1.2. THE NOTION OF LIMIT
13
without bound. We put off the study of the latter idea until further along in the course
when we will have some helpful calculus tools for understanding the end behavior of
functions. Here, we focus on what it means to say that “a function f has limit L as x
approaches a.” To begin, we think about a recent example.
In Preview Activity 1.2, you saw that for the given function g, as x gets closer and
closer (but not equal) to 0, g(x) gets as close as we want to the value 4. At first, this may
feel counterintuitive, because the value of g(0) is 1, not 4. By their very definition, limits
regard the behavior of a function arbitrarily close to a fixed input, but the value of the
function at the fixed input does not matter. More formally 1 , we say the following.
Definition 1.1. Given a function f , a fixed input x = a, and a real number L, we say that
f has limit L as x approaches a, and write
lim
x→a
f (x) = L
provided that we can make f (x) as close to L as we like by taking x sufficiently close (but
not equal) to a. If we cannot make f (x) as close to a single value as we would like as x
approaches a, then we say that f does not have a limit as x approaches a.
For the function g pictured in Figure 1.5, we can make the following observations:
lim
x→−1
g(x) = 3, lim
x→0
g(x) = 4, and lim
x→2
g(x) = 1,
but g does not have a limit as x → 1. When working graphically, it suffices to ask if the
function approaches a single value from each side of the fixed input, while understanding
that the function value right at the fixed input is irrelevant. This reasoning explains the
values of the first three stated limits. In a situation such as the jump in the graph of g at
x = 1, the issue is that if we approach x = 1 from the left, the function values tend to get
as close to 3 as we’ d like, but if we approach x = 1 from the right, the function values get
as close to 2 as we’ d like, and there is no single number that all of these function values
approach. This is why the limit of g does not exist at x = 1.
For any function f , there are typically three ways to answer the question “does f have
a limit at x = a, and if so, what is the limit?” The first is to reason graphically as we
have just done with the example from Preview Activity 1.2. If we have a formula for f (x),
there are two additional possibilities: (1) evaluate the function at a sequence of inputs that
approach a on either side, typically using some sort of computing technology, and ask if
the sequence of outputs seems to approach a single value; (2) use the algebraic form of the
function to understand the trend in its output as the input values approach a. The first
approach only produces an approximation of the value of the limit, while the latter can
1 What follows here is not what mathematicians consider the formal definition of a limit. To be completely
precise, it is necessary to quantify both what it means to say “as close to L as we like” and “sufficiently close
to a.” That can be accomplished through what is traditionally called the epsilon-delta definition of limits.
The definition presented here is sufficient for the purposes of this text.
13
without bound. We put off the study of the latter idea until further along in the course
when we will have some helpful calculus tools for understanding the end behavior of
functions. Here, we focus on what it means to say that “a function f has limit L as x
approaches a.” To begin, we think about a recent example.
In Preview Activity 1.2, you saw that for the given function g, as x gets closer and
closer (but not equal) to 0, g(x) gets as close as we want to the value 4. At first, this may
feel counterintuitive, because the value of g(0) is 1, not 4. By their very definition, limits
regard the behavior of a function arbitrarily close to a fixed input, but the value of the
function at the fixed input does not matter. More formally 1 , we say the following.
Definition 1.1. Given a function f , a fixed input x = a, and a real number L, we say that
f has limit L as x approaches a, and write
lim
x→a
f (x) = L
provided that we can make f (x) as close to L as we like by taking x sufficiently close (but
not equal) to a. If we cannot make f (x) as close to a single value as we would like as x
approaches a, then we say that f does not have a limit as x approaches a.
For the function g pictured in Figure 1.5, we can make the following observations:
lim
x→−1
g(x) = 3, lim
x→0
g(x) = 4, and lim
x→2
g(x) = 1,
but g does not have a limit as x → 1. When working graphically, it suffices to ask if the
function approaches a single value from each side of the fixed input, while understanding
that the function value right at the fixed input is irrelevant. This reasoning explains the
values of the first three stated limits. In a situation such as the jump in the graph of g at
x = 1, the issue is that if we approach x = 1 from the left, the function values tend to get
as close to 3 as we’ d like, but if we approach x = 1 from the right, the function values get
as close to 2 as we’ d like, and there is no single number that all of these function values
approach. This is why the limit of g does not exist at x = 1.
For any function f , there are typically three ways to answer the question “does f have
a limit at x = a, and if so, what is the limit?” The first is to reason graphically as we
have just done with the example from Preview Activity 1.2. If we have a formula for f (x),
there are two additional possibilities: (1) evaluate the function at a sequence of inputs that
approach a on either side, typically using some sort of computing technology, and ask if
the sequence of outputs seems to approach a single value; (2) use the algebraic form of the
function to understand the trend in its output as the input values approach a. The first
approach only produces an approximation of the value of the limit, while the latter can
1 What follows here is not what mathematicians consider the formal definition of a limit. To be completely
precise, it is necessary to quantify both what it means to say “as close to L as we like” and “sufficiently close
to a.” That can be accomplished through what is traditionally called the epsilon-delta definition of limits.
The definition presented here is sufficient for the purposes of this text.