Chapter 1.
Functions and Limits
Section 1.5: The Limit of a function
• Finding limits numerically and Graphically
• One-Sided Limits
• How can a limit fail to exist?
• Infinite limits, vertical asymptote
Chapter 1. Functions and Limits
Example. Consider the function
𝑥𝑥 − 1
𝑓𝑓(𝑥𝑥) =
𝑥𝑥 2 − 1
Q1. What is the domain of definition of the function 𝑓𝑓(𝑥𝑥)?
Q2. What happens to the values of the function 𝑓𝑓(𝑥𝑥) when evaluated at values of 𝑥𝑥
getting closer and closer to 1? but not equal to 1.
We see that from the table and the graph of 𝑓𝑓(𝑥𝑥) the closer 𝑥𝑥 is to 1 (on either side of 1),
the closer 𝑓𝑓(𝑥𝑥) is to 0.5.
𝑥𝑥−1
We say that “the limit of 𝑓𝑓(𝑥𝑥) = as 𝑥𝑥 approaches 1 is equal to 0.5”.
𝑥𝑥 2 −1
Notation:
𝑥𝑥 − 1
lim = 0.5
𝑥𝑥→1 𝑥𝑥 2 − 1
Suppose a function 𝒇𝒇 is defined for all 𝒙𝒙 in an open interval containing 𝒂𝒂 (except possibly at
𝑥𝑥 = 𝑎𝑎).
If 𝑓𝑓(𝑥𝑥) is arbitrarily close to some number 𝐿𝐿 for all 𝑥𝑥 sufficiently close (but not equal) to 𝑎𝑎,
then we write
lim 𝑓𝑓(𝑥𝑥) = 𝐿𝐿
𝑥𝑥→𝑎𝑎
and we say the limit of 𝒇𝒇(𝒙𝒙) as 𝒙𝒙 approaches 𝒂𝒂 equals 𝑳𝑳.
Chapter 1. Functions and Limits
One-Sided Limits
Right-Sided limit:
Suppose the function 𝒇𝒇 is defined for all 𝒙𝒙 near 𝒂𝒂 with 𝒙𝒙 > 𝒂𝒂. If 𝑓𝑓(𝑥𝑥) is arbitrarily close to 𝐿𝐿
for all 𝑥𝑥 sufficiently close to 𝑎𝑎 with 𝑥𝑥 > 𝑎𝑎, we write
lim+ 𝑓𝑓(𝑥𝑥) = 𝐿𝐿
𝑥𝑥→𝑎𝑎
and we say the limit of 𝒇𝒇 as 𝒙𝒙 approaches 𝒂𝒂 from the right equals 𝑳𝑳.
Left-Sided limit:
Suppose the function 𝒇𝒇(𝒙𝒙) is defined for all 𝒙𝒙 near 𝒂𝒂 with 𝒙𝒙 < 𝒂𝒂. If 𝑓𝑓(𝑥𝑥) is arbitrarily close to
𝐿𝐿 for all 𝑥𝑥 sufficiently close to 𝑎𝑎 with 𝑥𝑥 < 𝑎𝑎, we write
lim− 𝑓𝑓(𝑥𝑥) = 𝐿𝐿
𝑥𝑥→𝑎𝑎
and we say the limit of 𝒇𝒇(𝒙𝒙) as 𝒙𝒙 approaches 𝒂𝒂 from the left equals 𝑳𝑳.
Two-Sided Limits:
The limit lim 𝑓𝑓(𝑥𝑥) = 𝐿𝐿 (two-sided limit) exists if and only if:
𝑥𝑥→𝑎𝑎
lim 𝑓𝑓(𝑥𝑥) = 𝐿𝐿 𝑎𝑎𝑎𝑎𝑎𝑎 lim− 𝑓𝑓(𝑥𝑥) = 𝐿𝐿
𝑥𝑥→𝑎𝑎+ 𝑥𝑥→𝑎𝑎
Remark.
lim 𝑓𝑓(𝑥𝑥) does not exist (DNE) if and only if
𝑥𝑥→𝑎𝑎
• lim 𝑓𝑓(𝑥𝑥) or lim− 𝑓𝑓(𝑥𝑥) does not exist, or
𝑥𝑥→𝑎𝑎+ 𝑥𝑥→𝑎𝑎
lim 𝑓𝑓(𝑥𝑥) ≠ lim− 𝑓𝑓(𝑥𝑥)
𝑥𝑥→𝑎𝑎+ 𝑥𝑥→𝑎𝑎
Example.
0 𝑖𝑖𝑖𝑖 𝑡𝑡 < 0
The Heaviside function defined by: 𝐻𝐻(𝑡𝑡) = �
1 𝑖𝑖𝑖𝑖 𝑡𝑡 ≥ 0
lim 𝐻𝐻(𝑡𝑡) = 1, lim− 𝐻𝐻(𝑡𝑡) = 0,
𝑡𝑡→0+ 𝑡𝑡→0
The notation 𝑡𝑡 → 0− indicates that we consider only values of 𝑡𝑡 that are less than 0. Likewise,
𝑡𝑡 → 0+ indicates that we consider only values of 𝑡𝑡 that are greater than 0.
