1.2. THE NOTION OF LIMIT
17
(b) lim
x→0
(2 + x) 3 − 8
x
(c) lim
x→0
√
x + 1 − 1
x
⊳
This concludes a rather lengthy introduction to the notion of limits. It is important to
remember that our primary motivation for considering limits of functions comes from our
interest in studying the rate of change of a function. To that end, we close this section by
revisiting our previous work with average and instantaneous velocity and highlighting the
role that limits play.
Instantaneous Velocity
Suppose that we have a moving object whose position at time t is given by a function s. We
know that the average velocity of the object on the time interval [a, b] is AV [a,b] =
s(b)−s(a)
b−a .
We define the instantaneous velocity at a to be the limit of average velocity as b approaches
a. Note particularly that as b → a, the length of the time interval gets shorter and
shorter (while always including a). In Section 1.3, we will introduce a helpful shorthand
notation to represent the instantaneous rate of change. For now, we will write IV t=a for
the instantaneous velocity at t = a, and thus
IV t=a = lim
b→a
AV [a,b] = lim
b→a
s(b) − s(a)
b − a
.
Equivalently, if we think of the changing value b as being of the form b = a + h, where h is
some small number, then we may instead write
IV t=a = lim
h→0
AV [a,a+h] = lim
h→0
s(a + h) − s(a)
h
.
Again, the most important idea here is that to compute instantaneous velocity, we take a
limit of average velocities as the time interval shrinks. Two different activities offer the
opportunity to investigate these ideas and the role of limits further.
Activity 1.5.
Consider a moving object whose position function is given by s(t) = t 2 , where s is
measured in meters and t is measured in minutes.
(a) Determine the most simplified expression for the average velocity of the object
on the interval [3, 3 + h], where h > 0.
(b) Determine the average velocity of the object on the interval [3, 3.2]. Include
units on your answer.
17
(b) lim
x→0
(2 + x) 3 − 8
x
(c) lim
x→0
√
x + 1 − 1
x
⊳
This concludes a rather lengthy introduction to the notion of limits. It is important to
remember that our primary motivation for considering limits of functions comes from our
interest in studying the rate of change of a function. To that end, we close this section by
revisiting our previous work with average and instantaneous velocity and highlighting the
role that limits play.
Instantaneous Velocity
Suppose that we have a moving object whose position at time t is given by a function s. We
know that the average velocity of the object on the time interval [a, b] is AV [a,b] =
s(b)−s(a)
b−a .
We define the instantaneous velocity at a to be the limit of average velocity as b approaches
a. Note particularly that as b → a, the length of the time interval gets shorter and
shorter (while always including a). In Section 1.3, we will introduce a helpful shorthand
notation to represent the instantaneous rate of change. For now, we will write IV t=a for
the instantaneous velocity at t = a, and thus
IV t=a = lim
b→a
AV [a,b] = lim
b→a
s(b) − s(a)
b − a
.
Equivalently, if we think of the changing value b as being of the form b = a + h, where h is
some small number, then we may instead write
IV t=a = lim
h→0
AV [a,a+h] = lim
h→0
s(a + h) − s(a)
h
.
Again, the most important idea here is that to compute instantaneous velocity, we take a
limit of average velocities as the time interval shrinks. Two different activities offer the
opportunity to investigate these ideas and the role of limits further.
Activity 1.5.
Consider a moving object whose position function is given by s(t) = t 2 , where s is
measured in meters and t is measured in minutes.
(a) Determine the most simplified expression for the average velocity of the object
on the interval [3, 3 + h], where h > 0.
(b) Determine the average velocity of the object on the interval [3, 3.2]. Include
units on your answer.