Unit 1: Limit of a Function
Lesson 1.1
Evaluating Limits through Table of Values
Contents
Introduction
1
Learning Objectives
2
Warm Up
2
Learn about It!
5
Intuitive Definition of the Limit of a Function
One-Sided Limits
Infinite Limits
8
15
22
Key Points
30
Check Your Understanding
32
Challenge Yourself
33
Photo Credits
34
Bibliography
34
Key to Try It!
35
Unit 1: Limit of a Function
Lesson 1.1
Evaluating Limits through Table of
Values
Introduction
When you ride a car, do you notice its speed? Do you notice the movement of the
speedometer? The car may have different speeds at any given moment. Let us say you went
out of town for a vacation and the trip lasted five hours. We can estimate the average speed
of your car for five hours from your house to your destination. We can also estimate the
average speed of your car in a particular hour or even in a particular minute. However, is it
possible to estimate the speed of your car in a particular instance?
1.1. Evaluating Limits through Table of Values
1
Unit 1: Limit of a Function
To answer this question, we need the concept of limits. What comes to your mind when you
hear the word “limit”? It could be the speed limit at a highway, laws, restrictions, a maximum
value, or a boundary. Calculus is a branch of mathematics with a foundation on the limits of
functions. Thus, a solid grasp of the different types of functions is necessary. Limits are
essential for other important Calculus topics such as derivatives and integrals.
In this lesson, we will discover more about limit of a function using table of values.
Learning Objectives
In this lesson, you should be able to do the
following:
●
Define the limit of a function.
●
Define one-sided limits.
●
Define infinite limits.
●
Estimate the limit of a function
DepEd Competency
Illustrate the limit of a function
using a table of values
(STEM_BC11LC-IIIa-1).
using tables of values.
Warm Up
Approaching the Area of the Unit Circle
15 minutes
In this activity, we will approximate the area of the unit circle using areas of polygons.
Materials
●
worksheet
●
computer
●
pen
1.1. Evaluating Limits through Table of Values
2
Unit 1: Limit of a Function
Procedure
1. Answer Part A of the worksheet. This part serves as a review on the area of a circle.
2. In your computer, open the GeoGebra applet through this link:
https://www.geogebra.org/m/q59yJf4t
3. Explore the applet by changing the values the radius 𝑟 of the circle and the number
of sides 𝑛 of the inscribed polygon. Observe the values for the area of the circle, the
area of the inscribed polygon, and the difference between their areas.
4. Set 𝑟 to 1.
5. Complete the table in Part B of the worksheet by changing the values of 𝑛 and writing
the area of the resulting polygon. Make sure that the radius of the circle is constant
and equal to 1.
6. Answer the guide questions.
Guide Questions
1. What happens to the area of the polygon as the number of sides 𝑛 increases?
2. What do you think is the value being approached by the areas of the polygons as the
number of sides 𝑛 increases?
3. If you increase the value of 𝑛 further to larger values, do you think that the area of the
resulting polygon will be equal to your answer in No. 2? Explain your answer.
1.1. Evaluating Limits through Table of Values
3
Unit 1: Limit of a Function
WORKSHEET: Approximating the Area of a Circle by Finding the Area of Inscribed Polygons
A. Answer the following questions.
1. What is the formula for the area of a circle? ___________________
2. A circle with its radius equal to 1 is called a unit circle. What is the area of a unit
circle? ___________________
B. Open the GeoGebra applet with the following link on your computer:
Inscribed Polygon to Approximate Area of Circle
Jake Aust, “Inscribed Polygon to Approximate Area of Circle,”
GeoGebra, https://www.geogebra.org/m/q59yJf4t, last
accessed on December 13, 2019.
C. Set the radius 𝒓 of the circle to 1. Afterwards, complete the table below by writing
the areas of the polygons given the number of sides 𝑛.
Data Table
Table 1.1.1. Areas of the polygons given the number of sides 𝑛
Polygon
Number of sides 𝒏
Triangle
3
Square
4
Pentagon
5
Decagon
10
20-gon
20
1.1. Evaluating Limits through Table of Values
Area of polygon
4
Unit 1: Limit of a Function
30-gon
30
40-gon
40
50-gon
50
Learn about It!
In the warm-up activity, we talked about approximating the area of a unit circle by increasing
the number of sides of the inscribed polygon and getting the areas. As the number of sides
of the inscribed polygon increases, its area becomes closer to the area of the unit circle.
In this lesson, we will talk about finding the value that is being approached by a sequence of
values. We will call this the limit. Thus, the area of the unit circle is the limit of the areas of
the polygons as the number of sides increases indefinitely. Note that the area of the inscribed
polygon cannot be greater than 𝜋, which is the area of the unit circle. This makes sense visually
since the polygons are inscribed in the unit circle.
1.1. Evaluating Limits through Table of Values
5
Unit 1: Limit of a Function
Let us look at another example on the idea of approaching certain values. The square below
has an area of one square unit.
1
We can divide the square into two such that each part is 2 square unit. We can further divide
1
1
one half of the square to produce two 4-square-unit parts. We can divide 4 of the square into
1
two 8-square-unit parts, and so on. Note that there is no end to this process since we can
always divide the next fraction by two. The resulting figure is as follows.
1.1. Evaluating Limits through Table of Values
6
Unit 1: Limit of a Function
We can write these fractions as terms in an infinite series.
