PDF- -The Calculus Tutoring Book By Carol Ash Robert B Ash - Ebooks - calculus tutoring book.pdf


ANTIDERIVATIVE TABLES Forms Involving axe + bx + c'dx


2ax + b +

2ax + b

- 4ac < 0

- 4ac = 0

J+Ibx + c)'

2ax + b (r

- 1) (4ac

- 3)a + (r

- 1) (4ac

(2c + bx)

- 1) (4ac

Rational Forms Involving a + bu F2(a+bu-alnja+bul]+C

Ja+bu 6

Ju2du a + bu

7' J (a

Inja + buIJ + C b2 La + bu +

J u2(a + bu)

In au + a2

_ 1 1 du 1 u(a + bu)2 a(a + bu)

Forms Involving 12

JuVa + bu du



(b) f(x) > x for all x (for example,

f(5) is a number that must be larger than 5)

(a) Sketch the power functions x-',

(b) Sketch the power functions x,

A function f is said to be even if f (-x) = f (x) for all x

A function is odd if f(-x) =

The functions cos x and x2 are even,

(a) Figure 17 shows the graph of a function f(x) for x plete the graph for x 0

If f is even,

Find f(x) if the graph off is the line AB where A = (1,2) and B = (2,5)

Letf(t) be the position of a particle on a number line at time I

Describe the motion if

(a) f is a constant function (c) f is a decreasing function (d) f(t) > 0 for all t (b) f(i) = t

The Trigonometric Functions

We continue with the development of the basic functions listed in Section 1

The functions are entitled to be called basic because of their many applications,

two of which (vibrations and electron flow) are described later in the section

We assume that you have studied trigonometry before starting calculus and therefore this section contains only a summary of the main results

A list of trigonometric identities and formulas is included at the end of the section for reference

Definition of sine,

Figure 1 shows a positive 0 corresponding to a counterclockwise rotation away from the positive x-axis

A negative 0 corresponds to a clockwise rotation

The distance r is always positive,

but the signs of x and y depend on the quadrant

If 90° < 0 < 180°,

so that 0 is a second quadrant angle,

The Trigonometric Functions

x is negative and y is positive

while cos 9 and tan 0 are negative

In general,

cos 8 and tan 0 for 0 in the various quadrants


z Degrees versus radians An angle of 180° is called it radians

More generally,

number of radians _ IT number of degrees 180


number of radians = 180 x number of degrees

One radian is a bit more than 57°

Tables 1 and 2 list some important angles

and the corresponding functional values

Table I

Table 2





1' N/2-

3ar/2 2w

In most situations not involving calculus,

it makes no difference whether we use radians or degrees,

it will be better to use radian measure

One geometric instance where radians are preferable involves arc length

Suppose a central angle 8 cuts off arc length s'on a circle of radius r (Fig

The entire circumference of the circle is 21rr

arc length s'is just a fraction of the entire circumference,

the fraction 9/360 if 0 is measured in degrees,

and 8/21r if 8 is measured in radians


2ar = r8

which is not as If degrees are used,

the formula is s'= 360 attractive as (5)

Reference angles Trig tables list sin 0,

cos 0 and tan 0 for 0 < 0 < 90°

To find the functions for other angles,

we use knowledge of the appropriate signs given in Fig

as illustrated in the following examples

If 0 is a second quadrant angle,

so 150° has reference angle 30° (Fig

If 0 is in the third quadrant,

- 180°,

so 210° has reference angle 30° (Fig


If 0 is in quadrant IV,

so 330° has reference angle 30° (Fig


Right triangle trigonometry In the right triangle in Fig

cos 0 = opposite leg adjacent leg

The Trigonometric Functions

Graphs of sin x,

cos x and tan x Figures 8-10 give the graphs of the functions,

The graphs show that sin x and cos x


so that each function has amplitude 1

On the other hand,

the tangent function assumes all values,

has range Note that sin x and cos x are defined

but tan x is not defined at x = tv/2,


The graph of a sin(bx + c) The function sin x has period 2rr and amplitude 1

The function 3 sin 2x has period it and amplitude 3 (Fig

In general,

has amplitude a and period 21r/b

For example,

The graph of a sin(bx + c) not only involves the same change of period

and amplitude as a sin bx but is also shifted

As an example,

To sketch the graph,

first plot a few points to get your bearings

For this purpose,

the most convenient values of x are those which

Then continue on to make the amplitude 1 and the period r as shown in Fig

Sin 0 = 0

Application to simple harmonic motion If a cork is pushed down in a bucket of water and then released (or,

