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9. Graph each of the lines in Exercise 8 on its own set of axes. Label your
graphs.
10. Which of the following is a function and which is not? Give a reason in
each case.
CHAPTER 1 Basics 53

(a) f assigns to each person his biological father
(b) g assigns to each man his dog
(c) h assigns to each real number its square root
(d) f assigns to each positive integer its cube
(e) g assigns to each car its driver
(f) h assigns to each toe its foot
(g) f assigns to each rational number the greatest integer that does not
exceed it
(h) g assigns to each integer the next integer
(i) h assigns to each real number its square plus six
11. Graph each of these functions on a separate set of axes. Label your graph.
f (x) = 3x 2 ’ x
(a)
x+2
g(x) =
(b)
x
h(x) = x 3 ’ x2
(c)
f (x) = 3x + 2
(d)
g(x) = √ ’ 2x
x2
(e)
h(x) = x + 3
(f)
12. Calculate each of the following trigonometric quantities.
(a) sin(8π/3)
(b) tan(’5π/6)
(c) sec(7π/4)
(d) csc(13π/4)
(e) cot(’15π/4)
(f) cos(’3π/4)
13. Calculate the left and right sides of the twelve fundamental trigonometric
identities for the values θ = π/3 and ψ = ’π/6, thus con¬rming the
identities for these particular values.
14. Sketch the graphs of each of the following trigonometric functions.
f (x) = sin 2x
(a)
g(x) = cos(x + π/2)
(b)
h(x) = tan(’x + π )
(c)
f (x) = cot(3x + π )
(d)
g(x) = sin(x/3)
(e)
h(x) = cos(’π + [x/2])
(f)
15. Convert each of the following angles from radian measure to degree
measure.
54 CHAPTER 1 Basics

= π/24
(a) θ
= ’π/3
(b) θ
= 27π/12
(c) θ
= 9π/16
(d) θ
=3
(e) θ
= ’5
(f) θ
16. Convert each of the following angles from degree measure to radian
measure.
= 65—¦
(a) θ
= 10—¦
(b) θ
= ’75—¦
(c) θ
= ’120—¦
(d) θ
= π—¦
(e) θ
= 3.14—¦
(f) θ
17. For each of the following pairs of functions, calculate f —¦ g and g —¦ f .
f (x) = x 2 + 2x + 3 g(x) = (x ’ 1)2
(a) √

f (x) = x + 1 g(x) = x 2 ’ 2
3
(b)
f (x) = sin(x + 3x 2 ) g(x) = cos(x 2 ’ x)
(c)
f (x) = ex+2 g(x) = ln(x ’ 5)
(d)
f (x) = sin(x 2 + x) g(x) = ln(x 2 ’ x)
(e)
g(x) = e’x
2 2
f (x) = ex
(f)
f (x) = x(x + 1)(x + 2) g(x) = (2x ’ 3)(x + 4)
(g)
18. Consider each of the following as functions from R to R and say whether
the function is invertible. If it is, ¬nd the inverse with an explicit formula.
f (x) = x 3 + 5
(a)
g(x) = x 2 ’ x √
(b)
h(x) = (sgn x) · |x|, where sgn x is +1 if x is positive, ’1 if x is
(c)
negative, 0 if x is 0.
f (x) = x 5 + 8
(d)
g(x) = e’3x
(e)
h(x) = sin x
(f)
f (x) = tan x
(g)
g(x) = (sgn x) · x 2 , where sgn x is +1 if x is positive, ’1 if x is
(h)
negative, 0 if x is 0.
19. For each of the functions in Exercise 18, graph both the function and its
inverse in the same set of axes.
CHAPTER 1 Basics 55

