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

### ANTIDERIVATIVE TABLES Forms Involving axe + bx + c'dx

7477=167

• tax + b

2ax + b +

2ax + b

• ifb2-4ac>0

- 4ac < 0

- 4ac = 0

• 2ax+bC ZalnIax'+bx+c,
• -2aJax2+bx+c

#### J+Ibx + c)'

2ax + b (r

- 1) (4ac

• - b2) (ax2 + bx + c)'-

- 3)a + (r

- 1) (4ac

• - b2) J (ax 2 + bx + cy-i 1
• (axe + bx + c)'

(2c + bx)

- 1) (4ac

• - b2) (ax2 + bx + c)'-'
• - b2) I(ax2 + bx + c)'

Ja+bu 6

## Ju2du a + bu

• = b2[J(a + bu)2
• - 2a(a + bu)1 + a2 Inca + bud] + C a

7' J (a

# Inja + buIJ + C b2 La + bu +

• r Ju(a+bu)a

## J u2(a + bu)

In au + a2

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

• - 2a lnja + bul] + C
• (au+dbur = bs I a + bu
• a + bu In I

Forms Involving 12

JuVa + bu du

• + bu)" + C

=2(bu-2a)

• 2(Sbu5- 2a) (a

-+IraI+C

• ifa>0 ifa 0 for all x

(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-',

• x'12 on the same set of axes

(b) Sketch the power functions x,

• x' on the same set of axes

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

• for example,
• f (7) = 3
• and f(-7) = 3,
• f(-4) =
• -2 and f(4) =
• and so on

## A function is odd if f(-x) =

• -f(x) for all x
• for example,
• -12 and f(-3) = 12,f(-6) _
• -2 and f(6) = 2,
• and so on

### The functions cos x and x2 are even,

• sin x and x' are odd,
• 2x + 3 and x2 + x are neither

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

• (b) Complete the graph in Fig
• 17 if f is odd

# 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

• 1 by considering the six trigonometric functions

# 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,

• cosine and tangent Using Fig
• we define (1)
• tang=r x

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

• thus sin 0 is positive,

while cos 9 and tan 0 are negative

### In general,

• 2 indicates the sign of sin 0,

cos 8 and tan 0 for 0 in the various quadrants

• 516N OFcOS 9 41( N OF tan 6

## SIGN OF 5iA A

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

More generally,

• to convert back and forth use

number of radians _ IT number of degrees 180

### Equivalently

• number of degrees =

number of radians = 180 x number of degrees

### Tables 1 and 2 list some important angles

• in both radians and degrees,

and the corresponding functional values

Table 2

## Degrees

### Degrees

• cos IV-2

1' N/2-

• 270° 360°

3ar/2 2w

• tan 1/Vs 1

In most situations not involving calculus,

it makes no difference whether we use radians or degrees,

• but it turns out (Section 3
• 3) that for the calculus of the trigonometric functions,

it will be better to use radian measure

• on a circle

## 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

• the indicated

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

• namely,

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

and 8/21r if 8 is measured in radians

## Therefore,

• with 0 in radian measure,

2ar = r8

• 1/Functions
• 21rr = 180r8,

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

• 2 plus reference angles,

as illustrated in the following examples

If 0 is a second quadrant angle,

• its reference angle is 180°

so 150° has reference angle 30° (Fig

• cos 150° =
• -cos 30° =
• sin 150° = sin 30° = 1,
• tan 150° _

#### If 0 is in the third quadrant,

• its reference angle is 0

- 180°,

so 210° has reference angle 30° (Fig

• sin 210° =
• -sin 30° =
• cos 210° =
• -cos 30° =

-,11\/-3,

• tan 210° = tan 30° = 1/V3_

#### If 0 is in quadrant IV,

• its reference angle is 360°

so 330° has reference angle 30° (Fig

• sin 330° =
• -sin 30° =
• cos 330° = cos 30° _ Imo,
• tan 330° =
• -tan 30° =

# Right triangle trigonometry In the right triangle in Fig

• sin 0 =
• opposite leg hypotenuse '
• tan 0 =

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,

• with x measured in radians

The graphs show that sin x and cos x

• have period 21r (that is,
• they repeat every 2ir units),
• while tan x has period it

