six

Rates of Change and Limits
Average and Instantaneous Speed
A moving body’s average speed during an interval of time is found by dividing the distance
covered by the elapsed time. The unit of measure is length per unit time—kilometers
per hour, feet per second, or whatever is appropriate to the problem at hand.
EXAMPLE 1 Finding an Average Speed
A rock breaks loose from the top of a tall cliff. What is its average speed during the first
2 seconds of fall?
SOLUTION
Experiments show that a dense solid object dropped from rest to fall freely near the surface
of the earth will fall
y  16t 2
feet in the first t seconds. The average speed of the rock over any given time interval is
the distance traveled, y, divided by the length of the interval t. For the first 2 seconds
of fall, from t  0 to t  2, we have



y
t 

162
2
2


0
1602
 32
s
f
e
t
c 
. Now try Exercise 1.
EXAMPLE 2 Finding an Instantaneous Speed
Find the speed of the rock in Example 1 at the instant t  2.
SOLUTION
Solve Numerically We can calculate the average speed of the rock over the interval
from time t  2 to any slightly later time t  2  h as



y
t 
 . (1)
We cannot use this formula to calculate the speed at the exact instant t  2 because that
would require taking h  0, and 00 is undefined. However, we can get a good idea of
what is happening at t  2 by evaluating the formula at values of h close to 0. When we
do, we see a clear pattern (Table 2.1 on the next page). As h approaches 0, the average
speed approaches the limiting value 64 ft/sec.
162  h2  1622

h
Section 2.1 Rates of Change and Limits 59
2.1
What you’ll learn about
• Average and Instantaneous
Speed
• Definition of Limit
• Properties of Limits
• One-sided and Two-sided
Limits
• Sandwich Theorem
. . . and why
Limits can be used to describe
continuity, the derivative, and the
integral: the ideas giving the
foundation of calculus.
Free Fall
Near the surface of the earth, all bodies
fall with the same constant acceleration.
The distance a body falls after it is released
from rest is a constant multiple
of the square of the time fallen. At least,
that is what happens when a body falls
in a vacuum, where there is no air to
slow it down. The square-of-time rule
also holds for dense, heavy objects like
rocks, ball bearings, and steel tools during
the first few seconds of fall through
air, before the velocity builds up to
where air resistance begins to matter.
When air resistance is absent or insignificant
and the only force acting on
a falling body is the force of gravity, we
call the way the body falls free fall.
continued
Confirm Algebraically If we expand the numerator of Equation 1 and simplify, we
find that



y
t 
 

64h 
h
16h2
 64  16h.
For values of h different from 0, the expressions on the right and left are equivalent and
the average speed is 64  16h ft/sec.We can now see why the average speed has the
limiting value 64  16(0)  64 ft/sec as h approaches 0. Now try Exercise 3.
Definition of Limit
As in the preceding example, most limits of interest in the real world can be viewed as numerical
limits of values of functions. And this is where a graphing utility and calculus
come in. A calculator can suggest the limits, and calculus can give the mathematics for
confirming the limits analytically.
Limits give us a language for describing how the outputs of a function behave as the
inputs approach some particular value. In Example 2, the average speed was not defined at
h  0 but approached the limit 64 as h approached 0. We were able to see this numerically
and to confirm it algebraically by eliminating h from the denominator. But we cannot always
do that. For instance, we can see both graphically and numerically (Figure 2.1) that
the values of f (x)  (sin x)x approach 1 as x approaches 0.
We cannot eliminate the x from the denominator of (sin x)x to confirm the observation
algebraically. We need to use a theorem about limits to make that confirmation, as you will
see in Exercise 75.
164  4h  h2  64

h
162  h2  1622

h
60 Chapter 2 Limits and Continuity
Figure 2.1 (a) A graph and (b) table of
values for f x  sin xx that suggest the
limit of f as x approaches 0 is 1.
Table 2.1 Average Speeds over
Short Time Intervals Starting at
t  2



y
t 

Length of Average Speed
Time Interval, for Interval
h (sec) yt (ft/sec)
1 80
0.1 65.6
0.01 64.16
0.001 64.016
0.0001 64.0016
0.00001 64.00016
162  h2  1622