20. Determine whether each of the following functions, on the given domain
S, is invertible. If it is, then ¬nd the inverse explicitly.
f (x) = x 2 , S = [2, 7]
(a)
g(x) = ln x, S = [1, ∞)
(b)
h(x) = sin x, S = [0, π/2]
(c)
f (x) = cos x, S = [0, π]
(d)
g(x) = tan x, S = (’π/2, π/2)
(e)
h(x) = x 2 , S = [’2, 5]
(f)
f (x) = x 2 ’ 3x, S = [4, 7]
(g)
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CHAPTER 2



Foundations of
Calculus
2.1 Limits
The single most important idea in calculus is the idea of limit. More than 2000 years
ago, the ancient Greeks wrestled with the limit concept, and they did not succeed. It
is only in the past 200 years that we have ¬nally come up with a ¬rm understanding
of limits. Here we give a brief sketch of the essential parts of the limit notion.
Suppose that f is a function whose domain contains two neighboring intervals:
f : (a, c) ∪ (c, b) ’ R. We wish to consider the behavior of f as the variable x
approaches c. If f (x) approaches a particular ¬nite value as x approaches c, then
we say that the function f has the limit as x approaches c. We write
lim f (x) = .
x’c

The rigorous mathematical de¬nition of limit is this:

De¬nition 2.1 Let a < c < b and let f be a function whose domain contains
(a, c) ∪ (c, b). We say that f has limit at c, and we write limx’c f (x) = when
this condition holds: For each > 0 there is a δ > 0 such that
|f (x) ’ | <
whenever 0 < |x ’ c| < δ.
It is important to know that there is a rigorous de¬nition of the limit concept, and
any development of mathematical theory relies in an essential way on this rigorous
de¬nition. However, in the present book we may make good use of an intuitive

57
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58 CHAPTER 2 Foundations of Calculus

understanding of limit. We now develop that understanding with some carefully
chosen examples.
EXAMPLE 2.1
De¬ne
3’x if x < 1
f (x) =
x2 + 1 if x > 1
See Fig. 2.1. Calculate limx’1 f (x).




Y
FL
Fig. 2.1
AM

SOLUTION
Observe that, when x is to the left of 1 and very near to 1 then f (x) = 3 ’ x
is very near to 2. Likewise, when x is to the right of 1 and very near to 1 then
TE



f (x) = x 2 + 1 is very near to 2. We conclude that
lim f (x) = 2.
x’1
We have successfully calculated our ¬rst limit. Figure 2.1 con¬rms the conclusion
that our calculations derived.
EXAMPLE 2.2
De¬ne
x2 ’ 4
g(x) = .
x’2
Calculate limx’2 g(x).
SOLUTION
We observe that both the numerator and the denominator of the fraction
de¬ning g tend to 0 as x ’ 2 (i.e., as x tends to 2). Thus the question seems
to be indeterminate.
However, we may factor the numerator as x 2 ’ 4 = (x ’ 2)(x + 2).
As long as x = 2 (and these are the only x that we examine when we
59
CHAPTER 2 Foundations of Calculus

calculate limx’2 ), we can then divide the denominator of the expression
de¬ning g into the numerator. Thus
g(x) = x + 2 for x = 2.
Now
lim g(x) = lim x + 2 = 4.
x’2 x’2




Fig. 2.2

The graph of the function g is shown in Fig. 2.2. We encourage the reader to
use a pocket calculator to calculate values of g for x near 2 but unequal to 2 to
check the validity of our answer. For example,

x g(x) = [x 2 ’ 4]/[x ’ 2]
1.8 3.8
1.9 3.9
1.99 3.99
1.999 3.999
2.001 4.001
2.01 4.01
2.1 4.1
2.2 4.2

We see that, when x is close to 2 (but unequal to 2), then g(x) is close (indeed,
as close as we please) to 4.
x 3 ’ 3x 2 + x ’ 3
You Try It: Calculate the limit limx’3 .
x’3
Math Note: It must be stressed that, when we calculate limx’c f (x), we do not
evaluate f at c. In the last example it would have been impossible to do so. We want
to determine what we anticipate f will do as x approaches c, not what value (if any)
f actually takes at c. The next example illustrates this point rather dramatically.
60 CHAPTER 2 Foundations of Calculus




Fig. 2.3


EXAMPLE 2.3
De¬ne
if x = 7
3
h(x) =
if x = 7
1

Calculate limx’7 h(x).