Furthermore,

• -1 s'sin x s'1 and
• -1 s'cos x < 1,

so that each function has amplitude 1

On the other hand,

the tangent function assumes all values,

• that is,

has range Note that sin x and cos x are defined

• for all x,

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

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

### In general,

• a sin bx,
• for positive a and b,

has amplitude a and period 21r/b

# For example,

• 5 sin Ix has period 4ar and amplitude 5
• 1/Functions

### 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,

• consider sin(2x

### 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

• make the angle 2x
• - 3r a multiple of r/2
• the table in Fig
• 12 chooses angles 0 and r/4 to produce points (0,
• 0) on the graph

### 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,

• similarly,

a spring is stretched and released),

• it bobs up and down

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

• riME t=1T FIG
• flME 1:=2
• tmEt=3'7r
• 3 The Trigonometric Functions

#### It is approximately 3

• 14 minutes after time 0
• ) More generally,
• the amplitude,

frequency and shift depend on the cork,

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

• but the oscillation,

called simple harmonic motion,

• always has the
• form a sin(bt + c),
• or equivalently a cos(bt + c)

• ) in a wire

# Electrons flow back and forth,

• and if i(t) is the current,
• that is,

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,

• 10 units of charge per second flow in the given direction
• at time t = it/2,
• the flow momentarily stops
• at time t = jr,
• 10 units of charge per second flow opposite to the given direction

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 =

• -2 sin x also lies between those lines

the effect of the negative factor

• -2 is to change the signs of y-coordinates,

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

# Furthermore,

• 1/Functions

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

• 15 shows unbounded oscillations

### In general,

to sketch the graph off (x) sin x,

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

• its reflection in the x-axis,
• to serve as the envelope

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

Secant,

cosecant and cotangent By definition,

• sec x =
• csc x =
• cot x =
• cos x sin x

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

• 17 In a right triangle (Fig
• sec 0 = (8)

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

• 3 The Trigonometric functions

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

• and sin x2 to mean sin(x2)

# Similar notation holds for the other trigonometric functions

Standard trigonometric identities Negative angle formulas (9)

• sin(-x) =

-sin x,

• csc(-x) =

-csc x,

• cos(-x) = cos x,
• sec(-x) = sec x,
• tan(-x) _

-tan x,

• cot(-x) _

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

• - y) = sin x cos y
• - cos x sin y cos(x + y) = cos x cos y
• - sin x sin y cos(x
• - y) = cos x cos y + sin x sin y Double angle formulas
• sin 2x = 2 sin x cos x (11)
• cos 2x = cos2x
• - sin2x = 1
• - 2 sin2x = 2 cos(x

2 tan x

• tan 2x =

- tan2x

# Pythagorean identities

• sin2x + cos2x = 1 (12)
• 1 + tan2x = sec2x 1 + Cot2x = csc2x

### Half-angle formulas (13)

• sin2yx = cos2gx
• - cos x 2 1
• + cos x 2

# Product formulas

• sin xcos y =
• sin(x + y) + sin(x
• 1/Functions
• cos x sin y = cos x cos y =
• sin x sin y =
• sin(x + y)

- sin(x

• cos(x + y) + cos(x
• - cos(x + y) 2

Factoring formulas

• sin x + sin y = 2 cos (15)
• - sin y = 2 cos x 2 y sin cos x + cos y = 2 cos x +YCOSX
• - cos y = 2 sin XY sin

Reduction formulas

• - 0) = sin 0 sin(Tr
• - 0) = cos 0 cos(Tr
• -cos 0 sin(7r
• - 0) = sin 0 Law of Sines (Fig

#### Law of Cosines (Fig

• 19) 2 = a 2 b 2
• - 2ab cos C c

## Area formula (Fig

19) (19)

area of triangle ABC = iab sin C

# Convert from radians to degrees

• (a) it/5 (b) 51r/6 (c)

-7r/3 2

• (a) IT (b)
• -90° (c) 100°

#### 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

• (a) sin
• x (b) tan 4x
• (d) 5 sin(''px + a)
• (e) 2 cos(3x
• (c) 3 cos

Let sin x = a,

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

• if possible
• (a) sin(-x) (d)
• -cos y (b) cos(-y) (e) sin2x
• (f) sin x2

#### 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

• (a) x sin x
• (b) xs sin x
• 4 Inverse Functions and the Inverse

### 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

• - 32) we have the inverse function which produces C,
• given 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