a spring is stretched and released),

Experiments show that if a particular cork oscillates between 3 units above and 3 units below the water level with the timing indicated in Fig

its height h at time t is given by h(t) = 3 sin 't

TpMa t = it

i1ME to

(Note that there is nothing strange about time ir

It is approximately 3

frequency and shift depend on the cork,

the medium and the size and timing of the initial push down,

called simple harmonic motion,

Another instance of simple harmonic motion involves the flow of the alternating current (a

Electrons flow back and forth,

the amount of charge per second flowing in a given direction at time t,

then i(t) is of the form a sin(bt + c) or a cos(bt + c)

If i(t) = 10 cos t then at time t = 0,

The graph of f(x) sin x First consider two special cases

The graph of y = 2 sin x has amplitude 2 and lies between the pair of lines y = ±2 (Fig

although usually we do not actually sketch the lines

The lines,

which are reflections of one another in the x-axis,

are called the envelope of 2 sin x

The graph ofy =

the effect of the negative factor

so the graph is the reflection in the x-axis of the graph of 2 sin x (Fig

I'+ Similarly,

the graph of x' sin x is sandwiched between the curves y = ±x' which we sketch as guides (Fig

The curves,

called the envelope of x' sin x,

are reflections of one another in the x-axis


whenever x' is negative (as it is to the left of the y-axis) we not only change the amplitude but also reflect sine in the x-axis to obtain x' sin x

The result in Fig

In general,

to sketch the graph off (x) sin x,

first draw the curve y = f (x) and the curve y = f (x),

Then change

the height of the sine curve so that it fits within the envelope,

and in addition reflect the sine curve in the x-axis whenever f(x) is negative


cosecant and cotangent By definition,

In each case,

the function is defined for all values of x such that the denominator is nonzero

For example,

csc x is not defined for x = 0,

The graphs are given in Figs

hypotenuse hypotenuse csc 0 = opposite leg' adjacent leg' adjacent leg cot B = opposite leg

Notation It is standard practice to write sin(x for (sin x)2,

Similar notation holds for the other trigonometric functions

Standard trigonometric identities Negative angle formulas (9)

-sin x,

-csc x,

-tan x,

Addition formulas (10)

sin(x + y) = sin x cos y + cos x sin y sin(x

2 tan x

- tan2x

Pythagorean identities

Half-angle formulas (13)

Product formulas

- sin(x

Factoring formulas

Reduction formulas

Law of Cosines (Fig

Area formula (Fig

19) (19)

area of triangle ABC = iab sin C

Problems for Section 1

Convert from radians to degrees

-7r/3 2

Convert from degrees to radians

Inverse Functions and the Inverse Trigonometric Functions

Evaluate without using a calculator

(a) sin 210° (b) cos 31r (c) tan 51T/4

Sketch the graph

Let sin x = a,

cos y = b and evaluate the expression in terms of a and b,

In each of (a) and (b),

use right triangle trigonometry to find an exact answer,

rather than tables or a calculator which will give only approximations

(a) Find cos 0 if 0 is an acute angle and sin 0 = 2/3

(b) Find sin 0 if 0 is acute and tan 0 = 7/4

Sketch the graph

Trigonometric Functions

If a function maps a to b we may wish to switch the point of view and consider the inverse function which sends b to a

For example,

the function defined by F = IC + 32 gives the Fahrenheit temperature F as a function

If we solve the equation for C to obtain C = 9(F

If the original function is useful,

the inverse is probably also useful

In this section,

we discuss inverses in general,

and three inverse trigonometric functions in particular


The inverse function Let f be a one-to-one function

The inverse of f,

is defined as follows: if f (a) = b then f

-'(b) = a

In other words,

the inverse maps "backwards" (Fig

Only one-to-one functions have inverses because reversing a non-one-to-one function creates a pairing that is not a function (Fig

Given a table of values for f,

A partial table for f (x) = 3x and the corresponding partial table for its inverse are given below