Chapter 1. Functions and Limits
Example. Use the graph of the function 𝑓𝑓 to state the values (if they exist) of the following:
a) lim− 𝑔𝑔(𝑥𝑥) =
𝑥𝑥→2
b) lim+ 𝑔𝑔(𝑥𝑥) =
𝑥𝑥→2
c) lim 𝑔𝑔(𝑥𝑥) =
𝑥𝑥→2
d) lim− 𝑔𝑔(𝑥𝑥) =
𝑥𝑥→5
e) lim+ 𝑔𝑔(𝑥𝑥) =
𝑥𝑥→5
f) lim 𝑔𝑔(𝑥𝑥) =
𝑥𝑥→5
How Can a Limit Fail to Exist?
1
Example. Find lim if it exists.
𝑥𝑥→0 𝑥𝑥 2
2 1
As 𝑥𝑥 becomes close to 0, 𝑥𝑥 also becomes close to 0, and becomes very large. (see table
𝑥𝑥 2
1
below). The lim does not exist ( see graph)
𝑥𝑥→0 𝑥𝑥 2
Chapter 1. Functions and Limits
Infinite Limit ; Vertical asymptote
Definition: Let 𝑓𝑓(𝑥𝑥) be a function defined on both sides of 𝑎𝑎, except possibly at 𝑎𝑎
itself. Then
lim 𝑓𝑓(𝑥𝑥) = ∞
𝑥𝑥→𝑎𝑎
Means that the values of 𝑓𝑓(𝑥𝑥) can be made arbitrarily large by taking 𝑥𝑥 sufficiently
close to 𝑎𝑎 but not equal to 𝑎𝑎.
Another notation for lim 𝑓𝑓(𝑥𝑥) = ∞ is 𝑓𝑓(𝑥𝑥) → ∞ as 𝑥𝑥 → 𝑎𝑎
𝑥𝑥→𝑎𝑎
The symbol ∞ is not a number. But the expression lim 𝑓𝑓(𝑥𝑥) = ∞ is often read:
𝑥𝑥→𝑎𝑎
“the limit of 𝒇𝒇(𝒙𝒙), as 𝒙𝒙 approaches 𝒂𝒂, is infinity”
Or
“𝒇𝒇(𝒙𝒙) becomes infinite, as 𝑥𝑥 approaches 𝑎𝑎 “
This definition is illustrated graphically in the following Figure
𝐥𝐥𝐥𝐥𝐥𝐥 𝒇𝒇(𝒙𝒙) = ∞
𝒙𝒙→𝒂𝒂
A similar sort of limit, for functions that become large negative as x gets close to a,
illustrated in Figure below
𝐥𝐥𝐥𝐥𝐥𝐥 𝒇𝒇(𝒙𝒙) = −∞
𝒙𝒙→𝒂𝒂
Chapter 1. Functions and Limits
Definition: Let 𝑓𝑓(𝑥𝑥) be a function defined on both sides of 𝑎𝑎, except possibly at 𝑎𝑎
itself. Then
lim 𝑓𝑓(𝑥𝑥) = −∞
𝑥𝑥→𝑎𝑎
This means that the values of 𝑓𝑓(𝑥𝑥) can be made arbitrarily large negative by taking 𝑥𝑥
sufficiently close to 𝑎𝑎 but not equal to 𝑎𝑎.
Another notation for lim 𝑓𝑓(𝑥𝑥) = −∞ is 𝑓𝑓(𝑥𝑥) → −∞ as 𝑥𝑥 → 𝑎𝑎
𝑥𝑥→𝑎𝑎
The symbol -∞ is not a number. But the expression lim 𝑓𝑓(𝑥𝑥) = −∞ is often read:
𝑥𝑥→𝑎𝑎
“the limit of 𝒇𝒇(𝒙𝒙), as 𝒙𝒙 approaches 𝒂𝒂, is negative infinity”
Or
“𝒇𝒇(𝒙𝒙) decreases without bound infinite, as 𝑥𝑥 approaches 𝑎𝑎 “
Example: As an example, we have:
1
lim �− � = −∞ ∎
𝑥𝑥→0 𝑥𝑥 2
Remark:
Similar definitions can be given for the one-sided infinite limits
lim 𝑓𝑓(𝑥𝑥) = ∞ lim 𝑓𝑓(𝑥𝑥) = ∞
𝑥𝑥→𝑎𝑎− 𝑥𝑥→𝑎𝑎+
lim 𝑓𝑓(𝑥𝑥) = −∞ lim 𝑓𝑓(𝑥𝑥) = − ∞
𝑥𝑥→𝑎𝑎− 𝑥𝑥→𝑎𝑎+
Chapter 1. Functions and Limits
Illustrations of these four cases are given in the Figure below:
Definition: The line 𝑥𝑥 = 𝑎𝑎 is called a vertical asymptote of the curve 𝑦𝑦 = 𝑓𝑓(𝑥𝑥)
If at least one of the following statements is true:
lim 𝑓𝑓(𝑥𝑥) = ∞ lim 𝑓𝑓(𝑥𝑥) = ∞ lim 𝑓𝑓(𝑥𝑥) = ∞
𝑥𝑥→𝑎𝑎 𝑥𝑥→𝑎𝑎− 𝑥𝑥→𝑎𝑎+
lim 𝑓𝑓(𝑥𝑥) = −∞ lim 𝑓𝑓(𝑥𝑥) = −∞ lim 𝑓𝑓(𝑥𝑥) = − ∞
𝑥𝑥→𝑎𝑎 𝑥𝑥→𝑎𝑎− 𝑥𝑥→𝑎𝑎+
Chapter 1. Functions and Limits
2𝑥𝑥
Example: The curve 𝑦𝑦 = has a vertical asymptote 𝑥𝑥 = 3
𝑥𝑥−3