h
[–2p, 2p] by [–1, 2]
(a)
X
Y1 = sin(X)/X
–.3
–.2
–.1
0
.1
.2
.3
.98507
.99335
.99833
ERROR
.99833
.99335
.98507
Y1
(b)
The sentence limx?c f x  L is read, “The limit of f of x as x approaches c equals L.”
The notation means that the values f (x) of the function f approach or equal L as the values
of x approach (but do not equal) c. Appendix A3 provides practice applying the definition
of limit.
We saw in Example 2 that limh?0 64  16h  64.
As suggested in Figure 2.1,
lim
x?0

sin
x
x
 1.
Figure 2.2 illustrates the fact that the existence of a limit as x?c never depends on how
the function may or may not be defined at c. The function f has limit 2 as x?1 even though
f is not defined at 1. The function g has limit 2 as x?1 even though g1  2. The function
h is the only one whose limit as x?1 equals its value at x  1.
DEFINITION Limit
Assume f is defined in a neighborhood of c and let c and L be real numbers. The
function f has limit L as x approaches c if, given any positive number e, there is a
positive number d such that for all x,
0 x  c d ?  f x  L
.
We write
lim
x?c
f x  L.
Section 2.1 Rates of Change and Limits 61
THEOREM 1 Properties of Limits
If L, M, c, and k are real numbers and
lim
x?c
f x  L and lim
x?c
gx  M, then
1. Sum Rule: lim
x?c
 f x  gx  L  M
The limit of the sum of two functions is the sum of their limits.
2. Difference Rule: lim
x?c
 f x  gx  L  M
The limit of the difference of two functions is the difference of their limits.
3. Product Rule: lim
x?c
 f x • gx  L • M
The limit of a product of two functions is the product of their limits.
4. Constant Multiple Rule: lim
x?c
k • f x  k • L
The limit of a constant times a function is the constant times the limit of the
function.
5. Quotient Rule: lim
x?c
g
f 

x
x

 
 
M
L
, M  0
The limit of a quotient of two functions is the quotient of their limits, provided
the limit of the denominator is not zero.
continued
Properties of Limits
By applying six basic facts about limits, we can calculate many unfamiliar limits from
limits we already know. For instance, from knowing that
lim
x?c
k  k Limit of the function with constant value k
and
lim
x?c
x  c, Limit of the identity function at x  c
we can calculate the limits of all polynomial and rational functions. The facts are listed in
Theorem 1.
2
1
–1 0 1
y
x
2
1
–1 0 1
y
x
2
1
–1 0 1
y
x
(a) f(x) =
,
x (b) 2 – 1
x – 1
x2 – 1
x – 1
x ? 1
1, x = 1
g(x) = (c) h(x) = x + 1
Figure 2.2 lim
x?1
f x  lim
x?1
gx  lim
x?1
hx  2
Here are some examples of how Theorem 1 can be used to find limits of polynomial
and rational functions.
EXAMPLE 3 Using Properties of Limits
Use the observations limx?c k  k and limx?c x  c, and the properties of limits to
find the following limits.
(a) lim
x?c
x3  4x2  3 (b) lim
x?c
 x
4
x

2
x

2
5
1

SOLUTION
(a) lim
x?c
x3  4x2  3  lim
x?c
x3  lim
x?c
4x2  lim
x?c
3 Sum and Difference Rules
 c3  4c2  3 Product and Constant
(b) lim
x?c
 x
4
x

2
x

2
5
1
 
lim
x?c
lim
x?
x
c
4
x

2
x

2
5


1
 Quotient Rule
 Sum and Difference Rules
 
c 4
c

2
c

2
5
1
 Product Rule
Now try Exercises 5 and 6.
Example 3 shows the remarkable strength of Theorem 1. From the two simple observations
that limx?c k  k and limx?c x  c, we can immediately work our way to limits of
polynomial functions and most rational functions using substitution.
lim
x?c
x4  lim
x?c
x2  lim
x?c
1