1 1 1 1
1
1
1
+ + +
+
+
+
+⋯
2 4 8 16 32 64 128
It can be observed that the terms in the series are decreasing and getting closer and closer to
zero, but it will never reach zero. Is there a way to get the value of this infinite series? It would
be impossible to add all the terms. However, let us see if we can estimate the value of the
infinite series by observing the value of the first 𝑛 terms using a table of values.
Table 1.1.2. Value of the infinite series from the first 𝑛 terms
Series
Value
1 1
+
2 4
0.75
1 1 1
+ +
2 4 8
0.875
1 1 1 1
+ + +
2 4 8 16
0.9375
1 1 1 1
1
+ + +
+
2 4 8 16 32
0.96875
1 1 1 1
1
1
+ + +
+
+
2 4 8 16 32 64
0.984375
1 1 1 1
1
1
1
+ + +
+
+
+
2 4 8 16 32 64 128
0.9921875
1 1 1
1
1
1
1
1
+ + +
+
+
+
+
2 4 8 16 32 64 128 256
0.99609375
1 1 1 1
1
1
1
1
1
+ + +
+
+
+
+
+
2 4 8 16 32 64 128 256 512
0.998046875
1 1 1 1
1
1
1
1
1
1
+ + +
+
+
+
+
+
+
2 4 8 16 32 64 128 256 512 1024
0.9990234375
1 1 1
1
1
1
1
1
1
1
1
+ + +
+
+
+
+
+
+
+
2 4 8 16 32 64 128 256 512 1024 2048
0.99951171875
1.1. Evaluating Limits through Table of Values
7
Unit 1: Limit of a Function
Notice that the value of the series gets closer and closer to 1 as the number of terms
increases. However, it will never reach 1. Recall that the area of the given square is 1 square
unit. Since the terms in the series are the areas of non-overlapping parts of the square, the
sum of the infinite series must be equal to 1. We can say that the limit of the value of the
infinite series is 1.
We have seen how the areas of inscribed polygons approach the area of a unit circle as the
number of sides increases and how the value of the first 𝑛 terms in an infinite series
approaches 1. In this lesson, we will observe how the value of a function approach a limit as
the independent variable gets closer to a certain value by using tables of values.
Is it correct to say that we can only get an estimate
of a limit? Why do you say so?
Intuitive Definition of the Limit of a Function
Let us investigate what happens to the value of the linear function 𝑓 (𝑥) = 𝑥 + 4 as 𝑥
approaches 2. The given table shows 𝑥-values that are very close to 2 from the left and right
sides of the number line. What happens to the value of 𝑓(𝑥) as 𝑥 approaches 2?
𝒙<𝟐
1.900
1.950
1.990
1.995
1.999
𝟐
𝒙>𝟐
2.001
2.005
2.010
2.050
2.100
𝒇(𝒙)
5.900
5.950
5.990
5.995
5.999
𝟔
𝒇(𝒙)
6.001
6.005
6.010
6.050
6.100
Notice that as the value of 𝑥 gets closer and closer to 2 from both sides, the value of 𝑓(𝑥) gets
closer and closer to 6 from both sides. Thus, we say that “the limit of the function 𝑓(𝑥) = 𝑥 + 4
as 𝑥 approaches 2 is equal to 6.”
1.1. Evaluating Limits through Table of Values
8
Unit 1: Limit of a Function
Intuitive Definition of a Limit
Suppose the function 𝑓(𝑥) is defined when 𝑥 is near 𝑐. If 𝑓 (𝑥) gets closer to a real
number 𝐿 as 𝑥 gets closer to 𝑐 (both from left and right of 𝑐), then we say that “the
limit of 𝑓(𝑥) as 𝑥 approaches 𝑐 is equal to 𝐿.” This is written as
𝐥𝐢𝐦 𝒇(𝒙) = 𝑳.
𝒙→𝒄
In this case, we say that the limit exists.
The number 𝑐 may or may not be in the domain of the function 𝑓 (𝑥).
Thus, we can write the limit of the function 𝑓(𝑥) = 𝑥 + 4 as
lim (𝑥 + 4) = 6.
𝑥→2
Based on the intuitive definition of a limit, it is important to note that the function need not
be defined at 𝑎 for a limit to exist at 𝑎.
Let’s Practice!
Example 1
Estimate the limit of the quadratic function 𝑔(𝑥) = 𝑥 2 − 6𝑥 + 14 as 𝑥 approaches 4 using tables
of values.
Solution
The domain of a quadratic function is the set of real numbers. Therefore, the limit of the given
function at 𝑥 = 4 exists. Using tables, let us find the values of 𝑔(𝑥) for values of 𝑥 that are very
close to 4. Note that we can use arbitrary 𝑥 values as long as we get as close as possible to
𝑥 = 4.
1.1. Evaluating Limits through Table of Values
9
Unit 1: Limit of a Function
Step 1:
Construct two tables with arbitrary 𝑥 values that are very close to the value
𝑥 = 4 from the left and right sides.
𝒙<𝟒
3.979
3.985
3.989
3.995
3.999
𝟒
𝒈(𝒙)
𝒙>𝟒
4.001
4.005
4.01
4.015
4.02
𝒈(𝒙)
Step 2:
Complete the table by solving the value of 𝑔(𝑥) for each 𝑥 value.
𝒙<𝟒
3.979
3.985
3.989
3.995
3.999
𝒈(𝒙)
5.958
5.97
5.978
5.99
5.998
Step 3:
𝟒
𝒙>𝟒
4.001
4.005
4.01
4.015
4.02
𝒈(𝒙)
6.002
6.01
6.02
6.03
6.04
Estimate the values that are being approached by 𝑔(𝑥) as 𝑥 approaches 4 from
the left and right sides.
