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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


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


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

Chapter 1, 1.1, P2


Essential calculus ch01 functions limits

Chapter 1, 1.1, P2


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

Chapter 1, 1.1, P2


Essential calculus ch01 functions limits

Chapter 1, 1.1, P2


Essential calculus ch01 functions limits

Chapter 1, 1.1, P2


Essential calculus ch01 functions limits

  • 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


Essential calculus ch01 functions limits

EXAMPLE 3 Find the domain of each function.

Chapter 1, 1.1, P4


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

Chapter 1, 1.1, P5


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

Chapter 1, 1.1, P5


Essential calculus ch01 functions limits

EXAMPLE 5 Sketch the graph of the

absolute value function f(x)=│X│.

Chapter 1, 1.1, P6


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

Chapter 1, 1.1, P6


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

Chapter 1, 1.1, P6


Essential calculus ch01 functions limits

Chapter 1, 1.1, P6


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

EXAMPLE 7 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


Essential calculus ch01 functions limits

Chapter 1, 1.1, 07


Essential calculus ch01 functions limits

A function f is called 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


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

Chapter 1, 1.1, 08


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

Chapter 1, 1.1, 08


Essential calculus ch01 functions limits

3–6 ■ 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


Essential calculus ch01 functions limits

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

Chapter 1, 1.1, 10


Essential calculus ch01 functions limits

A function P is called a 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


Essential calculus ch01 functions limits

Chapter 1, 1.2, 14


Essential calculus ch01 functions limits

Chapter 1, 1.2, 14


Essential calculus ch01 functions limits

Chapter 1, 1.2, 14


Essential calculus ch01 functions limits

Chapter 1, 1.2, 14


Essential calculus ch01 functions limits

Chapter 1, 1.2, 14


Essential calculus ch01 functions limits

Chapter 1, 1.2, 14


Essential calculus ch01 functions limits

Chapter 1, 1.2, 14


Essential calculus ch01 functions limits

Chapter 1, 1.2, 14


Essential calculus ch01 functions limits

Chapter 1, 1.2, 14


Essential calculus ch01 functions limits

Chapter 1, 1.2, 15


Essential calculus ch01 functions limits

A 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


Essential calculus ch01 functions limits

Chapter 1, 1.2, 15


Essential calculus ch01 functions limits

Chapter 1, 1.2, 15


Essential calculus ch01 functions limits

-1≤ son x≤1 -1≤ cos x≤1

Chapter 1, 1.2, 15


Essential calculus ch01 functions limits

Chapter 1, 1.2, 16


Essential calculus ch01 functions limits

Chapter 1, 1.2, 16


Essential calculus ch01 functions limits

Chapter 1, 1.2, 16


Essential calculus ch01 functions limits

Sin(x+2π)=sin x cos(x+2π)=cos x

Chapter 1, 1.2, 16


Essential calculus ch01 functions limits

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

Chapter 1, 1.2, 16


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

■ 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


Essential calculus ch01 functions limits

VERTICAL AND HORIZONTAL SHIFTS

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


Essential calculus ch01 functions limits

VERTICAL AND HORIZONTAL STRETCHING AND REFLECTING

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


Essential calculus ch01 functions limits

Chapter 1, 1.2, 17


Essential calculus ch01 functions limits

Chapter 1, 1.2, 17


Essential calculus ch01 functions limits

EXAMPLE 2 Given the graph of y= , use transformations to graph y= -2 , y= , y=- , y=2 , and y=

Chapter 1, 1.2, 18


Essential calculus ch01 functions limits

Chapter 1, 1.2, 18


Essential calculus ch01 functions limits

Chapter 1, 1.2, 18


Essential calculus ch01 functions limits

Chapter 1, 1.2, 18


Essential calculus ch01 functions limits

Chapter 1, 1.2, 18


Essential calculus ch01 functions limits

Chapter 1, 1.2, 18


Essential calculus ch01 functions limits

Chapter 1, 1.2, 18


Essential calculus ch01 functions limits

EXAMPLE 3 Sketch the graph of the function

y=1-sin x.

Chapter 1, 1.2, 18


Essential calculus ch01 functions limits

Chapter 1, 1.2, 18


Essential calculus ch01 functions limits

Chapter 1, 1.2, 18


Essential calculus ch01 functions limits

(f+g)(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 ∩ B

Chapter 1, 1.2, 18


Essential calculus ch01 functions limits

(fg)(x)=f(x)g(x)

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


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

Chapter 1, 1.2, 19


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

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

Chapter 1, 1.2, 20


Essential calculus ch01 functions limits

  • 17. 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


Essential calculus ch01 functions limits

18. 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


Essential calculus ch01 functions limits

19 The graph of f is given. Use it to graph

the following functions.

