever x1< x2 in I

It is called decreasing on I if

f (x1)> f (x2) whenever x1 < x2 in I

Chapter 1, 1.1, 07


1. domain, then f is called an 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


Chapter 1, 1.1, 08 domain, then f is called an


2. The graphs of f and g are given. domain, then f is called an

(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


Chapter 1, 1.1, 08 domain, then f is called an


3 domain, then f is called an –6 ■ 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


53 domain, then f is called an –54 ■ Graphs of f and g are shown. Decide whether each function is even, odd, or neither. Explain your reasoning.

Chapter 1, 1.1, 10


A function P is called a domain, then f is called an polynomial if

P(x)=anxn+an-1xn-1+‧‧‧+a2x2+a1x+a0

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=(-∞,∞)

If the leading coefficient an≠0, then the degree of the polynomial is n.

Chapter 1, 1.2, 13


Chapter 1, 1.2, 14 domain, then f is called an


Chapter 1, 1.2, 14 domain, then f is called an


Chapter 1, 1.2, 14 domain, then f is called an


Chapter 1, 1.2, 14 domain, then f is called an


Chapter 1, 1.2, 14 domain, then f is called an


Chapter 1, 1.2, 14 domain, then f is called an


Chapter 1, 1.2, 14 domain, then f is called an


Chapter 1, 1.2, 14 domain, then f is called an


Chapter 1, 1.2, 14 domain, then f is called an


Chapter 1, 1.2, 15 domain, then f is called an


A domain, then f is called an rational function fis 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


Chapter 1, 1.2, 15 domain, then f is called an


Chapter 1, 1.2, 15 domain, then f is called an


-1≤ son x≤1 -1≤ cos x≤1 domain, then f is called an

Chapter 1, 1.2, 15


Chapter 1, 1.2, 16 domain, then f is called an


Chapter 1, 1.2, 16 domain, then f is called an


Chapter 1, 1.2, 16 domain, then f is called an


Sin(x+2 domain, then f is called an π)=sin x cos(x+2π)=cos x

Chapter 1, 1.2, 16


The domain, then f is called an exponential functions are the functions of the form f(x)=ax , where the base is a positive constant.

Chapter 1, 1.2, 16


The domain, then f is called an 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


domain, then f is called an Figure 15 illustrates these shifts by showing how the graph of y=(x+3)2+1 is obtained from the graph of the parabola y=x2: Shift 3 units to the left and 1 unit upward.

Y=(x+3)2+1

Chapter 1, 1.2, 17


VERTICAL AND HORIZONTAL SHIFTS domain, then f is called an

Suppose c>0. To obtain the graph of

Y= f(x)+c, shift the graph of y=f(x) a distance c units c units upward

Y= f(x)- c, shift the graph of y=f(x) a distance c units c units downward

Y= f(x- c), shift the graph of y=f(x) a distance c units c units to the right

Y=f(x+ c), shift the graph of y=f(x) a distance c units c units to the left

Chapter 1, 1.2, 17


VERTICAL AND HORIZONTAL STRETCHING AND REFLECTING domain, then f is called an

Suppose c>1. To obtain the graph of

y=cf(x), stretch the graph of y=f(x) vertically by a factor of c

y=(1/c)f(x), compress the graph of y=f(x) vertically by a factor of c

Y=f(cx), compress the graph of y=f(x) horizontally by a factor of c

Y=f(x/c), stretch the graph of y=f(x) horizontally by a factor of c

Y=-f(x), reflect the graph of y=f(x) about the x-axis

Y=f(-x), reflect the graph of y=f(x) about they-axis

Chapter 1, 1.2, 17


Chapter 1, 1.2, 17 domain, then f is called an


Chapter 1, 1.2, 17 domain, then f is called an


EXAMPLE 2 domain, then f is called an Given the graph of y= , use transformations to graph y= -2 , y= , y=- , y=2 , and y=

Chapter 1, 1.2, 18


Chapter 1, 1.2, 18 domain, then f is called an


Chapter 1, 1.2, 18 domain, then f is called an


Chapter 1, 1.2, 18 domain, then f is called an


Chapter 1, 1.2, 18 domain, then f is called an


Chapter 1, 1.2, 18 domain, then f is called an


Chapter 1, 1.2, 18 domain, then f is called an


EXAMPLE 3 domain, then f is called an Sketch the graph of the function

y=1-sin x.