lim
x?c
x2  lim
x?c
5
62 Chapter 2 Limits and Continuity
6. Power Rule: If r and s are integers, s  0, then
lim
x?c
 f xrs  Lrs
provided that Lrs is a real number.
The limit of a rational power of a function is that power of the limit of the function,
provided the latter is a real number.
THEOREM 2 Polynomial and Rational Functions
1. If f x  anxn  an1xn1 …  a0 is any polynomial function and c is any
real number, then
lim
x?c
f x  f c  ancn  an1cn1 …  a0.
2. If f x and g(x) are polynomials and c is any real number, then
lim
x?c
g
f 

x
x

 
 
g
f 

c
c

 
, provided that gc  0.
Multiple Rules
EXAMPLE 4 Using Theorem 2
(a) lim
x?3
x22  x  322  39
(b) lim
x?2
 x
2 
x 
2x
2
4
  
22 
2 
22
2
 4
  
1
4
2
  3
Now try Exercises 9 and 11.
As with polynomials, limits of many familiar functions can be found by substitution at
points where they are defined. This includes trigonometric functions, exponential and logarithmic
functions, and composites of these functions. Feel free to use these properties.
EXAMPLE 5 Using the Product Rule
Determine lim
x?0

tan
x
x
.
SOLUTION
Solve Graphically The graph of f x  tan xx in Figure 2.3 suggests that the limit
exists and is about 1.
Confirm Analytically Using the analytic result of Exercise 75, we have
lim
x?0

tan
x
x
 lim
x?0 (
sin
x
x
•
co
1
sx  ) tan x = 
c
s
o
in
s
x
x 
 lim
x?0

sin
x
x
• lim
x?0

co
1
sx  Product Rule
 1 •
co
1
s 0
 1 • 
1
1
  1.
Now try Exercise 27.
Sometimes we can use a graph to discover that limits do not exist, as illustrated by
Example 6.
EXAMPLE 6 Exploring a Nonexistent Limit
Use a graph to show that
lim
x?2
 x
x
3


2
1
does not exist.
SOLUTION
Notice that the denominator is 0 when x is replaced by 2, so we cannot use substitution
to determine the limit. The graph in Figure 2.4 of f (x)  (x3  1x  2) strongly suggests
that as x?2 from either side, the absolute values of the function values get very
large. This, in turn, suggests that the limit does not exist.
Now try Exercise 29.
One-sided and Two-sided Limits
Sometimes the values of a function f tend to different limits as x approaches a number c
from opposite sides. When this happens, we call the limit of f as x approaches c from the
Section 2.1 Rates of Change and Limits 63
[–p, p] by [–3, 3]
Figure 2.3 The graph of
f x  tan xx
suggests that f x?1 as x?0. (Example 5)
[–10, 10] by [–100, 100]
Figure 2.4 The graph of
f (x)  (x3  1x  2)
obtained using parametric graphing to produce
a more accurate graph. (Example 6)
right the right-hand limit of f at c and the limit as x approaches c from the left the lefthand
limit of f at c. Here is the notation we use:
right-hand: lim
x?c
f x The limit of f as x approaches c from the right.
left-hand: lim
x?c
f x The limit of f as x approaches c from the left.
EXAMPLE 7 Function Values Approach Two Numbers
The greatest integer function f (x)  int x has different right-hand and left-hand limits at
each integer, as we can see in Figure 2.5. For example,
lim
x?3
int x  3 and lim
x?3
int x  2.
The limit of int x as x approaches an integer n from the right is n, while the limit as x approaches
n from the left is n – 1.
Now try Exercises 31 and 32.
We sometimes call limx?c f x the two-sided limit of f at c to distinguish it from the
one-sided right-hand and left-hand limits of f at c. Theorem 3 shows how these limits are
related.
64 Chapter 2 Limits and Continuity
THEOREM 3 One-sided and Two-sided Limits
A function f (x) has a limit as x approaches c if and only if the right-hand and lefthand
limits at c exist and are equal. In symbols,
lim
x?c
f x  L?lim
x?c
f x  L and lim
x?c
f x  L.
Thus, the greatest integer function f (x)  int x of Example 7 does not have a limit as
x?3 even though each one-sided limit exists.
EXAMPLE 8 Exploring Right- and Left-Hand Limits
All the following statements about the function y  f (x) graphed in Figure 2.6 are true.
At x  0: lim
x?0
f