-'(x) = 36

Note that we may also think of 36 as the "original" function with inverse U

In general,

Figure I shows that if f and f

or vice versa) the result is a "circular" trip which returns to the starting

In other words,

For example,

multiplying a number by 3 and then multiplying that result by 1/3 produces the original number

Example 1 In functional notation,

the centigrade/fahrenheit equations show that if f(x) = 23x + 32 then f-'(x) = y(x

The graph off `(x) One of the advantages of an inverse function is that its properties,

often follow easily from the properties of the original function

Comparing the graphs off and f

The points are reflections of one another in the line y = x

In general,

so that the pair of graphs is symmetric with respect to the line

If f(x) = x2,

and x >_ 0 so that f is one-to-one,

The symmetry of the two graphs is displayed in Fig

The inverse sine function Unfortunately,

the sine function as a whole doesn't have an inverse because it isn't one-to-one

But various pieces of the

any section between a low and a high point passes the horizontal line test and can be inverted

By convention,


sin-'a = b if and only if sin b = a and

The graph of sin-'x is found by reflecting sin x,

The domain of sin-'x is [-1,


The sin-' function is also denoted by Sin-' and arcsin

In computer programming,

the abbreviation ASN of arcsin is often used

Example 2 Find sin-' 2

Solution: Let x = sin-' 2

We know that sin 30° = 2,

We must choose the angle between

Example 3 Find sin-'(-1)

Of all the angles whose sine is

the one in the interval [- ir/2,


Inverse Functions and the Inverse Trigonometric Functions

The angles

as rotations from the positive x-axis,

they terminate in the same place

Although (1) states that f

This is because sin-' is not the inverse of sine unless the angle is between

The sine function maps 200°,


The sin-' function maps in reverse to the particular angle between


The inverse cosine function The cosine function,

By convention,

one-to-one piece between 0 and a,

and let cos-'x be the inverse of this abbreviated cosine function (Fig

cos''x is the angle between 0 and it whose cosine is x


cos-'a = b if and only if cos b = a and 0 < b _< ir

The domain of cos-'x is [-1,

The cos-' function is also denoted by Cos-',

Example 4 Find cos-'(-2)

Solution: The angle between 0° and 180° whose cosine is

Warning The graphs of sin x and cos x wind forever along the x-axis,

but the graphs of sin-'x and cos-'x (reflections of portions of sin x and cos x) do not continue forever up and down the y-axis

They are shown in entirety in Figs

5 and 6

(If either curve did continue winding,

the result would be a nonfunction

The inverse tangent function The tan-' function is the inverse of the branch of the tangent function through the origin (Fig

In other words,


tan-'a = b if and only if tan b = a and

The tan-' function is also denoted Tan"',

For example,

Example 5 The equation y = 2 tan 3x does not have a unique solution for x

Restrict x suitably so that there is a unique solution and then solve for x


restrict x so that the function 2 tan 3x is one-to-one,

and then find the inverse function

Solution: To use tan-' as the inverse of tangent,

must be restricted to the interval (-111r,


With this restriction,

(divide both sides of the original equation by 2)

(take tan-' on both sides) (divide by 3)


Exponential and Logarithm Functions

Problems for Section 1

Suppose f is one-to-one so that it has an inverse

If f(3) = 4 and f(5) = 2,

Find the inverse by inspection,

If f(x) = 2x


Show that an increasing function always has an inverse and then decide if the inverse is decreasing

True or False

? If f is continuous and invertible then f

Are the following pairs of functions inverses of one another

Find the function value

Estimate tan-'1000000

True or False

? (a) If sin a = b then sin-'b = a (b) If sin-'c = d'then sin d'= c

Place restrictions on 8 so that the equation has a unique solution for 8,

(a) z = 3 + s'sin ar8 (b) x = 5 cos(28

Odd and even functions were defined in Problem 8,

Section 1

Do odd (resp

Exponential and Logarithm Functions

This section completes the discussion of the basic functions listed in Section 1

As with the other basic functions,

they have important physical applications,

Exponential functions Functions such as 2',

as opposed to power functions x2,

In gen-

an exponential function has the form b',

Negative bases create a problem

If f(x) _ (-4)' then f(j) = and ,


the domain of (-4)' is too riddled with gaps to be useful in calculus

(The power function x' also has a restricted domain,

but at least the domain is an entire interval

we do not consider exponential functions with negative bases

To sketch the graph of 2',

we first make a table of values

For convenience,

we used integer values of x in the table,

but 2' is also defined when x is not an integer

For example,

E = 231110 =

and the graph of 2' also contains the points (2/3,

4) and (3

1,'2 V7)