#### NOT A FWeTiON

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

The inverse of f,

• denoted by 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 table of values for f
• -' can be constructed by interchanging columns

Clearly,

-'(x) = 36

# In general,

• f and f
• -' are inverses of each other

Figure I shows that if f and f

• -' are applied successively (first f and then f

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

### In other words,

• 1/Functions
• f-'(f(x)) = x and f(f-'(x)) = x

#### 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,

• such as its graph,

often follow easily from the properties of the original function

### Comparing the graphs off and f

• -' amounts to com-
• - a(7,2) paring points such as (2,
• 7) and (7,2) (Fig

## In general,

• the graph off
• -' is the reflection of the graph off in the line y = x,

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,

• then f"'(x) = Vx

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

• sine graph are one-to-one,
• in particular,

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

# By convention,

• we use the part between
• - 7r/2 and a/2 and let sin-'x be the inverse of this abbreviated sine function
• that is,
• sin-'x is the angle between
• - 1T12 and 7r/2 whose sine is x

Equivalently,

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

• -7r/2 s'b < ir/2

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

• -ir/2 s'x < zr/2,
• in the line y = x (Fig

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

• 1] and the range is (2)

[-zr/2,

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

# In computer programming,

the abbreviation ASN of arcsin is often used

## Solution: Let x = sin-' 2

• then sin x = 2

# We know that sin 30° = 2,

• sin(-330°) = 2,
• sin 150° = 2,

We must choose the angle between

• -90° and 90°
• therefore sin-' 2 = 30°,
• sin-' 2 = zr/6

Example 3 Find sin-'(-1)

Of all the angles whose sine is

the one in the interval [- ir/2,

Therefore,

• sin-'(-1) =

• 5 Warning 1

## The angles

• -wr/2 and 3ir/2 are coterminal angles
• that is,

as rotations from the positive x-axis,

they terminate in the same place

• -ir/2 and 3ir/2 are not the same angle or the same number,
• and aresin(-1) is
• not 3ir/2

## Although (1) states that f

• -'(f (x)) = x,
• sin-(sin 200°) is not 200°

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

• -90° and 90°

The sine function maps 200°,

• along with many other angles,
• such as 560°,

-160°,

• all to the same output

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

• -90° and 90°

Therefore,

• sin-'(sin 200°) =

The inverse cosine function The cosine function,

• like the sine function,
• has no inverse,
• because it is not one-to-one

By convention,

• we consider the

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

### Equivalently,

• 1/Functions

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

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

• 1) and the range is [0,

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

• arccos and ACN

Example 4 Find cos-'(-2)

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

• -2 is 120°
• -2= Therefore,
• cos-'(-2) = 120°,
• cos'(-)

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-'x is the angle between
• -ir/2 and a/2 whose tangent is x

#### Equivalently,

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

• -a/2 < b < 1r/2

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

• arctan and ATN

# For example,

• tan-'(-1) _
• -ir/4 because
• -Ir/4 is between
• -1T/2 and 7r/2 and tan(- 7r/4) =

## 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

Equivalently,

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,

• the angle,
• which is 3x
• in this problem,

must be restricted to the interval (-111r,

• that is,
• -2ir < 3x < 2Tr

Consequently,

• we choose
• -ir/6 < x < Tr/6

#### With this restriction,

• 2y = tan 3x tan ' 2y = 3x
• tan-' 2y = x

(divide both sides of the original equation by 2)

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

#### Equivalently,

• if f(x) = 2 tan 3x and
• -Ir/6 < x < it/6,
• then f''(x) _ I3 tan'2x

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,

• if possible,

## Find the inverse by inspection,

• if it exists
• (c) 1 /x
• (b) Int x

If f(x) = 2x

• - 9 find a formula for f

-'(f(17))

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

• -' is also continuous

## Are the following pairs of functions inverses of one another

• ? (a) x2 and (b) xs and 8

### Find the function value

• (a) cos-'O
• (b) sin''0
• (f) tan-'I
• (c) sin`2
• (g) tan-'(-1)
• (d) cos-'(-§\) 9