The value of 𝑔(𝑥) approaches 6 from both sides. Therefore, we estimate the
limit to be equal to 6.
𝒙<𝟒
3.979
3.985
3.989
3.995
3.999
𝟒
𝒙>𝟒
4.001
4.005
4.01
4.015
4.02
𝒈(𝒙)
5.958
5.97
5.978
5.99
5.998
𝟔
𝒈(𝒙)
6.002
6.01
6.02
6.03
6.04
Thus, we write lim (𝑥 2 − 6𝑥 + 14) = 6. This is read as “the limit of (𝑥 2 − 6𝑥 + 14) as 𝑥
𝑥→4
approaches 4 is equal to 6.”
1 Try It!
Estimate the limit of the function 𝑔(𝑥) = 5𝑥 + 8 as 𝑥 approaches 1 using tables of
values.
1.1. Evaluating Limits through Table of Values
10
Unit 1: Limit of a Function
Example 2
Estimate lim √𝑥 using tables of values.
𝑥→9
Solution
Let ℎ(𝑥) = √𝑥. The domain of the radical function ℎ(𝑥) = √𝑥 is the set containing zero and all
positive real numbers. Thus, we can get the limit as 𝑥 approaches 9 from both sides.
Step 1:
Construct two tables with arbitrary 𝑥 values that are very close to the value
𝑥 = 9 from the left and right sides.
Let us select 𝑥 values that are close to 𝑥 = 9. The goal is to examine the behavior
of the value of ℎ(𝑥) as 𝑥 approaches 9 from both sides, so the number of
decimal places that we need for the 𝑥 values need not be as many as possible.
𝒙<𝟗
8.95
8.96
8.97
8.98
8.99
𝟗
𝒉(𝒙)
𝒙>𝟗
9.01
9.02
9.03
9.04
9.05
9.04
9.05
𝒉(𝒙)
Step 2:
Complete the table by solving the values of ℎ(𝑥) for each 𝑥 value.
𝒙<𝟗
8.95
8.96
8.97
𝒉(𝒙)
2.992
2.993
2.995 2.997 2.998
8.98
8.99
1.1. Evaluating Limits through Table of Values
𝟗
𝒙>𝟗
9.01
9.02
9.03
𝒉(𝒙)
3.002
3.003
3.005 3.007 3.008
11
Unit 1: Limit of a Function
Step 3:
Estimate the values that are being approached by ℎ(𝑥) as 𝑥 approaches 9
from the left and right sides.
Notice that the value of ℎ(𝑥) approaches 3.
𝒙<𝟗
8.95
8.96
8.97
8.99
𝟗
𝒙>𝟗
9.01
9.02
9.03
𝒉(𝒙)
2.992
2.993
2.995 2.997 2.998
𝟑
𝒉(𝒙)
3.002
3.003
3.005 3.007 3.008
8.98
9.04
9.05
From the tables, we can say that
lim √𝑥 = 3.
𝑥→9
2
Try It!
Estimate lim √𝑥 − 4 using tables of values.
𝑥→13
Example 3
Estimate lim
sin 𝑥
𝑥→0 𝑥
.
Solution
Let 𝑚(𝑥) =
sin 𝑥
𝑥
. The function 𝑚(𝑥) =
sin 𝑥
𝑥
is defined everywhere except at 𝑥 = 0. Note that in
finding the limit, we are only concerned about the value being approached by 𝑚(𝑥) as 𝑥
approaches zero. The function need not be defined at 𝑥 = 0.
Step 1:
Construct two tables with arbitrary 𝑥 values that are very close to 0 from the
left and right sides.
1.1. Evaluating Limits through Table of Values
12
Unit 1: Limit of a Function
The goal is to examine the behavior of the values of 𝑚(𝑥) =
sin 𝑥
𝑥
as 𝑥 approaches
0 from both sides, so the number of decimal places that we need for the 𝑥
values need not be as many as possible.
𝒙<𝟎
𝒎(𝒙)
–1
1
– 0.5
0.5
– 0.1
0.1
– 0.01
0.01
– 0.001
0.001
𝒎(𝒙)
𝟎
𝟎
Step 2:
𝒙>𝟎
Complete the table by solving the value of function 𝑚(𝑥) for each 𝑥 value.
𝒙<𝟎
𝒎(𝒙)
𝒙>𝟎
𝒎(𝒙)
–1
0.84147
1
0.84147
– 0.5
0.95885
0.5
0.95885
– 0.1
0.99833
0.1
0.99833
– 0.01
0.99998
0.01
0.99998
– 0.001
0.99999
0.001
0.99999
𝟎
1.1. Evaluating Limits through Table of Values
𝟎
13
Unit 1: Limit of a Function
Step 3:
Estimate the values that are being approached by 𝑚(𝑥) as 𝑥 approaches 0 from
the left and right sides.
𝒙<𝟎
𝒎(𝒙)
𝒙>𝟎
𝒎(𝒙)
–1
0.84147
1
0.84147
– 0.5
0.95885
0.5
0.95885
– 0.1
0.99833
0.1
0.99833
– 0.01
0.99998
0.01
0.99998
– 0.001
0.99999
0.001
0.99999
𝟎
𝟏
𝟎
𝟏
Therefore, we can say that
sin 𝑥
= 1.
