ge
of applications, especially in the sciences and in engineering. What
makes these methods and applications part of calculus is that they all
rely ultimately on the concept of a limit. We will see throughout the
text how limits allow us to make computations and solve problems
that cannot be solved using algebra alone.
This chapter introduces the limit concept and focuses on those
aspects of limits that will be used in Chapter 3 to develop differen-
tial calculus. The rst section, intended as motivation, discusses how
limits arise in the study of rates of change and tangent lines.
2.1
Limits, Rates of Change, and Tangent Lines
Rates of change play a role whenever we study the relationship between two changing
quantities. Velocity is a familiar example (the rate of change of position with respect to
time), but there are many others, such as: The infection rate of an epidemic (newly infected individuals per month) Mileage of an automobile (miles per gallon) Rate of change of atmospheric temperature with respect to altitude
Roughly speaking, if y and x are related quantities, the rate of change should tell us how
much y changes in response to a unit change in x. However, this has a clear meaning only
if the rate of change is constant. If a car travels at a constant velocity of 40 mph, then its
position changes by 40 miles for each unit change in time (the unit being 1 hour). If the
trip lasts only half an hour, its position changes by 20 miles, and in general, the change
in position is 40t miles, where t is the length of time in hours. In other words,
Change in position
= velocity � time
This simple formula is no longer valid if the velocity is changing and, in fact, it is not
even meaningful. Indeed, if the automobile is accelerating or decelerating, which velocity
would we use in the formula? It is possible to compute the change in position when
velocity is changing, but this involves both differential and integral calculus. Differential
calculus uses the limit concept to dene the instantaneous velocity of an object, and
integral calculus tells us how to compute the change in position in terms of instantaneous
velocity. This is just one example of how calculus provides the concepts and tools for
modeling real-world phenomena.
In this section, we discuss velocity and other rates of change, emphasizing the graph-
ical interpretation in terms of tangent lines. At this stage, we cannot dene tangent lines
41 42
C H A P T E R 2
L I M I T S
precisely. This will have to wait until Chapter 3. In the meantime, you can think of a
tangent line as a line that skims a curve at a point as in Figure 1.
(A)
(B)
(C)
FIGURE 1
The line is tangent in (A) and
(B) but not in (C).
HISTORICAL
PERSPECTIVE
Philosophy is written in this grand bookI mean the
universewhich stands continually open to our gaze,
but it cannot be understood unless one rst learns to
comprehend the language
. . . in which it is written.
It is written in the language of mathematics
. . .
G
ALILEO
G
ALILEI
, 1623
The scientic revolution of the sixteenth and
seventeenth centuries reached its high point in
the work of Isaac Newton (16431727), who
was the rst scientist to show that the physical
world, despite its complexity and diversity, is
governed by a small number of universal laws,
such as the laws of motion and gravitation. One
of Newtons great insights was that the univer-
sal laws do not describe the world as it is, now
or at any particular time, but rather how the
world changes over time in response to forces.
To express the laws in mathematical form, New-
ton invented calculus, which he understood to
be the mathematics of change.
More than fty years before the work
of Newton, the astronomer Johannes Kepler
(15711630) discovered his three laws of plan-
etary motion, including the law stating that the
path of a planet around the sun is an ellipse.
Kepler found these laws through a painstak-
ing analysis of astronomical data, but he could
not explain why they were true. Newtons laws
explain the motion of any objectplanet or
pebblein terms of the forces acting on it. Ac-
cording to Newton, the planets, if left undis-
turbed, will travel in straight lines. Since their
paths are elliptical, some forcein this case, the
gravitational force of the sunmust be acting to
make them change direction continuously. In a
show of unmatched mathematical skill, Newton
proved that Keplers laws follow from Newtons
own inverse square law of gravity. Ironically,
the proof Newton chose to publish in 1687 in
his magnum opus Principia Mathematica uses
geometrical arguments rather than the calculus
Newton himself had invented.
Velocity
Consider an object traveling in a straight line (linear motion). When we speak of its ve-
The terms speed and velocity, though
sometimes used interchangeably, do not
have the same meaning. For linear motion,
velocity may be positive or negative
(indicating the direction of motion). Speed
is the absolute value of velocity and is
always positive.
locity, we usually mean the instantaneous velocity, which tells us the speed and direction
of the object at a particular moment. As suggested above, however, instantaneous veloc-
ity does not have a straightforward denition. It is simpler to dene the average velocity
over a given time interval as the ratio
Average velocity
= change in position
length of time interval
For example, if a car travels 200 miles in 4 hours, then its average velocity during this
4-hour period is
200
4
= 50 mph. At any given moment the car may be going faster or
slower than the average.