### 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,

• and then solve

(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

• even) functions have inverses
• ? If inverses exist,
• must they also be odd (resp

Exponential and Logarithm Functions

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

• 1 by considering the exponential functions and their inverses,
• the logarithm functions

#### As with the other basic functions,

they have important physical applications,

• such as exponential growth,
• discussed in Section 4

# Exponential functions Functions such as 2',

• (4)' and 7' are called exponential functions,

as opposed to power functions x2,

• x"' and x'

#### In gen-

an exponential function has the form b',

• and is said to have base b

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

• which are not real

#### Similarly,

• there is no (real) f (f'),
• f (17),

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

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

• namely [0,

but at least the domain is an entire interval

• ) Because of this difficulty,

we do not consider exponential functions with negative bases

### To sketch the graph of 2',

we first make a table of values

• (Remember
• that 2',
• for example,
• is defined as 1/2',
• and 2° is 1

### For convenience,

we used integer values of x in the table,

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

For example,

• 1/Functions 22"3

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,

• the most popular base is 10,

while computer science often favors base 2

#### However,

for reasons to be given in Section 3

• calculus uses base e,

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,

• and correspondingly,

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

# In practice,

• a value of e',
• such as e2,

may be approximated with tables or a calculator

### Section 8

• 9 will indicate one method for evaluating e' directly

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

• e2 is somewhat less than 9

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

• that is,

when x is an infinite nonrepeating decimal,

• such as ir

### For example,

and by connecting the points to make a continuous curve,

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

• (o) ex = e-x

-in x = 4

• (p) x In x = 0
• (g) In(-x) = 4
• (q) xex + 2ex = 0
• (h) esx+3 = e2x
• (r) ex In x = 0 25

### In In x =

• (j) aresin ex = it/6 6
• 2 +1n 3x = 5

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,

• and then solve for x itself

## True or False

• ? (a) If a = b,
• then e° = e'
• (b) If a + b = c,
• then e° + e" = e`

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

## We know that (s)2 < 4,

• so ln(Y)2 < In 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),

• in general,
• inequalx s'X 2x 1 > 0,

we ities are trickier than equations

### For example,

• to solve
• 1/Functions

want to multiply on both sides by x

• - 5 to eliminate fractions

# But if x < 5,

• - 5 is negative and multiplication by x
• - 5 reverses the inequality
• if x > 5,
• - 5 is positive,

and the inequality is not reversed

• (For equations,

this type of difficulty doesn't arise

• ) This section

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

• 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,

• or vice

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

### Therefore,

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 =

• -3 and x = 2,

and is discontinuous only at x = 5,

so that within the open intervals (-w,

• (-3,2),(2,5) and (5,-),f is nonzero and continuous

### One possibility is shown in Fig

#### In general,

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

• that is,
• for solving the inequalities
• f(x)>0,f(x)0

x2-2x+1 5,

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

### Equivalently,

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

Indeed,

• in this example,

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

The graph in Fig

• 2 shows what is happening

At x = 1,

f touches the x-axis but does not cross,

• so there is no sign change

## At x = 5,

f happens to jump over the axis,

• so there is a sign change
• z Problems for Section 1

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

- 10x2 9(x

• - 3)2 (b)x+i
• e` (d) x
• (e)x2+x-6
• (c) x2-x+2 2

### Solve (a) L6

• (b) 2x+
• 1/Functions

Graphs of Translations,

Reflections,

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,

• negative,
• increasing,
• decreasing,
• one-to-one,
• discontinuous,
• and so on,

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

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

# 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

Similarly,

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,

• coming later

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,

• given the individual graphs

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

• - I is given in Fig

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

• - 7)' + 3(x

look for a connection between the two tables of values

Y = x' + 3x2

2' + 3(22)

- 1 = 19

53 + 3(52)

- 1 = 199

• y=(x-7)'+3(x-7)2-1
• 2'+3(22)-1=19

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,

# Reflections,

Expansions and Sums

• original equation

#### Similarly,

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,

• b) is in the new table

Now that we have a connection between the tables,

• how are the graphs related
• ? The new point (9,
• 19) is 7 units to the right of the old point (2,

### In general,

• given the (old) graph
• of y = f (x),
• the (new) graph of y = f (x
• - 7) is obtained by translating (i