𝑥→0 𝑥
lim
3 Try It!
Estimate lim
𝑥→−1
𝑥 2+3𝑥+2
𝑥+1
using tables of values.
Remember
In finding the limit, we are only concerned about the value being
approached by the function 𝑓(𝑥) a 𝑥 approaches a number 𝑐. The
function need not be defined at 𝑥 = 𝑐.
1.1. Evaluating Limits through Table of Values
14
Unit 1: Limit of a Function
Does the limit of a function always exist?
One-Sided Limits
We find the limit of a function by observing the value it approaches from two sides. We can
also talk about the limit of a function 𝑓(𝑥) as 𝑥 approaches a certain value from either left or
right side only. This is called a one-sided limit.
One-Sided Limits
Left-hand Limit: Suppose the function 𝑓 (𝑥) is defined when 𝑥 is near 𝑐 from
the left. If 𝑓 (𝑥) gets closer to 𝑀 as 𝑥 gets closer to 𝑐 from the left, then we say
that “the limit of 𝑓(𝑥) as 𝑥 approaches 𝑐 from the left is equal to 𝑀.” This can be
written as
𝐥𝐢𝐦 𝒇(𝒙) = 𝑴.
𝒙→𝒄−
Right-hand Limit: Suppose the function 𝑓 (𝑥) is defined when 𝑥 is near 𝑐 from
the right. If 𝑓(𝑥) gets closer to 𝑁 as 𝑥 gets closer to 𝑐 from the right, then we say
that “the limit of 𝑓(𝑥) as 𝑥 approaches 𝑐 from the right is equal to 𝑁.” This can
be written as
𝐥𝐢𝐦 𝒇(𝒙) = 𝑵.
𝒙→𝒄+
The number 𝑐 may or may not be in the domain of the function 𝑓 (𝑥).
1.1. Evaluating Limits through Table of Values
15
Unit 1: Limit of a Function
Given a function 𝑓 (𝑥) and a number 𝑎, the values of 𝑀 and 𝑁 above may or may not be equal.
If the values of 𝑀 and 𝑁 are equal, then the limit of 𝒇(𝒙) as 𝒙 approaches 𝒄 exists. If the
values of 𝑀 and 𝑁 are not equal, then the limit of 𝒇(𝒙) as 𝒙 approaches 𝒄 does not exist.
Let’s Practice!
Example 4
Estimate the limit of the signum function 𝑠(𝑥) as 𝑥 approaches zero from the left.
−1,
𝑠(𝑥) = { 0,
1,
𝑥<0
𝑥=0
𝑥>1
Solution
Since we are looking for the limit of 𝑠(𝑥) as 𝑥 approaches zero from the left, we solve the
left-hand limit.
Step 1:
Construct a table with arbitrary 𝑥 values that are very close to 0 from the left
side.
𝒙<𝟎
𝒔(𝒙)
– 0.1
– 0.01
– 0.001
𝟎
1.1. Evaluating Limits through Table of Values
16
Unit 1: Limit of a Function
Step 2:
Complete the table by solving the value of 𝑠(𝑥) for each 𝑥 value.
Notice that for any negative value of 𝑥, the value of 𝑠(𝑥) is equal to −1.
𝒙<𝟎
𝒔(𝒙)
– 0.1
–1
– 0.01
–1
– 0.001
–1
𝟎
Step 3:
Estimate the values that are being approached by 𝑠(𝑥) as 𝑥 approaches 0
from the left.
𝒙<𝟎
𝒔(𝒙)
– 0.1
–1
– 0.01
–1
– 0.001
–1
𝟎
–𝟏
For any negative 𝑥 value, 𝑠(𝑥) = −1. Thus, we say that “the limit of 𝑠(𝑥) as 𝑥
approaches zero from the left is −1.” This can be written as
lim 𝑠(𝑥) = −1.
𝑥→0−
1.1. Evaluating Limits through Table of Values
17
Unit 1: Limit of a Function
4 Try It!
Given the piecewise function 𝑡(𝑥) below, estimate lim− 𝑡(𝑥) using a table of values.
𝑥→1
𝑡(𝑥) = {
2𝑥 + 3,
𝑥 2 − 4,
𝑥<1
𝑥≥1
Example 5
Given the piecewise function 𝑚(𝑥) below, estimate lim+ 𝑚(𝑥) using a table of values.
𝑥→1
𝑚(𝑥) = {
𝑥 + 4,
𝑥 2 − 2𝑥 + 2,
𝑥<1
𝑥≥1
Solution
Step 1:
Construct a table with arbitrary 𝑥 values that are close to 1 from the right side.
𝒙>𝟏
𝒎(𝒙)
1.05
1.04
1.03
1.02
1.01
𝟏
1.1. Evaluating Limits through Table of Values
18
Unit 1: Limit of a Function
Step 2:
Complete the table by solving the value of the function for each 𝑥 value.
𝒙>𝟏
𝒎(𝒙)
1.05
1.0025
1.04
1.0016
1.03
1.0009
1.02
1.0004
1.01
1.0001
𝟏
Step 3:
Estimate the values that are being approached by 𝑚(𝑥) as 𝑥 approaches 1
from the right.
Notice that as 𝑥 approaches 1 from the right, 𝑚(𝑥) approaches 1.
1.1. Evaluating Limits through Table of Values
𝒙>𝟏
𝒎(𝒙)
1.05
1.0025
1.04
1.0016
1.03
1.0009
1.02
1.0004
1.01
1.0001
𝟏
𝟏
19
Unit 1: Limit of a Function
Therefore, we write
lim 𝑚(𝑥) = 1.