We plot the points from the table,

and when the pattern seems clear,

connect them to obtain the final graph (Fig

The connecting process assumes that 2' is continuous

t Figure 1 also contains the graphs of (11)' and 3' for comparison

I The exponential function e' In algebra,

while computer science often favors base 2


for reasons to be given in Section 3

a particular irrational number (that is,

an infinite nonrepeating decimal) between 2

71 and 2

the official definition will be given in that section

Because calculus concentrates on base e,

the function e' is often referred to as the exponential function

It is sometimes written as expx

programming languages use EXP(X)

Figure 2 shows the graph of e',

along with 2' and 3' for comparison

Note that 2 < e < 3,

the graph of e' lies between the graphs of 2' and 3

We continue to assume that exponential functions are continuous

In practice,

may be approximated with tables or a calculator

Section 8

A rough estimate of e2 can be obtained by noting that since e is slightly less than 3,

tThe connecting process also provides a definition of 2' for irrational x,

when x is an infinite nonrepeating decimal,

For example,

and by connecting the points to make a continuous curve,

we are defining 2' by the following sequence of inequalities: 234 5

-in x = 4

In In x =

Show that In sV simplifies to

-s In 2

A scientist observes the temperature T and the volume V in an experiment and finds that In T always equals

-1 In V

Show that TV2" must therefore be constant

The equation 4 In x + 2(In x)2 = 0 can be considered as a quadratic equation in the variable In x

Solve for In x,

True or False

Find the mistake in the following "proof"that 2 < 1

We know that (s)2 < 4,

Thus 2 In s'< In 1

Cancel In s'to get 2 < 1

Solving Inequalities Involving Elementary Functions

This section contains algebra needed in Chapters 3 and 4

A simple inequality such as 2x + 3 > 11 is solved with the same maneuvers as the

equation 2x + 3 = 11 (the solution is x > 4),

we ities are trickier than equations

For example,

want to multiply on both sides by x

But if x < 5,

and the inequality is not reversed

this type of difficulty doesn't arise

offers a straightforward method for solving inequalities of the form f (x) > 0,

or equivalently for deciding where a function is positive and where it is negative

In order for a function f to change from positive to negative,

its graph must either cross or jump over the x-axis


a nonzero continuous f cannot change signs

its graph must lie entirely on one side of the x-axis

Suppose f is 0 only at x =

and is discontinuous only at x = 5,

so that within the open intervals (-w,

Then in each interval f cannot change signs and is either entirely positive or entirely negative

One possibility is shown in Fig

In general,

we have the following method for determining the sign of a function f,

x2-2x+1 5,

and negative for x < 1 and for 1 < x < 5


the solution to the first inequality in (1) is x > 5,

and the solution to the second inequality is x < 1 or 1 < x < 5

Note that Steps 1 and 2 locate points where the function either jumps or touches the x-axis

These are places where f might (but doesn't have to) change sign by crossing or jumping over the x-axis


f changes sign at x = 5 but not at x = 1

The graph in Fig

At x = 1,

f touches the x-axis but does not cross,

At x = 5,

f happens to jump over the axis,

Decide where the function f is positive and where it is negative

- 10x2 9(x

Solve (a) L6

Graphs of Translations,


Expansions and Sums

Considerable time is spent in mathematics finding graphs of functions because graphs can be extremely useful

It is possible to see from a graph where a function is positive,

when it may be very hard to do this from a formula

Suppose that the graph of y = f(x) is known

We will develop efficient techniques for finding the graphs of certain variations off

For example,

in trigonometry it is shown that the graph of sin 2x can be obtained easily from the graph of sin x by changing the period to or


the graph of 2 sin x can be derived from the graph of sin x by changing the amplitude to 2

We will generalize these ideas to arbitrary graphs

In each case,

the problem will be to find the graph of a variation off,

assuming that we have the graph off

We are not concerned here with how the original graph was obtained

Perhaps it was found by plotting many points,

possibly it was generated by a computer,

it may be a standard curve such asy = e' or it may have been drawn using techniques of calculus,