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

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

Chapter 1, 1.2, 22


Essential calculus ch01 functions limits

  • 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


Essential calculus ch01 functions limits

Chapter 1, 1.3, 25


Essential calculus ch01 functions limits

Chapter 1, 1.3, 25


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

Chapter 1, 1.3, 26


Essential calculus ch01 functions limits

Chapter 1, 1.3, 26


Essential calculus ch01 functions limits

Chapter 1, 1.3, 26


Essential calculus ch01 functions limits

Chapter 1, 1.3, 26


Essential calculus ch01 functions limits

Chapter 1, 1.3, 26


Essential calculus ch01 functions limits

Chapter 1, 1.3, 28


Essential calculus ch01 functions limits

Chapter 1, 1.3, 28


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

Chapter 1, 1.3, 30


Essential calculus ch01 functions limits

Chapter 1, 1.3, 30


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

Chapter 1, 1.3, 30


Essential calculus ch01 functions limits

Chapter 1, 1.3, 31


Essential calculus ch01 functions limits

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 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


Essential calculus ch01 functions limits

Chapter 1, 1.3, 32


Essential calculus ch01 functions limits

Chapter 1, 1.31, 32


Essential calculus ch01 functions limits

Chapter 1, 1.3, 32


Essential calculus ch01 functions limits

Chapter 1, 1.3, 32


Essential calculus ch01 functions limits

Chapter 1, 1.3, 32


Essential calculus ch01 functions limits

Chapter 1, 1.3, 33


Essential calculus ch01 functions limits

  • 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


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

  • 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


Essential calculus ch01 functions limits

  • 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


Essential calculus ch01 functions limits

Sum Law

Difference Law

Constant Multiple Law

Product Law

Quotient Law

Chapter 1, 1.4, 36


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

6. lim[f(x)]n=[limf(x)]nwhere n is a positive integer

X→a X→a

Chapter 1, 1.4, 36


Essential calculus ch01 functions limits

7. lim c=c 8. lim x=a

X→a X→a

Chapter 1, 1.4, 36


Essential calculus ch01 functions limits

9. lim xn=an where n is a positive integer

X→a

Chapter 1, 1.4, 36


Essential calculus ch01 functions limits

10. lim = where n is a positive integer

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

X→a

Chapter 1, 1.4, 36


Essential calculus ch01 functions limits

11.Lim = 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


Essential calculus ch01 functions limits

DIRECT SUBSTITUTION PROPERTY 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


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

FIGURE 2

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

Chapter 1, 1.4, 39


Essential calculus ch01 functions limits

2 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


Essential calculus ch01 functions limits

Chapter 1, 1.4, 40


Essential calculus ch01 functions limits

Chapter 1, 1.4, 40


Essential calculus ch01 functions limits

Chapter 1, 1.4, 40


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

Chapter 1, 1.4, 41


Essential calculus ch01 functions limits

Chapter 1, 1.4, 41


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

■ 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


Essential calculus ch01 functions limits

Chapter 1, 1.5, 46


Essential calculus ch01 functions limits

  • DEFINITION A function f is continuous at a

  • number a if

  • lim f(X)=f(a)

X→a

Chapter 1, 1.5, 46


Essential calculus ch01 functions limits

  • 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


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

  • 4. 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


Essential calculus ch01 functions limits

  • 5. THEOREM

  • 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


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

Chapter 1, 1.5, 52


Essential calculus ch01 functions limits

Chapter 1, 1.5, 52


Essential calculus ch01 functions limits

Chapter 1, 1.5, 53


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

4. From the graph of g , state the intervals on

which g is continuous.

Chapter 1, 1.5, 54


Essential calculus ch01 functions limits

1 DEFINITION 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


Essential calculus ch01 functions limits

Chapter 1, 1.6, 57


Essential calculus ch01 functions limits

Chapter 1, 1.6, 57


Essential calculus ch01 functions limits

Chapter 1, 1.6, 57


Essential calculus ch01 functions limits

Chapter 1, 1.6, 57


Essential calculus ch01 functions limits

Chapter 1, 1.6, 57


Essential calculus ch01 functions limits

Chapter 1, 1.6, 57


Essential calculus ch01 functions limits

2. DEFINITION 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


Essential calculus ch01 functions limits

Chapter 1, 1.6, 58


Essential calculus ch01 functions limits

3. DEFINITION 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


Essential calculus ch01 functions limits

Chapter 1, 1.6, 59


Essential calculus ch01 functions limits

Chapter 1, 1.6, 59


Essential calculus ch01 functions limits

Chapter 1, 1.6, 59


Essential calculus ch01 functions limits

Chapter 1, 1.6, 60


Essential calculus ch01 functions limits

Chapter 1, 1.6, 60


Essential calculus ch01 functions limits

Chapter 1, 1.6, 60


Essential calculus ch01 functions limits

4. DEFINITION 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


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

5. If n is a positive integer, then

lim =0 lin =0

X→∞

X→∞

Chapter 1, 1.6, 61


Essential calculus ch01 functions limits

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)=∞

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


Essential calculus ch01 functions limits

7. DEFINITION 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


Essential calculus ch01 functions limits

Chapter 1, 1.6, 65


Essential calculus ch01 functions limits

Chapter 1, 1.6, 65


Essential calculus ch01 functions limits

8. DEFINITION 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


Essential calculus ch01 functions limits

  • 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→∞

X→ -1+

X→ -∞

Chapter 1, 1.6, 66


Essential calculus ch01 functions limits

Chapter 1, 1.6, 66


Essential calculus ch01 functions limits

  • 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→ -∞

X→∞

X→3

X→ 0

X→ -2+

Chapter 1, 1.6, 67


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

Chapter 1, Review, 70


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

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


Essential calculus ch01 functions limits

  • 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→∞

X→ -∞

Chapter 1, Review, 71


Essential calculus ch01 functions limits

Chapter 1, Review, 71


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