𝑥→1+
5 Try It!
Estimate the limit of the signum function 𝑠(𝑥) as 𝑥 approaches zero from the right.
−1,
𝑠(𝑥) = { 0,
1,
𝑥<0
𝑥=0
𝑥>1
Example 6
Given the function 𝑓(𝑥) below, estimate lim − 𝑓 (𝑥), lim + 𝑓 (𝑥), and lim 𝑓 (𝑥).
𝑥→−3
𝑥→−3
𝑥→−3
𝑥 + 4,
𝑥 < −3
𝑓 (𝑥) = { 2
𝑥 − 2𝑥 + 6, 𝑥 ≥ −3
Solution
Step 1:
Construct two tables with arbitrary 𝑥 values that are very close to the value
𝑥 = −3 from the left and right sides.
𝒙 < −𝟑
𝒇(𝒙)
𝒙 > −𝟑
−3.005
−2.995
−3.004
−2.996
−3.003
−2.997
−3.002
−2.998
−3.001
−2.999
−𝟑
1.1. Evaluating Limits through Table of Values
𝒇(𝒙)
−𝟑
20
Unit 1: Limit of a Function
Step 2:
Complete the tables by solving the value of the function for each 𝑥 value.
𝒙 < −𝟑
𝒇(𝒙)
𝒙 > −𝟑
𝒇(𝒙)
−3.005
0.995
−2.995
20.960
−3.004
0.996
−2.996
20.968
−3.003
0.997
−2.997
20.976
−3.002
0.998
−2.998
20.984
−3.001
0.999
−2.999
20.992
−𝟑
−𝟑
Step 3:
Estimate the values that are being approached by 𝑓(𝑥) as 𝑥 approaches −3
from the left and right sides.
𝒙 < −𝟑
𝒇(𝒙)
𝒙 > −𝟑
𝒇(𝒙)
−3.005
0.995
−2.995
20.960
−3.004
0.996
−2.996
20.968
−3.003
0.997
−2.997
20.976
−3.002
0.998
−2.998
20.984
−3.001
0.999
−2.999
20.992
−𝟑
𝟐𝟏
−𝟑
𝟏
Therefore, we estimate that lim − 𝑓 (𝑥) = 1 and lim + 𝑓 (𝑥) = 21. Since the one𝑥→−3
𝑥→−3
sided limits are not equal, lim 𝑓 (𝑥) does not exist.
𝑥→−3
1.1. Evaluating Limits through Table of Values
21
Unit 1: Limit of a Function
6 Try It!
Given the function 𝑔(𝑥) below, estimate lim− 𝑔(𝑥), lim+ 𝑔(𝑥), and lim 𝑔(𝑥).
𝑥→2
1
,
(
)
𝑔 𝑥 = {𝑥 + 5
𝑥 3 + 5,
𝑥→2
𝑥→2
𝑥<2
𝑥≥2
Infinite Limits
A function 𝑓(𝑥) may not have a limit as 𝑥 approaches a certain value because it increases or
decreases indefinitely. In this case, we will use the concept of infinity.
Did You Know?
The symbol for infinity, ∞, was credited to be introduced by John
Wallis, an English mathematician. Take note that infinity is not a
number, but a symbol used to indicate that a quantity is decreasing
or increasing without bound.
1.1. Evaluating Limits through Table of Values
22
Unit 1: Limit of a Function
Infinite Limits
Suppose the function 𝑓 (𝑥) is defined when 𝑥 is as near as possible to 𝑐 on
both sides.
If 𝑓(𝑥) increases without bound as 𝑥 approaches 𝑐, then we write
𝐥𝐢𝐦 𝒇(𝒙) = ∞.
𝒙→𝒄
If 𝑓(𝑥) decreases without bound as 𝑥 approaches 𝑐, then we write
𝐥𝐢𝐦 𝒇(𝒙) = − ∞.
𝒙→𝒄
In both cases, the limit does not exist.
Note that for infinite limits, the limit does not exist because the values of 𝑓(𝑥) do not
approach a specific value.
We can also talk about one-sided infinite limits. If 𝑓(𝑥) increases or decreases without bound
as 𝑥 approaches 𝑐 from the left, then we write 𝐥𝐢𝐦− 𝒇(𝒙) = ∞ or 𝐥𝐢𝐦− 𝒇(𝒙) = − ∞, respectively.
𝒙→𝒄
𝒙→𝒄
If 𝑓 (𝑥) increases or decreases without bound as 𝑥 approaches 𝑐 from the right, then we write
𝐥𝐢𝐦 𝒇(𝒙) = ∞ or 𝐥𝐢𝐦+ 𝒇(𝒙) = − ∞, respectively.
𝒙→𝒄+
𝒙→𝒄
1.1. Evaluating Limits through Table of Values
23
Unit 1: Limit of a Function
Let’s Practice!
Example 7
Estimate lim
1
𝑥→3 (𝑥−3)2
using tables of values.
Solution
Step 1:
Construct two tables with arbitrary 𝑥 values that are very close to 3 from the
left and right sides.
1
Let 𝑞(𝑥) = (𝑥−3)2 . The tables below show the values of the function
1
𝑞(𝑥) = (𝑥−3)2 as 𝑥 approaches 3 from both sides.