Unlike average velocity, we cannot dene instantaneous velocity as a ratio because
we would have to divide by the length of the time interval (which is zero). However, S E C T I O N 2.1
Limits, Rates of Change, and Tangent Lines
43
we can estimate instantaneous velocity by computing average velocity over successively
smaller time intervals. The guiding principle is the following: Average velocity over a very
small time interval is very close to instantaneous velocity. To explore this idea further,
we introduce some notation.
The Greek letter
(delta) is commonly used to denote the change in a function or
variable. If s
(t) is the position of an object (distance from the origin) at time t and [t
0
, t
1
]
is a time interval, we set
s
= s(t
1
) s(t
0
) = change in position
t
= t
1
t
0
= change in time (length of interval)
For t
1
= t
0
,
Average velocity over
[t
0
, t
1
] = st = s(t
1
) s(t
0
)
t
1
t
0
The change in position
s is also called the displacement, or net change in position.
One motion we will study is the motion of an object falling to earth under the in-
t
= 0
t
= 0.5
0
16
4
t
= 1
FIGURE 2
Distance traveled by a falling
object after t seconds is s
(t) = 16t
2
feet.
uence of gravity (assuming no air resistance). Galileo discovered that if the object is
released at time t
= 0 from a state of rest, then the distance traveled after t seconds is
given by the formula (see Figure 2)
s
(t) = 16t
2
ft
E X A M P L E 1
A stone is released from a state of rest and falls to earth. Estimate the
instantaneous velocity at t
= 0.5 s by calculating average velocity over several small time
intervals.
Solution The distance traveled by the stone after t seconds is s
(t) = 16t
2
by Galileos
formula. We use this to compute the average velocity over the ve time intervals listed in
TABLE 1
Time
Average
Interval
Velocity
[0.5, 0.6]
17.6
[0.5, 0.55]
16.8
[0.5, 0.51]
16.16
[0.5, 0.505]
16.08
[0.5, 0.5001]
16.0016
Table 1. To nd the stones average velocity over the rst interval
[t
0
, t
1
] = [0.5, 0.6], we
compute the changes:
s
= s(0.6) s(0.5) = 16(0.6)
2
16(0.5)
2
= 5.76 4 = 1.76 ft
t
= 0.6 0.5 = 0.1 s
The average velocity over
[0.5, 0.6] is the ratio
s
t =
s
(0.6) s(0.5)
0
.6 0.5
= 1.76
0
.1 = 17.6 ft/s
Table 1 shows the results of similar calculations for several intervals of successively
Note that there is nothing special about the
particular time intervals shown in Table 1.
We are looking for a trend and we could
have chosen any intervals
[0.5, t] for values
of t approaching 0
.5.
shorter lengths. It is clear that the average velocities get closer to 16 ft
/s as the time
interval shrinks:
17
.6, 16.8, 16.16, 16.08, 16.0016
We express this by saying that the average velocity converges to 16 or that 16 is the limit
of average velocity as the length of the time interval shrinks to zero. This suggests that
the instantaneous velocity at t
= 0.5 is 16 ft/s.
The conclusion in this example is correct, but we need to base it on more formal
and precise reasoning. This will be done in Chapter 3, after we have developed the limit
concept. 44
C H A P T E R 2
L I M I T S
Graphical Interpretation of Velocity
The idea that average velocity converges to instantaneous velocity as we shorten the time
interval has a vivid interpretation in terms of secant lines where, by denition, a secant
line is a line through two points on a curve.
Consider the graph of position s
(t) for an object traveling in a straight line as in
Figure 3. The ratio we use to dene average velocity over
[t
0
, t
1
] is nothing more than the
slope of the secant line through the points
(t
0
, s(t
0
)) and (t
1
, s(t
1
)). For t
1
= t
0
,
Average velocity
= slope of secant line = st = s(t
1
) s(t
0
)
t
1
t
0
Secant line
t (time)
s (t
0
)
s (t
1
)
s = s(t
1
)
s(t
0
)
t = t
1

t
0
t
1
t
0
s (position)
FIGURE 3
Average velocity over
[t
0
, t
1
]
is equal to the slope of the secant line.
By interpreting average velocity as a slope, we can visualize what happens as the
time interval gets smaller. Figure 4 shows the graph of s
(t) = 16t
2
together with the
secant lines for the time interval