1.2. THE NOTION OF LIMIT
11
1.2 The notion of limit
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is the mathematical notion of limit and what role do limits play in the study
of functions?
• What is the meaning of the notation lim
x→a
f (x) = L?
• How do we go about determining the value of the limit of a function at a point?
• How do we manipulate average velocity to compute instantaneous velocity??
Introduction
Functions are at the heart of mathematics: a function is a process or rule that associates
each individual input to exactly one corresponding output. Students learn in courses prior
to calculus that there are many different ways to represent functions, including through
formulas, graphs, tables, and even words. For example, the squaring function can be
thought of in any of these ways. In words, the squaring function takes any real number x
and computes its square. The formulaic and graphical representations go hand in hand,
as y = f (x) = x 2 is one of the simplest curves to graph. Finally, we can also partially
represent this function through a table of values, essentially by listing some of the ordered
pairs that lie on the curve, such as (−2, 4), (−1, 1), (0, 0), (1, 1), and (2, 4).
Functions are especially important in calculus because they often model important
phenomena – the location of a moving object at a given time, the rate at which an
automobile is consuming gasoline at a certain velocity, the reaction of a patient to the
size of a dose of a drug – and calculus can be used to study how these output quantities
change in response to changes in the input variable. Moreover, thinking about concepts
like average and instantaneous velocity leads us naturally from an initial function to a
related, sometimes more complicated function. As one example of this, think about the
falling ball whose position function is given by s(t) = 64 − 16t 2 and the average velocity of
the ball on the interval [1, x]. Observe that
AV [1, x] =
s(x) − s(1)
x − 1
=
(64 − 16x 2 ) − (64 − 16)
x − 1
=
16 − 16x 2
x − 1
.
Now, two things are essential to note: this average velocity depends on x (indeed, AV [1, x]
is a function of x), and our most focused interest in this function occurs near x = 1, which
is where the function is not defined. Said differently, the function g(x) =
16−16x 2
x−1
tells us
the average velocity of the ball on the interval from t = 1 to t = x, and if we are interested
11
1.2 The notion of limit
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is the mathematical notion of limit and what role do limits play in the study
of functions?
• What is the meaning of the notation lim
x→a
f (x) = L?
• How do we go about determining the value of the limit of a function at a point?
• How do we manipulate average velocity to compute instantaneous velocity??
Introduction
Functions are at the heart of mathematics: a function is a process or rule that associates
each individual input to exactly one corresponding output. Students learn in courses prior
to calculus that there are many different ways to represent functions, including through
formulas, graphs, tables, and even words. For example, the squaring function can be
thought of in any of these ways. In words, the squaring function takes any real number x
and computes its square. The formulaic and graphical representations go hand in hand,
as y = f (x) = x 2 is one of the simplest curves to graph. Finally, we can also partially
represent this function through a table of values, essentially by listing some of the ordered
pairs that lie on the curve, such as (−2, 4), (−1, 1), (0, 0), (1, 1), and (2, 4).
Functions are especially important in calculus because they often model important
phenomena – the location of a moving object at a given time, the rate at which an
automobile is consuming gasoline at a certain velocity, the reaction of a patient to the
size of a dose of a drug – and calculus can be used to study how these output quantities
change in response to changes in the input variable. Moreover, thinking about concepts
like average and instantaneous velocity leads us naturally from an initial function to a
related, sometimes more complicated function. As one example of this, think about the
falling ball whose position function is given by s(t) = 64 − 16t 2 and the average velocity of
the ball on the interval [1, x]. Observe that
AV [1, x] =
s(x) − s(1)
x − 1
=
(64 − 16x 2 ) − (64 − 16)
x − 1
=
16 − 16x 2
x − 1
.
Now, two things are essential to note: this average velocity depends on x (indeed, AV [1, x]
is a function of x), and our most focused interest in this function occurs near x = 1, which
is where the function is not defined. Said differently, the function g(x) =
16−16x 2
x−1
tells us
the average velocity of the ball on the interval from t = 1 to t = x, and if we are interested