𝒙<𝟑
𝒒(𝒙)
𝒙>𝟑
2.9
3.1
2.95
3.05
2.99
3.01
2.999
3.001
2.9999
3.0001
𝟑
1.1. Evaluating Limits through Table of Values
𝒒(𝒙)
𝟑
24
Unit 1: Limit of a Function
Step 2:
Complete the table by solving the value of the function for each 𝑥 value.
Notice how the value of 𝑞(𝑥) increases without bound as 𝑥 approaches 3 from
either side.
𝒙<𝟑
𝒒(𝒙)
𝒙>𝟑
𝒒(𝒙)
2.9
100
3.1
100
2.95
400
3.05
400
2.99
10 000
3.01
10 000
2.999
1 000 000
3.001
1 000 000
2.9999
100 000 000
3.0001
100 000 000
𝟑
𝟑
Step 3:
Estimate the values that are being approached by 𝑞(𝑥) as 𝑥 approaches 3
from the left and right sides.
𝒙<𝟑
𝒒(𝒙)
𝒙>𝟑
𝒒(𝒙)
2.9
100
3.1
100
2.95
400
3.05
400
2.99
10 000
3.01
10 000
2.999
1 000 000
3.001
1 000 000
2.9999
100 000 000
3.0001
100 000 000
𝟑
∞
1.1. Evaluating Limits through Table of Values
𝟑
∞
25
Unit 1: Limit of a Function
1
We say that “the limit of 𝑞(𝑥) = (𝑥−3)2 as 𝑥 approaches 3 is infinity” or
1
“𝑞(𝑥) = (𝑥−3)2 increases without bound as 𝑥 approaches 3.” In symbols, we can
write this as
1
= ∞.
𝑥→3 (𝑥 − 3)2
lim
7 Try It!
Estimate lim
𝑥+5
𝑥→0 𝑥 2
using tables of values.
Example 8
𝑥+6
Estimate lim 𝑥−2 using tables of values.
𝑥→2
Solution
Step 1:
Construct two tables with arbitrary 𝑥 values that are very close to 2 from the
left and right sides.
𝑥+6
𝑥+6
Let 𝑟(𝑥) = 𝑥−2. The tables below show the values of 𝑟(𝑥) = 𝑥−2 as 𝑥 approaches
2 from both sides.
𝒙<𝟐
𝒓(𝒙)
𝒙>𝟐
1.999
2.001
1.9995
2.0005
1.9999
2.0001
1.99995
2.00005
1.99999
2.00001
𝟐
1.1. Evaluating Limits through Table of Values
𝒓(𝒙)
𝟐
26
Unit 1: Limit of a Function
Step 2:
Complete the table by solving the value of the function for each 𝑥 value.
𝒙<𝟐
𝒓(𝒙)
𝒙>𝟐
𝒓(𝒙)
1.999
– 7 999
2.001
8 001
1.9995
– 15 999
2.0005
16 001
1.9999
– 79 999
2.0001
80 001
1.99995
– 159 999
2.00005
160 001
1.99999
– 799 999
2.00001
800 001
𝟐
𝟐
Step 3:
Estimate the values that are being approached by 𝑟(𝑥) as 𝑥 approaches 2
from the left and right sides.
𝒙<𝟐
𝒓(𝒙)
𝒙>𝟐
𝒓(𝒙)
1.999
– 7 999
2.001
8 001
1.9995
– 15 999
2.0005
16 001
1.9999
– 79 999
2.0001
80 001
1.99995
– 159 999
2.00005
160 001
1.99999
– 799 999
2.00001
800 001
𝟐
−∞
𝟐
∞
Observe that the value of 𝑟(𝑥) decreases without bound as 𝑥 approaches 2
from the left and increases without bound as 𝑥 approaches 2 from the right.
Thus,
1.1. Evaluating Limits through Table of Values
27
Unit 1: Limit of a Function
lim−
𝑥→2
𝑥+6
= −∞
𝑥−2
Therefore, lim
𝑥+6
𝑥→2 𝑥−2
and
lim+
𝑥→2
𝑥+6
= ∞.
𝑥−2
does not exist.
8 Try It!
Estimate lim
𝑥−6
𝑥→−7 𝑥+7
using tables of values.
Example 9
Estimate lim
tan 𝑥, lim+ tan 𝑥 and lim𝜋 tan 𝑥 using tables of values.
𝜋−
𝑥→
2
𝑥→
𝜋
2
𝑥→
2
Solution
Step 1:
Construct two tables with arbitrary 𝑥 values that are very close to
𝜋
2
from the
left and right sides.
Let 𝑞(𝑥) = tan 𝑥. The goal is to examine the behavior of the values of
𝜋
𝑞(𝑥) = tan 𝑥 as 𝑥 approaches 2 from either side. Note that
𝒙<
𝝅
𝟐
𝒒(𝒙)
𝒙>
𝝅
𝟐
1.570792
1.570801
1.570793
1.570800
1.570794
1.570799
1.570795
1.570798
1.570796
1.570797
𝝅
𝟐
1.1. Evaluating Limits through Table of Values
𝜋
2
≈ 1.5707963.
𝒒(𝒙)
𝝅
𝟐
28
Unit 1: Limit of a Function
Step 2:
Complete the table by solving the value of the function for each 𝑥 value.
𝒙<
𝝅
𝟐
𝒙>
𝒒(𝒙)
𝒒(𝒙)
1.570792
231 118
1.570801
−213 986
1.570793
300 590
1.570800
−272 242
1.570794
429 776
1.570799
−374 083
1.570795
753 696
1.570798
−597 655
1.570796
3 060 023
1.570797
−1 485 431
𝝅
𝟐
𝝅
𝟐
Step 3:
𝝅
𝟐
𝜋
Estimate the values that are being approached by 𝑞(𝑥) as 𝑥 approaches 2
from the left and right sides.