We will first consider three variations in which an operation is performed on the variable x in the equation y = f(x),

resulting in horizontal changes in the graph

Then we examine three variations obtained by operating on the entire right-hand side of the equation y = f(x),

resulting in vertical changes in the graph

Results are summarized in Table 1

Finally we consider the graph of a sum of functions,

Horizontal translation The graph of y = x' + 3x2

The problem is to draw the graph of the variation y = (x

look for a connection between the two tables of values

Y = x' + 3x2

2' + 3(22)

- 1 = 19

53 + 3(52)

- 1 = 199

53 + 3(52)

- 1 = 199

Substituting x = 9 into the new equation involves the same arithmetic (because 7 is immediately subtracted away) as substituting x = 2 in the

Graphs of Translations,


Expansions and Sums


x = 12 in the new equation produces the same

calculation as x = 5 in the old equation

In general,

b) is in the old table then (a + 7,

Now that we have a connection between the tables,

In general,

shifting) the old graph to the right by 7 units (Fig

This agrees with the familiar result that x2 + y2 = r2 is a circle with center at the origin,

the graph of y = f (x + 3) is found by translating y = f (x) to the left by 3 units

Horizontal expansion/contraction Consider the following two equations with their respective tables of values

Y = x3 + 3x2

23 + 3(22)

- 1 = 199

23 + 3(22)

- 1 = 199

Substituting x = 2/5 in the new equation produces the same calculation as x = 2 in the old equation (because each occurrence of 2/5 in the new equation is immediately multiplied by 5)

b) is in the old table then (a/5,

In general,

given the graph of y = f (x) (Fig

the graph of y = f (5x) is obtained by dividing x-coordinates by 5 so as to contract the graph horizontally (Fig


the graph of y = f (qx) is found by tripling x-coordinates so as to expand the graph off horizontally (Fig

Note that in the expansion (resp

points on the y-axis do not move,

but all other points move away from (resp

toward) the y-axis so as to triple widths (resp

The expansion/contraction rule says that the graph of y = sin 2x is drawn by halving x-coordinates and contracting the graph of y = sin x horizontally

This agrees with the standard result from trigonometry that y = sin 2x is drawn by changing the period on the sine curve from 21r to ir,

R FLr,LtoN iN r4E Y-AXIS

OeiZpNfAL HogizDn-' A L'coN W& 1,10N

M (b) 1

Horizontal reflection Consider the following two equations and their respective tables of values

23 + 3(22)

- 1 = 19

5' + 3(52)

- I = 199

- I = 119

Substituting x =

b) is in the old table then (-a,

In general,

given the graph of y = f(x) (Fig

the graph of y = f (-x) is obtained by reflecting the old graph in the y-axis (Fig

Vertical translation Consider the equations

y =x'+3x2- 1 and y =(x'+3x2- 1) + 10

For any fixed x,

they value for the second equation is 10 more than the first

In general,

the graph of y = f (x) + 10 is obtained by translating the original graph up by 10

Vertical expansion/contraction Consider the equations

y =x'+3x2- 1 and y =2(x'+3x2- 1)

For any fixed x,

the y value for the second equation is twice the first y

In general,

the graph of y = 2f (x) is obtained by doubling the y-coordinates so as to expand the original graph vertically


the graph of y = 3f(x) is found by multiplying heights by 2/3,

so as to contract the graph of f (x) vertically

tThe conclusion that y = f (x) + 10 is obtained by translating up by 10 may be compared with a corresponding result for circles,

provided that we rewrite the equation as (y

- 10) f(x)

The circle x2 + y4 = r4 has center at the origin,


Graphs of Translations,


Expansions and Sums

The familiar method for graphing y = 2 sin x (change the amplitude from 1 to 2) is a special case of the general method for y = 2f (x) (double all heights)

Vertical reflection Consider y = f (x) versus y =

- f (x)

The second y is always the negative of the first y

A special case appeared in Fig

Table 1


Variation of y = f (x)

How to obtain the graph from the original y = f (x)

An operation is performed on the variable x

Reflect the graph of y = f (x) in the

Halve the x-coordinates of the graph of y = f (x) so as to contract horizontally Multiply the x-coordinates of the graph of y = f (x) by 3 so as to expand horizontally Translate the graph of y = f (x) to the