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,

• - 7)2 + y2 = r2 is a circle centered at the point (7,0),
• that is,
• translated to the right by 7
• ? Similarly,

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 = 19 5' + 3(52)

- 1 = 199

• (5x)3 + 3(5x)2

23 + 3(22)

• - 1 = 19 53 + 3(52)

- 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,

• b) is in the new table

### 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

# Similarly,

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

• contraction),

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

• divide widths by 5)

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,

• a horizontal contraction
• - 1 /Functions

#### 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

• y = X' + 3x2

23 + 3(22)

- 1 = 19

5' + 3(52)

- I = 199

• y=(-x)'+3(-x)2- 1 2' + 3(22)
• - I = 19 5' + 3(52)

- I = 119

# Substituting x =

• -2 into the new equation results in the same calculation as x = 2 in the original

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

• b) is in the
• new table

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

• 3d) so as to change the sign of each x-coordinate

## 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,

• given the graph of y = f (x),

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

• t Similarly,
• the graph of y = f (x)
• - 4 is found by translating the graph of y = f (x) down by 4

## 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,

• given the graph of y = f (x),

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

### Similarly,

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,

• while x4 + (y
• - 10)4 = r2 is centered at the point (0,
• that is,
• translated up by 10

Similarly,

• the graph of (y
• - 10) = f (x) is obtained by translating y = f(x) up by 10

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

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

## A special case appeared in Fig

• 14 of Section 1
• 3 which showed the graphs of y = 2 sin x and y =
• -2 sin x as reflections of one another

# Table 1

#### Summary

Variation of y = f (x)

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

An operation is performed on the variable x

• y =f(-x)

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

• y = f (2x)
• y=f(x+2)

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

• y = f(x

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

• y = B36)
• left by 2
• right by 3

An operation is performed on f (x),

• on the entire righthand side

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

• y = 2f(x)

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

• y = SOX)

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

• vertically
• y =f(x)+2
• y = f(x)

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

## Six variations are given in Figs

Warning The graph off (x

• - 1) (note the minus sign) is obtained by translating f (x) to the right (in the positive direction)

### The graph off (x)

• - 1 (note the minus sign) is found by translating f (x) down (in the negative direction)
• 1/Functions
• :k T-i rRANS%A1E

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• 4zr fi coy-1 "
• x WAND Vj:RrJ cALLY

CONTRA( TN

#### HoRizoNr^uY

• - cog REFlkC1

YERToay

-ir FIG

10 FIG M

Graphs of Translations,

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

• to sketch y = f(x) + g(x),

add the heights from the separate graphs off and g,

• as shown in Fig

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

• f(x)+y(x)

### II To sketch y = cos x + sin x,

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

• and then add heights (Fig

For example,

height X to obtain the new height AD

• at x = ir,
• when the sine height is 0,

the corresponding point on the sum graph is point E,

• lying on the cosine curve

### Sketch the graph and,

• in each case,

include the graph of In x for comparison

• (a) In(-x) (d) In 2x (b)
• (e) In(x + 2)
• (c) 2 In x
• (f) 2 + In x

Figure 13 shows the graph of a function,

• which we denote by star x

### Sketch the following variations given on the next page

• 1/Functions
• 13 (d) star x
• - 2 (e) star(-x) star(x

- 2) (f)

-star x

• (a) star
• (b) s'star x (c)

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

• (d) y = el'I (e) y = Inlxl

Sketch each trio of functions on the same set of axes

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

• sin x,x + sin x 6

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) =

• (a) Find f(-4)

(b) For which values of x is f defined

? With these values as the domain,

• the range off
• (c) Find f(a2) and (f (a))2

(d) Sketch the graph off by plotting points

# Then sketch the graph off

• if it exists

## 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

• 8 is divided by 3,

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

• (a) Sketch the graph off

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

• -'(x) if it exists
• (d) Find f (f (x))

## 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

• (g) 2 cos(4x
• (d) 2 sin-x 8

# The functions sink x = Z(e"

• - e"`) and cosh x = 2te" + e hyperbolic sine and hyperbolic cosine,
• respectively