As 𝑥 approaches
𝜋
2
from the left, 𝑞(𝑥) = tan 𝑥 does not approach any value but
increases without bound. As 𝑥 approaches
𝜋
2
from the right, 𝑞(𝑥) = tan 𝑥 also
does not approach any value but decreases without bound.
𝒙<
𝝅
𝟐
𝒒(𝒙)
𝒙>
𝝅
𝟐
𝒒(𝒙)
1.570792
231 118
1.570801
−213 986
1.570793
300 590
1.570800
−272 242
1.570794
429 776
1.570799
−374 083
1.570795
753 696
1.570798
−597 655
1.570796
3 060 023
1.570797
−1 485 431
𝝅
𝟐
1.1. Evaluating Limits through Table of Values
∞
𝝅
𝟐
−∞
29
Unit 1: Limit of a Function
Therefore, we estimate the left-hand side limit of 𝑞(𝑥) to be ∞ and the righthand side limit to be −∞. We write
lim
tan 𝑥 = ∞
𝜋−
𝑥→
and
lim+ tan 𝑥 = − ∞.
𝑥→
2
𝜋
2
Since the left- and right-hand limits are infinite limits, we say that lim𝜋 tan 𝑥
𝑥→
2
does not exist.
9 Try It!
𝑥−3
𝑥−3
𝑥−3
Estimate lim− 𝑥 2 −5𝑥+6, lim+ 𝑥 2−5𝑥+6, and lim 𝑥 2 −5𝑥+6 using tables of values.
𝑥→2
𝑥→2
𝑥→2
Key Points
___________________________________________________________________________________________
● Suppose that the function 𝑓 (𝑥) is defined when 𝑥 is near 𝑐. If 𝑓 (𝑥) gets closer to 𝐿 from
both sides as 𝑥 gets closer to 𝑐, then we say that “the limit of 𝑓(𝑥) as 𝑥 approaches 𝑐 is
equal to 𝐿.” This is written as
𝐥𝐢𝐦 𝒇(𝒙) = 𝑳.
𝒙→𝒄
●
Suppose that the function 𝑓 (𝑥) is defined when 𝑥 is near 𝑐 from the left. Then, the lefthand limit of 𝑓(𝑥) as 𝑥 approaches 𝑐 from the left is equal to a number 𝑀. This can be
written as
𝐥𝐢𝐦 𝒇(𝒙) = 𝑴.
𝒙→𝒄−
1.1. Evaluating Limits through Table of Values
30
Unit 1: Limit of a Function
●
Suppose that the function 𝑓 (𝑥) is defined when 𝑥 is near 𝑐 from the right. Then, the
right-hand limit of 𝑓(𝑥) as 𝑥 approaches 𝑐 from the right is equal to a number 𝑁. This
can be written as
𝐥𝐢𝐦 𝒇(𝒙) = 𝑵.
𝒙→𝒄+
●
If the left- and right-hand limits of 𝑓(𝑥) as 𝑥 approaches a number 𝑐 are equal, then we
say that the limit of 𝒇(𝒙) as 𝒙 approaches 𝒄 exists. If this is not the case, then the
limit does not exist.
●
Suppose the function 𝑓 (𝑥) is defined when 𝑥 is as near as possible to 𝑐 on both sides.
If 𝑓(𝑥) increases without bound as 𝑥 approaches 𝑐, then we say that “the limit of 𝑓(𝑥)
as 𝑥 approaches 𝑐 is infinity.” This is written as
𝐥𝐢𝐦 𝒇(𝒙) = ∞.
𝒙→𝒄
If 𝑓(𝑥) decreases without bound as 𝑥 approaches 𝑐, then we say that “the limit of 𝑓(𝑥)
as 𝑥 approaches 𝑐 is negative infinity.” This is written as
𝐥𝐢𝐦 𝒇(𝒙) = − ∞.
𝒙→𝒄
●
To estimate the limit of a function 𝑓(𝑥) as 𝑥 approaches 𝑐, follow the steps below:
Step 1: Construct tables with arbitrary 𝑥 values that are very close to 𝑐 from the left
and right sides.
Step 2: Complete the table by solving the value of 𝑓(𝑥) for each 𝑥 value.
Step 3: Estimate the values that are being approached by 𝑓(𝑥) from the left and right
sides of 𝑥 = 𝑐.