Translate the graph of y = f (x) to the

An operation is performed on f (x),

Reflect the graph of y = f (x) in the

Double the y-coordinates of the graph of y = f (x) so as to expand

Multiply the y-coordinates of the graph of y = f (x) by 3 so as to contract vertically Translate the graph of y = f (x) up

Translate the graph of y = f (x) down by 3

Example 1 The graph of cos-'x is shown in Fig

Six variations are given in Figs

Warning The graph off (x

The graph off (x)

--- DOWN




-ir FIG

10 FIG M

Graphs of Translations,


Expansions and Sums

The graph of f(x) + g(x) Given the graphs of f(x) and g(x),

add the heights from the separate graphs off and g,

For example,

the new point D'is found by adding height

ff to height AC to obtain the new height AD

On the other hand,

since point P has a negative y-coordinate,

the newpooint R is found by subtracting

length PQ from 0 to get the new height

II To sketch y = cos x + sin x,

draw y = cos x and y = sin x on the same set of axes,

For example,

height X to obtain the new height AD

the corresponding point on the sum graph is point E,

Problems for Section 1

Sketch the graph and,

include the graph of In x for comparison

Figure 13 shows the graph of a function,

Sketch the following variations given on the next page

- 2) (f)

-star x

Find the new equation of the curve y = 2x7 + (2x + 3)6 if the curve is (a) translated left by 2 (b) translated down by 5

Sketch the graph

(a) y = Isin xl (b) y = Iln xl (c) y = le'I

Sketch each trio of functions on the same set of axes

In x,x + In x (b) x,

The variations sin2x,

sin3x and s's n x were not discussed in the section

Sketch their graphs by graphically squaring heights,

cubing heights and cuberooting heights on the sine graph


Letf(x) =

(b) For which values of x is f defined

? With these values as the domain,

(d) Sketch the graph off by plotting points

Then sketch the graph off

For this problem,

we need the idea of the remainder in a division problem

If 8 is divided by 3,

we say that the quotient is 2 and the remainder is 2

the quotient is 8 and the remainder is 2

If 27 is divided by 3,

the quotient is 9 and the remainder is 0

If x > 0,

let f(x) be the remainder when x is divided by 3

(b) Find the range of f (c) Find f

Chapter 1 Review Problems

Describe the graph off under each of the following conditions

(a) f(a) = a for all a (b) f(a) # f(b) if a 0 b (c) f (a + 7) = f (a) for all a 4

If log2x is the inverse of 2,

sketch the graphs of log2x and In x on the same set of axes

Find sin-'(--LV)

Solve for x

(a) y = 2 ln(3x + 4) (b) y = 4 + e3 7

Sketch the graph

(a) e- sin x (e) sin-' 2x (b) sin-'(x + 2) (f) sin 3irx (c) sin-x + 12or

The functions sink x = Z(e"

(a) Sketch their graphs by first drawing 2e' and 32e (b) Show that cosh2x

Solve the equation or inequality

- ln(2x

Simplify 5e21n3

Show that In x



We begin the discussion of limits with some examples

As you read them,

you will become accustomed to the new language and,

see how limit statements about a function correlate with the graph of the function

The examples will show how limits are used to describe discontinuities,

the "ends" of the graph where x

-- x or x

that is approached by the graph of f

) Limits will further be used in Sections 3

2 and 5

the two major concepts of calculus

A limit definition The graph of a function f is given in Fig

Note that as x gets closer to 2,

We write lim


This contrasts with f(2) itself which is 3

If point A in Fig

I is moved vertically or removed entirely,

In other words,

if the value off at x = 2 is changed from 3 to anything else ,

or if no value is assigned at all to f(2) ,

In general,

f (x) is forced to stay as close as we like,

One-sided limits In Fig

through values less than 3 such as 2

meaning that if x approaches 3 from the right,

through values greater than 3 such as 3

We call (1) a left-hand limit and (2) a right-hand limit

The symbols

they are symbols that are used only in the context of a limit statement to indicate from which direction 3 is approached