Chapter 1, 1.2, 18


Chapter 1, 1.2, 18 domain, then f is called an


Chapter 1, 1.2, 18 domain, then f is called an


(f+g)(x)=f(x)+g(x) (f-g)(x)=f(x)-g(x) domain, then f is called an

If the domain of f is A and the domain of g is B, then the domain of f + g is the intersection A ∩ B

Chapter 1, 1.2, 18


(fg)(x)=f(x)g(x) domain, then f is called an

The domain of fg is A ∩B, but we can’t divide by 0 and so the domain of f/g is

Chapter 1, 1.2, 18


DEFINITION domain, then f is called an 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


Chapter 1, 1.2, 19 domain, then f is called an


EXAMPLE 5 domain, then f is called an 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


EXAMPLE 6 domain, then f is called an Given F(x)=cos2(x+9) , find functions f ,g ,and h such that F=f。g。H.

Chapter 1, 1.2, 20


  • 17. domain, then f is called an The graph of y=f(x) is given. Match each equation with its graph and give reasons for your choices.

  • y=f(x-4)

  • y=f(x)+3

  • y= f(x)

  • y=-f(x+4)

  • y=2f(x+6)

Chapter 1, 1.2, 22


18. domain, then f is called an The graph of f is given. Draw the graphs of the

following functions.

(a)y=f(x+4) (b) y=f(x)+4

(c) y=2f(x) (d) y=- f(x)+3

Chapter 1, 1.2, 22


19 The graph of f is given. Use it to graph domain, then f is called an

the following functions.

(a) y=f(2x) (b) y=f( x)

(c) y=f(-x) (d)y=-f(-x)

Chapter 1, 1.2, 22


Chapter 1, 1.2, 22


Chapter 1, 1.3, 25 expression, or explain why it is undefined.


Chapter 1, 1.3, 25 expression, or explain why it is undefined.


1 DEFINITION expression, or explain why it is undefined.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


limf(x)=L expression, or explain why it is undefined.

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


Chapter 1, 1.3, 26 expression, or explain why it is undefined.


Chapter 1, 1.3, 26 expression, or explain why it is undefined.


Chapter 1, 1.3, 26 expression, or explain why it is undefined.


Chapter 1, 1.3, 26 expression, or explain why it is undefined.


Chapter 1, 1.3, 26 expression, or explain why it is undefined.


Chapter 1, 1.3, 28 expression, or explain why it is undefined.


Chapter 1, 1.3, 28 expression, or explain why it is undefined.


2. DEFINITION expression, or explain why it is undefined.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


Chapter 1, 1.3, 30 expression, or explain why it is undefined.


Chapter 1, 1.3, 30 expression, or explain why it is undefined.


3 expression, or explain why it is undefined.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


EXAMPLE 7 expression, or explain why it is undefined.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


Chapter 1, 1.3, 30 expression, or explain why it is undefined.


Chapter 1, 1.3, 31 expression, or explain why it is undefined.


FINITION expression, or explain why it is undefined.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 ε>0 there is a corresponding

number δ>0 such that

if 0<│x-a│<δ then │f(x)-L│<ε

Chapter 1, 1.3, 31


Chapter 1, 1.3, 32 expression, or explain why it is undefined.


Chapter 1, 1.31, 32 expression, or explain why it is undefined.


Chapter 1, 1.3, 32 expression, or explain why it is undefined.


Chapter 1, 1.3, 32 expression, or explain why it is undefined.


Chapter 1, 1.3, 32 expression, or explain why it is undefined.


Chapter 1, 1.3, 33 expression, or explain why it is undefined.


  • 3. expression, or explain why it is undefined.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


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


X→0- X→0+ X→0

X→2- X→2+ X→2

X→4

Chapter 1, 1.3, 33


  • LIMIT LAWS of each quantity, if it exists. If it does not exist, explain why. 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


Sum Law of each quantity, if it exists. If it does not exist, explain why.