___________________________________________________________________________________________
1.1. Evaluating Limits through Table of Values
31
Unit 1: Limit of a Function
Check Your Understanding
A. For each item, choose the appropriate expression from the box.
lim 𝑓 (𝑥) = 𝐿
lim 𝑓 (𝑥) = 𝐿
𝑥→𝑐 +
𝑥→𝑐
lim 𝑓 (𝑥) = −∞
𝑥→𝑐 +
lim 𝑓 (𝑥) = −∞
𝑥→𝑐 −
lim 𝑓 (𝑥) = 𝐿
𝑥→𝑐 −
lim 𝑓(𝑥) = ∞
𝑥→𝑐 +
lim 𝑓 (𝑥) = ∞
𝑥→𝑐
lim 𝑓(𝑥) = ∞
𝑥→𝑐 −
1. The limit of the function 𝑓(𝑥) as 𝑥 approaches 𝑐 is equal to 𝐿.
2. The value of the function 𝑓(𝑥) decreases without bound as 𝑥 approaches 𝑐.
3. The limit of the function 𝑓(𝑥) as 𝑥 approaches 𝑐 from the left is equal to 𝐿.
4. The limit of the function 𝑓(𝑥) as 𝑥 approaches 𝑐 from the right is infinity.
B. Estimate lim− 𝑟(𝑥), lim+ 𝑟(𝑥), and lim 𝑟(𝑥) given the table of values below.
𝑥→4
𝑥→4
𝑥→4
𝒙<𝟒
𝒓(𝒙)
𝒙>𝟒
𝒓(𝒙)
3.995
1.9987
4.005
16.0400
3.996
1.9990
4.004
16.0320
3.997
1.9992
4.003
16.0240
3.998
1.9995
4.002
16.0160
3.999
1.9997
4.001
16.0080
1.1. Evaluating Limits through Table of Values
32
Unit 1: Limit of a Function
C. For each item, use tables of values to estimate li𝑚− 𝑓 (𝑥), li𝑚+ 𝑓 (𝑥),
𝑥→𝑐
𝑥→𝑐
and li𝑚 𝑓(𝑥) . Then, determine 𝑓(𝑐).
𝑥→𝑐
1. 𝑓 (𝑥) = 𝑥 3 − 𝑥 2 ;
𝑎=2
𝑥 + 2,
𝑥<0
𝑥=0; 𝑎=0
2. 𝑓 (𝑥) = {3,
𝑥2 − 4 , 𝑥 > 0
3. 𝑓 (𝑥) =
1
√𝑥−2
; 𝑎=4
Challenge Yourself
A. Answer each item and explain in your own words.
1. Explain what the equation lim 𝑓(𝑥) = 6 means. Is it possible for the equation to be true
𝑥→2
and yet 𝑓(2) = 7?
2. Explain what the equation lim 𝑔(𝑥) = 3 means. Is it possible for the equation to be true
𝑥→7
and yet 𝑔(7) is undefined?
3. Suppose lim− ℎ(𝑥) = 0 and lim+ ℎ(𝑥) = 1. Is it possible for lim ℎ(𝑥) to exist?
𝑥→5
𝑥→5
𝑥→5
4. Suppose lim+ 𝑚(𝑥) = ∞. Is it possible for lim 𝑚(𝑥) to exist?
𝑥→10
𝑥→10
B. Determine all values of 𝑐 for which lim 𝑟(𝑥) does not exist.
𝑥→𝑐
𝑥−2
,
𝑥+5
4,
𝑟(𝑥) =
𝑥 2 − 7𝑥 + 4,
𝑥−3
,
{ (𝑥 − 8)2
1.1. Evaluating Limits through Table of Values
𝑥 ≤ −4
−4<𝑥 <0
0≤𝑥≤7
𝑥>7
33
Unit 1: Limit of a Function
C. For each item, create your own function 𝑓(𝑥) with the given properties and write
the value of 𝑐 that will make the statement true. Note that 𝑐 and 𝐿 are real
numbers.
1.
lim 𝑓(𝑥) and lim+ 𝑓(𝑥) both exist, but lim 𝑓(𝑥) does not exist.
𝑥→𝑐 −
𝑥→𝑐
𝑥→𝑐
2. lim 𝑓(𝑥) exists but 𝑓(𝑎) is undefined.
𝑥→𝑐
3. lim 𝑓(𝑥) = ∞
𝑥→𝑐
Photo Credits
Eriks airconditioned road trip car by Stig Nygaard is licensed under CC BY 2.0 via Flickr.
Bibliography
Edwards, C.H., and David E. Penney. Calculus: Early Transcendentals. 7th ed. Upper Saddle
River, New Jersey: Pearson/Prentice Hall, 2008.
Larson, Ron H., and Bruce H. Edwards. Essential Calculus: Early Transcendental Functions.
Boston: Houghton Mifflin, 2008.
Leithold, Louis. The Calculus 7. New York: HarperCollins College Publ., 1997.
Smith, Robert T., and Roland B. Milton. Calculus. New York: McGraw Hill, 2012.
Tan, Soo T. Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach.
Australia: Brooks/Cole Cengage Learning, 2012.
1.1. Evaluating Limits through Table of Values
34
Unit 1: Limit of a Function
Key to Try It!
1. lim (5𝑥 + 8) = 13
𝑥→1
2.
3.
4.
5.
6.
lim √𝑥 − 4 = 3
𝑥→13
lim
𝑥 2+3𝑥+2
𝑥+1
𝑥→−1
lim 𝑡(𝑥) = 5
𝑥→1−
lim 𝑠(𝑥) = 1
𝑥→0∓
1
lim− 𝑔(𝑥) = , lim 𝑔(𝑥) = 13, and lim 𝑔(𝑥) does not exist.
7 𝑥→2∓
𝑥→2
7. lim
𝑥+5
𝑥→0 𝑥2
8.
9.
=1
lim
=∞
𝑥−6
𝑥→−7 𝑥+7
lim
𝑥→2
does not exist.
𝑥−3
𝑥→2− 𝑥2 −5𝑥+6
= − ∞, lim+
𝑥−3
2
𝑥→2 𝑥 −5𝑥+6
1.1. Evaluating Limits through Table of Values
= ∞, and lim
𝑥−3
𝑥→2 𝑥2 −5𝑥+6
does not exist.
35