In this example,

if we are asked simply to find limas f(x),

we have to conclude that the limit does not exist

Since the left-hand and right-hand limits disagree,

there is no single limit to settle on


Infinite limits Let

There is no f(3),

3 we write

meaning that as x approaches 3 from the right,

f(x) becomes unboundedly large

There is no value for limx,3 f(x),

since the left-hand and right-hand limits do not agree

We do not write lim

In general,

lim_ f (x) = x means that for all x sufficiently close,

to a,f(x) can be forced to stay as large as we like


Limits as x


to indicate that as x becomes unboundedly large,

far out to the right on the graph,

the values of y get closer to 4

More precisely,

because the values of y are always less than 4 as they approach 4

Both (3) and (3') are correct,

but (3') supplies more information since it indicates that the graph off(x) approaches its asymptote,

For the same function,

_x f(x) = x because the graph rises unboundedly to the left

If a functionf(t) represents height,

x f(t) is called the steady state height,

and is sometimes denoted by f (x)

It is often interpreted as the eventual height,

speed reached after some transient disturbances have died out

Example 1 There is no limit of sin x as x

sin x just bounces up and down between

-1 and 1

Example 2 The graph of e" (Section 1


We sometimes abbreviate (4) by writing ex = x,

The left side of the graph of ex approaches the x-axis asymptotically (from above),


consider e" " = 1 /e e I /e to see that the limit is 0 (more precisely,

The result in (5) may be abbreviated by e-x = 0

Warning The limit of a function may be L'even though f never reaches L

The limit must be approached,

We have limx

_x ex = 0 although ex never reaches 0

Example 3 The graph of In x (Section 1

The graph of In x drops asymptotically toward the y-axis,

Limits of continuous functions If f is continuous at x = a so that its graph does not break,

For example,

If there is a discontinuity at x = a,

Example 4 The function x3


we can merely substitute x = 2 to get limx

- 2x) = 8

- 4 = 4

Some types of discontinuities Figure 1 shows a point discontinuity at x = 2,

In general,

a function f has a point discontinuity at x = a if lim,

f(x) is finite but not equal to f(a),

either because the two values are different or becausef(a) is not defined

The function has a jump discontinuity at x = a if the left-hand and right-hand limits are finite but unequal


f has an infinite discontinuity at x = a if at least one of the left-hand and right-hand limits is oc or

A function with an infinite discontinuity at x = a is said to blow up at x = a

Problems for Section 2

Find the limit

0 cos x

2(x2 + 3x

Find lim Int x as (a) x

--* 3+ 3

Find lim jxI/x as (a) x

-- 0+ 4

Find urn tan x as (a) x

(a) Draw the graph of a function f such that f is increasing,

(b) Draw the graph of a function f such that lim,

but f is not an increasing function

Identify the type of discontinuity and sketch a picture

f(x) = 2 and f(3) = 6 (b) lira,-,

Does lim,,

o f(2 + a) necessarily equal j(2)

Use limits to describe the asymptotic behavior of the function in Fig

Letf(x) = 0 if x is a power of 10,

For example,

Finding Limits of Combinations of Functions

Use the graph of f(x) to find limx

Finding Limits of Combinations of Functions

The preceding section considered problems involving individual basic functions,

We now examine limits of combinations

limits of elementary functions in general,

and continue to apply limits to curve sketching

Limits of combinations To find the limit of a combination of functions we find all the "sublimits" and put the results together sensibly,

as illustrated by the following example


Jim x2+5+lnx 2e'

We can't conveniently find the limit simply by looking at the graph of the function because we don't have the graph on hand

In fact,

finding the limit will help get the graph

The graph exists only for x > 0 because of the term In x,

" We find the limit by combining sublimits

- 0+ then

The sum of three numbers,

the second 5 and the third large and negative,


In the denominator,

A quotient with a large negative numerator and a denominator near 2 is still large and negative

We abbreviate all this by writing lim

In each limit problem involving combinations of functions,

find the individual limits and then put them together

The last section emphasized the former so now we concentrate on the latter,

especially for the more interesting and challenging cases where the individual limits to be combined involve the number 0 and/or the symbol x

Consider x/0-,

an abbreviation for a limit problem where the numerator grows unboundedly large and the denominator approaches 0 from the left

To put the pieces together,

In abbreviated notation,

Consider 2/x,

an abbreviation for a limit problem in which the numerator approaches 2 and the denominator grows unboundedly large