Difference Law

Constant Multiple Law

Product Law

Quotient Law

Chapter 1, 1.4, 36


1. of each quantity, if it exists. If it does not exist, explain why.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


6. of each quantity, if it exists. If it does not exist, explain why.lim[f(x)]n=[limf(x)]nwhere n is a positive integer

X→a X→a

Chapter 1, 1.4, 36


7. of each quantity, if it exists. If it does not exist, explain why.lim c=c 8. lim x=a

X→a X→a

Chapter 1, 1.4, 36


9. of each quantity, if it exists. If it does not exist, explain why.lim xn=an where n is a positive integer

X→a

Chapter 1, 1.4, 36


10. of each quantity, if it exists. If it does not exist, explain why.lim = where n is a positive integer

(If n is even, we assume that a>0.)

X→a

Chapter 1, 1.4, 36


11.Lim = of each quantity, if it exists. If it does not exist, explain why.where n is a positive integer

[If n is even, we assume that lim f(X)>0.]

X→a

X→a

X→a

Chapter 1, 1.4, 36


DIRECT SUBSTITUTION PROPERTY of each quantity, if it exists. If it does not exist, explain why.If f is a polynomial or a rational function

and is in the domain of f, then

lim f(X)>f(a)

X→a

Chapter 1, 1.4, 37


If f(x)=g(x) when x ≠ a, then lim f(x)=lim g(x), of each quantity, if it exists. If it does not exist, explain why.

provided the limits exist.

X→a

X→a

Chapter 1, 1.4, 38


FIGURE 2 of each quantity, if it exists. If it does not exist, explain why.

The graphs of the functions f (from Example 2) and g (from Example 3)

Chapter 1, 1.4, 39


2 of each quantity, if it exists. If it does not exist, explain why.THEOREM lim f(x)=L if and only if

lim f(x)=L=lim f(x)

X→a

X→a-

X→a+

Chapter 1, 1.4, 39


Chapter 1, 1.4, 40 of each quantity, if it exists. If it does not exist, explain why.


Chapter 1, 1.4, 40 of each quantity, if it exists. If it does not exist, explain why.


Chapter 1, 1.4, 40 of each quantity, if it exists. If it does not exist, explain why.


3 of each quantity, if it exists. If it does not exist, explain why.. 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


4. of each quantity, if it exists. If it does not exist, explain why.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


Chapter 1, 1.4, 41 of each quantity, if it exists. If it does not exist, explain why.


Chapter 1, 1.4, 41 of each quantity, if it exists. If it does not exist, explain why.


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)lim[f(x)+g(x)] (b) lim [f(x)+g(x)]

(c)lim [f(x)g(x)] (d) lim

(e)Lim[x3f(x)] (f) lim

X→2

X→1

X→0

X→ -1

X→2

X→1

Chapter 1, 1.4, 43


■ As illustrated in Figure 1, if f is continuous, each limit, if it exists. If the limit does not exist, explain why.

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


Chapter 1, 1.5, 46 each limit, if it exists. If the limit does not exist, explain why.


  • DEFINITION each limit, if it exists. If the limit does not exist, explain why. A function f is continuous at a

  • number a if

  • lim f(X)=f(a)

X→a

Chapter 1, 1.5, 46


X→a

X→a

Chapter 1, 1.5, 46


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


2. open interval containing a, except perhaps at a), we say that f is 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


3. DEFINITION open interval containing a, except perhaps at a), we say that f is 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


  • 4. open interval containing a, except perhaps at a), we say that f is THEOREM If f and g are continuous at a and c is a constant, then the following functions are also continuous at a :

  • f+g 2 f-g 3 cf

  • 4. fg 5. if g(a)≠0

Chapter 1, 1.5, 48


  • 5. THEOREM open interval containing a, except perhaps at a), we say that f is

  • Any polynomial is continuous everywhere; that is, it is continuous on R=(-∞,∞).

  • (b) Any rational function is continuous wherever it is defined; that is, it is

  • continuous on its domain.

Chapter 1, 1.5, 49


6. THEOREM open interval containing a, except perhaps at a), we say that f is 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


7. THEOREM open interval containing a, except perhaps at a), we say that f is 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


8. THEOREM open interval containing a, except perhaps at a), we say that f is 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


9. INTERMEDIATE VALUE THEOREM open interval containing a, except perhaps at a), we say that f is 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


Chapter 1, 1.5, 52 open interval containing a, except perhaps at a), we say that f is


Chapter 1, 1.5, 52 open interval containing a, except perhaps at a), we say that f is


Chapter 1, 1.5, 53 open interval containing a, except perhaps at a), we say that f is


3 (a) From the graph of f, state the numbers at open interval containing a, except perhaps at a), we say that f is

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


4. From the graph of g , state the intervals on open interval containing a, except perhaps at a), we say that f is

which g is continuous.