SOLUTION
It would be incorrect to simply plug the value 7 into the function h and
thereby to conclude that the limit is 1. In fact when x is near to 7 but unequal
to 7, we see that h takes the value 3. This statement is true no matter how close
x is to 7. We conclude that limx’7 h(x) = 3.

You Try It: Calculate limx’4 [x 2 ’ x ’ 12]/[x ’ 4].




2.1.1 ONE-SIDED LIMITS
There is also a concept of one-sided limit. We say that

lim f (x) =
x’c’

if the values of f become closer and closer to when x is near to c but on the left.
In other words, in studying limx’c’ f (x), we only consider values of x that are
less than c.
Likewise, we say that
lim f (x) =
x’c+

if the values of f become closer and closer to when x is near to c but on the right.
In other words, in studying limx’c+ f (x), we only consider values of x that are
greater than c.
61
CHAPTER 2 Foundations of Calculus

EXAMPLE 2.4
Discuss the limits of the function
2x ’ 4 if x < 2
f (x) =
if x ≥ 2
x2
at c = 2.

SOLUTION
As x approaches 2 from the left, f (x) = 2x ’ 4 approaches 0. As x
approaches 2 from the right, f (x) = x 2 approaches 4. Thus we see that f
has left limit 0 at c = 2, written
lim f (x) = 0,
x’2’
and f has right limit 4 at c = 2, written
lim f (x) = 4.
x’2+
Note that the full limit limx’2 f (x) does not exist (because the left and right
limits are unequal).
You Try It: Discuss one-sided limits at c = 3 for the function
±
x 3 ’ x if x < 3

f (x) = 24 if x = 3


4x + 1 if x > 3
All the properties of limits that will be developed in this chapter, as well as the
rest of the book, apply equally well to one-sided limits as to two-sided (or standard)
limits.


2.2 Properties of Limits
To increase our facility in manipulating limits, we have certain arithmetical and
functional rules about limits. Any of these may be veri¬ed using the rigorous de¬-
nition of limit that was provided at the beginning of the last section. We shall state
the rules and get right to the examples.
If f and g are two functions, c is a real number, and limx’c f (x) and
limx’c g(x) exist, then
Theorem 2.1
(a) limx’c (f ± g)(x) = limx’c f (x) ± limx’c g(x);
62 CHAPTER 2 Foundations of Calculus

(b) limx’c (f · g) (x) = (limx’c f (x)) · (limx’c g(x)) ;

limx’c f (x)
f
(x) = provided that limx’c g(x) = 0;
(c) lim
limx’c g(x)
g
x’c

(d) limx’c (± · f (x)) = ± · (limx’c f (x)) for any constant ±.
Some theoretical results, which will prove useful throughout our study of
calculus, are these:
Theorem 2.2
Let a < c < b. A function f on the interval {x : a < x < b} cannot have two
distinct limits at c.

Theorem 2.3
If
lim g(x) = 0
x’c
and
lim f (x) either does not exist or exists and is not zero
x’c
then
f (x)
lim
x’c g(x)

does not exist.

Theorem 2.4 (The Pinching Theorem)
Suppose that f, g, and h are functions whose domains each contain S = (a, c) ∪
(c, b). Assume further that
g(x) ¤ f (x) ¤ h(x)
for all x ∈ S. Refer to Fig. 2.4.
y = h(x)

y = f (x)


y = g (x)

a c b


Fig. 2.4

If
lim g(x) =
x’c
63
CHAPTER 2 Foundations of Calculus

and
lim h(x) =
x’c
then
lim f (x) = .
x’c

EXAMPLE 2.5

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