Compute fractions like


In abbreviated notation,

2/x = 0+

To provide further practice,

we list more limit results in abbreviated form

If you understood the preceding examples you will be able to do the following similar problems when they occur (without resorting to memorizing the list)


4°= 1 0+

x1/2 = x

The graph of a + be" Consider the function f(x) = 2

From Section 1

The result is a curve fairly similar to the graph of e',

The fastest way to determine the new location is to take limits as x

-- x and x

and perhaps plot one convenient additional point as a check: f(x)=2-ex=2-x=-x

The three computations lead to the graph in Fig

Example 2 Let

Then f is not defined at x = 5

Find lim,

f(x) and sketch the graph off in the vicinity of x = 5

Solution: We have limx,5

Finding Limits of Combinations of Functions

On closer examination,

mains larger than 5 as it approaches 5,

Thus lim 2 :

20+ = x

Since the left-hand and right-hand limits disagree,


the one-sided limits are valuable for revealing that f has an infinite discontinuity at x = 5 with the asymptotic behavior indicated I/

Warning A limit problem of the form 2/0 does not necessarily have the answer x


In general,

in a problem which is of the form (non 0)/0,

it is important to examine the denominator carefully

Example 3 Letf(x) = e where f is not defined

Determine the type of discontinuity at x = 0


1&2 (Fig

lim e- = e-""+ = e-' = 0+ "o Therefore f has a point discontinuity at x = 0

If we choose the natural definition f (O) = 0,

we can remove the discontinuity and make f continuous

In other words,

In general,

if a function g has a point discontinuity at x = a,

the discontinuity is called removable in the sense that we can define or redefine

g(a) to make the function continuous

On the other hand,

jump discontinuities and infinite discontinuities are not removable

There is no way to define f(5) in Example 2 (Fig

Problems for Section 2

Find (a)

(-'k') '

Find (a) lim(In x)2

- In x)


Find the limit and sketch the corresponding portion of the graph of the function:

Use limits to sketch the graph:

The function f (x) = e " has a discontinuity at x = 0 where it is not defined

Decide if the discontinuity is removable and,

remove it with an appropriate definition of f(0)

Letf(x) = sin 1/x

(a) Try to find the limit as x

In this case,

f has a discontinuity which is neither point nor jump nor infinite

The discontinuity is called oscillatory

(c) Use (a) and (b) to help sketch the graph off for x > 0

Indeterminate Limits

The preceding section considered many limit problems,

but deliberately avoided the forms 0/0,

This section discusses these forms and explains why they must be evaluated with caution

Consider 0/0,

function f(x) which approaches 0 as x

Unlike problems say of the form 0/3,

Suppose that as x

we have the following table of values:

Then the quotient approaches 1

But consider a second possible table of values:

In this case the quotient approaches 2

Or consider still another possible table of values:

Then the quotient approaches x

Because of this unpredictability,

the limit form 0/0 is called indeterminate

In general,

a limit form is indeterminate when

Indeterminate Limits

different problems of that form can have different answers

The characteristic of

an indeterminate form is a conflict between one function pulling one way and a second function pulling another way

In a 0/0 problem,

the small numerator is pulling the quotient toward 0,

while the small denominator is trying to make the quotient x or

result depends on how "fast" the numerator and denominator each approach 0

In a problem of the form x/x,

the large numerator is pulling the quotient toward x,

while the large denominator is pulling the quotient toward 0

The limit depends on how fast the numerator and denominator each approach x

In a problem of the form (0+)°,

which is positive and nearing 0,

is pulling the answer toward 0,

is pulling the answer toward 1

The final answer depends on the particular base and exponent,

In a problem of the form 0 x x,

the factor approaching 0 is trying to make the product small,

while the factor growing unboundedly large is trying to make the product unbounded

In an x° problem,

the base tugs the answer toward x while the exponent,

In a 1" problem,

while the exponent wants the answer to be x if the base is larger than 1,

or 0 if the base is less than 1

In a problem of the form x

the first term pulls toward x while the second term pulls toward

V and x are also indeterminate

Here is a list of indeterminate forms:

- x,(-x)

we do not accept "indeterminate" as a final answer

For example,

if a problem is of the form 0/0,

there is an answer (perhaps 0,

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