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

• - sinh2x = I for all x

Solve the equation or inequality

• (a) In x

- ln(2x

• - 3) = 4 ( b)
• (c) (d )
• 2e' + 8 < 0 x

# Simplify 5e21n3

## Show that In x

• - In 5x simplifies to
• are called the

2/LIMITS

### We begin the discussion of limits with some examples

you will become accustomed to the new language and,

• in particular,

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

• and asymptotes
• (An asymptote is a line,
• more generally,
• a curve,

that is approached by the graph of f

) Limits will further be used in Sections 3

2 and 5

• 2 where they are fundamental for the definitions of the derivative and the integral,

the two major concepts of calculus

## Note that as x gets closer to 2,

• but not equal to 2,
• f(x) gets closer to 5

### We write lim

• 2 f(x) = 5 and say that as x approaches 2,
• f(x) approaches 5

# Equivalently,

• -> 2 then f(x)

### I is moved vertically or removed entirely,

• the limit of f (x) as x
• -> 2 remains 5

In other words,

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

• including 5 ,

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

• we still have limx
• 2 f (x) = 5

In general,

• we write
• lim f (x) = L'x+n
• for all x sufficiently close,
• but not equal,

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

• and possibly equal,

# One-sided limits In Fig

• there is no f(3),
• but we write (1)
• lim f(x) = 4,
• x approaches 3 from the left,
• that is,

through values less than 3 such as 2

• then f (x) approaches 4
• lim f W = 5,

meaning that if x approaches 3 from the right,

• that is,

through values greater than 3 such as 3

• then f (x) approaches 5

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

The symbols

• 3- and 3+ are not new numbers

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

2/Limits

## Infinite limits Let

• x-3 A table of values and the graph are given in Fig

### There is no f(3),

3 we write

• lim f (x) = x

meaning that as x approaches 3 from the right,

f(x) becomes unboundedly large

• and we write
• lim Ax) =
• -x to convey that as x approaches 3 from the left,f(x) gets unboundedly large and negative

### 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

• 3 f(x) _ ±x

#### In general,

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

• but not equal,

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

### Similarly,

• a limit of
• -x means thatf(x) can be made to stay arbitrarily large and negative

#### Limits as x

• -x For the function in Fig
• we write

# Introduction

• lim f (x) = 4 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,

• lim f (x) = 4x

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,

• the line y = 4,
• from below

### For the same function,

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

# If a functionf(t) represents height,

• voltage,
• at time t,
• then lim,

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

• voltage,

and is sometimes denoted by f (x)

# It is often interpreted as the eventual height,

• voltage,

speed reached after some transient disturbances have died out

Example 1 There is no limit of sin x as x

• -+ no because as x increases without bound,

sin x just bounces up and down between

-1 and 1

### Example 2 The graph of e" (Section 1

• 2) rises unboundedly to the right,
• lim ex = x

### Alternatively,

• consider the values e ",
• to see that the limit

• lim ex = 0

## Alternatively,

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

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

The limit must be approached,

• but not necessarily attained

### We have limx

_x ex = 0 although ex never reaches 0

• for the function f in Fig
• 2 f(x) = 5 although f(x) never attains 5

### Example 3 The graph of In x (Section 1

• 3) rises unboundedly to the right,
• lim In x = x

• lim In x =

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

• then limx
• f (x) is simply f (a)

# For example,

• _1 f(x) =f(-1) = 2

# If there is a discontinuity at x = a,

• then either limx
• f (x) and f (a) disagree,
• or one or both will not exist

Example 4 The function x3

• - 2x is continuous (the elementary functions are continuous except where they are not defined) so to find

2/Limits

• the limit as x approaches 2,

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,

• 2 shows a jump discontinuity at x = 3 and Fig
• 3 shows an infinite
• discontinuity at x = 3

## 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

Finally,

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

• (a) lim,
• (b) lim,
• (f) lim,
• ,,2 tan x
• (c) lim

0 cos x

• (g) lim,

2(x2 + 3x

• (d) lim,,

Find lim Int x as (a) x

• -- 3- (b) x

--* 3+ 3

## Find lim jxI/x as (a) x

• -- 0- (b) x

-- 0+ 4

Find urn tan x as (a) x

• --> 1- (b) x

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

• but lim,
• f (x) is not x

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

• f (x) = x,

but f is not an increasing function

#### Identify the type of discontinuity and sketch a picture

• (a) lim,

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

• f(x) (c) lim
• f(x) = 4 and lim,
• 2_ f(x) = 7 (d) lim,
• -- and lim,
• ,_ f(x) = 5 7