Chapter 1, 1.5, 54


1 DEFINITION open interval containing a, except perhaps at a), we say that f is The notation

lim f(x)=∞

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


Chapter 1, 1.6, 57 open interval containing a, except perhaps at a), we say that f is


Chapter 1, 1.6, 57 open interval containing a, except perhaps at a), we say that f is


Chapter 1, 1.6, 57 open interval containing a, except perhaps at a), we say that f is


Chapter 1, 1.6, 57 open interval containing a, except perhaps at a), we say that f is


Chapter 1, 1.6, 57 open interval containing a, except perhaps at a), we say that f is


Chapter 1, 1.6, 57 open interval containing a, except perhaps at a), we say that f is


2. DEFINITION open interval containing a, except perhaps at a), we say that f is The line x = a is called a vertical asymptote of the curve y=f(x) if at least one of the following statements is true:

lim f(x)=∞

lim f(x)=∞

lim f(x)=∞

X→a+

X→a

X→a-

lim f(x)=-∞

lim f(x)=-∞

lim f(x)=-∞

X→a

X→a-

X→a+

Chapter 1, 1.6, 57


Chapter 1, 1.6, 58 open interval containing a, except perhaps at a), we say that f is


3. DEFINITION open interval containing a, except perhaps at a), we say that f is Let f be a function defined on some interval(a, ∞) . 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


Chapter 1, 1.6, 59 open interval containing a, except perhaps at a), we say that f is


Chapter 1, 1.6, 59 open interval containing a, except perhaps at a), we say that f is


Chapter 1, 1.6, 59 open interval containing a, except perhaps at a), we say that f is


Chapter 1, 1.6, 60 open interval containing a, except perhaps at a), we say that f is


Chapter 1, 1.6, 60 open interval containing a, except perhaps at a), we say that f is


Chapter 1, 1.6, 60 open interval containing a, except perhaps at a), we say that f is


4. DEFINITION open interval containing a, except perhaps at a), we say that f is The line y=L is called a horizontal asymptote of the curve y=f(x)

if either

lim f(x)=L or lim f(x)=L

X→∞

X→∞

Chapter 1, 1.6, 60


EXAMPLE 3 open interval containing a, except perhaps at a), we say that f is 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


5. If n is a positive integer, then open interval containing a, except perhaps at a), we say that f is

lim =0 lin =0

X→∞

X→∞

Chapter 1, 1.6, 61


6. DEFINITION open interval containing a, except perhaps at a), we say that f is Let f be a function defined on some open interval that contains the number a, except possibly at a itself. Then

lim f(x)=∞

means that for every positive number M there is a positive number δsuch that

if 0<│x-a│<δ then f(x)>M

X→ a

Chapter 1, 1.6, 64


7. DEFINITION open interval containing a, except perhaps at a), we say that f is Let f be a function defined on

some interval(a, ∞) . Then

lim f(x)=L

X→∞

means that for every ε>0 there is a corresponding

number N such that

if x>N then │f(x)-L│<ε

Chapter 1, 1.6, 65


Chapter 1, 1.6, 65 open interval containing a, except perhaps at a), we say that f is


Chapter 1, 1.6, 65 open interval containing a, except perhaps at a), we say that f is


8. DEFINITION open interval containing a, except perhaps at a), we say that f is Let f be a function defined on some interval(a, ∞) . Then

lim f(x)=∞

means that for every positive number M there is a corresponding positive number N such that

if x>N then f(x)>M

X→∞

Chapter 1, 1.6, 66


X→ 2

X→ -1-

X→∞

X→ -1+

X→ -∞

Chapter 1, 1.6, 66


Chapter 1, 1.6, 66 following.


X→ -∞

X→∞

X→3

X→ 0

X→ -2+

Chapter 1, 1.6, 67


1. following. 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



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


8. The graph of f is given. Draw the graphs of the following functions.

(a)y=f(x-8) (b)y=-f(x)

(c)y=2-f(x) (d)y= f(x)-1

Chapter 1, Review, 71


  • 21. The graph of f is given. following functions.

  • 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→∞

X→ -∞

Chapter 1, Review, 71


Chapter 1, Review, 71 following functions.


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