Does lim,,

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

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

• and letf(x) = I otherwise

### For example,

• f(100) = 0,((1000) = 0,
• f(983) = 1

Finding Limits of Combinations of Functions

• (a) limx
• es f (x) (b) lima
• (c) lim,

Use the graph of f(x) to find limx

• f(x) if (a) f(x) = x sin x
• (b) f(x) =

# Finding Limits of Combinations of Functions

The preceding section considered problems involving individual basic functions,

• such as e',
• sin x and In x

### We now examine limits of combinations

• of basic functions,
• that is,

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

Consider

#### 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,

• and finding the limit as x
• -+ 0+ will give information about how the graph "begins

" We find the limit by combining sublimits

- 0+ then

• 5 remains 5 and In x

## The sum of three numbers,

• the first near 0,

the second 5 and the third large and negative,

• is itself large and negative

Therefore,

• the numerator approaches

In the denominator,

• --> 1 so 2e'

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

We abbreviate all this by writing lim

• x2+5+lnx 0+5+ 2e ' = 2

# 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,

• examine say 100

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

• - 2/Limits 1

002001,

• to see that the limit is 0

# In abbreviated notation,

• 2/x = 0 or,
• more precisely,

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)

• 0x0=0 0+0=0

-2xx=-x

• x + x = x
• - xx (6-)xx=x
• xxx=x x

4°= 1 0+

• (0+)' = 0 cI = x

x1/2 = x

• (0+)' = 0 Example 1
• ex In x = x x x = x,
• ,,+ e' In x = l'x

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

# From Section 1

• 7 we know that the graph can be obtained from the graph of e' by reflection,
• contraction and translation

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

• but in a different location

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

• f(-x)=2-e-x2-02
• as a check,

# Example 2 Let

### Find lim,

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

# On closer examination,

• if x re-

mains larger than 5 as it approaches 5,

• - x remains less than 0 as it approaches 0

# Thus lim 2 :

• -x and (similarly)

20+ = x

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

• 5 f (x) does not exist

## However,

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

Rather,

• 2/0+ = x while 2/0- =

### 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

# Solution:

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,

• for all practical purposes,
• is 0 when x = 0

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

• 3) so as to remove the infinite discontinuity and make f continuous

Problems for Section 2

### Find (a)

• (f) e"= (g) 1/e"
• (b) (c)
• l \4/ 3
• (t) (J)

(-'k') '

# Find (a) lim(In x)2

• (d) lim e'-'
• (b) lim
• (e) lim ln(3x

- In x)

• (f) Inn

2/Limits

• (g) Iim x(x + 4)
• lim x cos
• (h) Jim e'
• lim 3 '
• /s sin x
• (k) h m
• e' In x

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

• (a) lim
• 2 (b) Iim I (c) Iim sin x

Use limits to sketch the graph:

• (a) e-'
• (b) 3 + 2e5=

### 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

• (b) Find the limit as x

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

### The preceding section considered many limit problems,

but deliberately avoided the forms 0/0,

• - x and a few others

# Consider 0/0,

• an abbreviation for lsm

function f(x) which approaches 0 as x

• -> a function g(s) which approaches 0 as x

Unlike problems say of the form 0/3,

• which all have the answer 0,
• 0/0 problems can produce a variety of answers

### Suppose that as x

we have the following table of values:

• numerator denominator

### Then the quotient approaches 1

But consider a second possible table of values:

• numerator 2/3
• denominator

In this case the quotient approaches 2

#### Or consider still another possible table of values:

• numerator denominator

# 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

#### In a problem of the form (0+)°,

• the base,

which is positive and nearing 0,

is pulling the answer toward 0,

• while the exponent,
• which is nearing 0,

is pulling the answer toward 1

# The final answer depends on the particular base and exponent,

• and on how "hard" they pull

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,

• which is nearing 0,
• pulls toward 1

### In a 1" problem,

• the base,
• which is nearing 1,
• pulls the answer toward I,

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:

• -xx=x0 x x x,0 x

- x,(-x)

• - (-x'),(0+)° 1" x° xa Every indeterminate limit problem can be done

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,

• or "no limit"),

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