11
Basic Calculus
Third Quarter
Module 1: The Limit of a
Function
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Republic of the Philippines
Department of Education
REGION VII-CENTRAL VISAYAS
SCHOOLS DIVISION OF SIQUIJOR
_________________________________________________________________________________
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11
Basic Calculus
Third Quarter
Module 1: The Limit of a
Function
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INTRODUCTION
This module is written in support of the K to 12 Basic Education Program to
ensure attainment of the standards expected of you as learner.
This learning material deals with the theorems that will enable us to directly
evaluate limits without the need for a table or a graph. We will learn how to compute
the limit of a function using the limit laws.
This includes the following activities/tasks:
▪
▪
▪
▪
▪
Expected Learning Outcome – This lays out the learning outcome that
you are expected to have accomplished at the end of the module.
Pre-test – This determines your prior learning on the particular lesson
you are about to take.
Discussion of the lesson – This provides you with the important
knowledge, principles and attitude that will help you meet the expected
learning outcome.
Learning Activities – These provide you with the application of the
knowledge and principles you have gained from the lesson and enable
you to further enhance your skills as you carry out prescribed tasks.
Post-test – This evaluates your overall understanding about the module.
With the different activities provided in this module, may you find this material
engaging and challenging as it develops you critical thinking skills.
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What I need to know
After going through this module, you will be able to:
➢ illustrate limit of a function using a table of values and the graph of the
function. (STEM_BC11LC-IIIa-1)
What I Know
Pre-Test
1. Using a calculator, complete the following table of values to investigate the
limits of the following functions.
a. lim(5 − 3x) = _______
x →2
x
f ( x) = 5 − 3 x
1
1.5
1.75
1.9
1.99
1.999
lim− (5 − 3x) = _______
x→2
x
f ( x) = 5 − 3 x
3
2.5
2.25
2.1
2.01
2.001
lim+ (5 − 3x) = _______
x→2
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2. Consider the graph below. Evaluate the following limits.
a. lim− g ( x) = ________
x→−2
lim g ( x) = ________
x→−2−
lim g ( x) = ________
x →−2 −
b. lim− g ( x) = ________
x→0
lim g ( x) = ________
x →0 −
lim g ( x) = ________
x→0−
c. lim− g ( x) = ________
x→2
lim g ( x) = ________
x→2−
lim g ( x) = ________
x→2−
Figure 1. Graph of y = f(x)
What’s In
Activity 1
Let us recall on the important notions about function. A function, f, is a special
type of a relation such that no two ordered pairs of the set have different second
coordinates corresponding to the same first coordinate. The set of all the first
coordinates (x) of the ordered pairs is the domain of the function. The set of all
second coordinated (y) of the ordered pairs is the range of the function.
If we have a rule or formula giving y in terms of x and there is no more than one
value of y for each value of x, then y is said to be a function of x. this can be written
in symbols, y = f(x) is read as “ y is a function of x” where y is called the value of
the function.
Let us recall on how to identify functions. Try doing the activity below. Write
your answers with complete solution on your notebook.
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1. Which of the following are graphs of a function?
2. Find the domain and range of the following set of ordered pairs.
a. {(2, - 1), (3, 4), (0, 2), (- 3, - 2)}
Domain: __________________________
Range: ___________________________
b. {(- 4, 0), (0, 0), (3, - 2), (1, 5), (- 3, - 3)}
Domain: ____________________________
Range: _____________________________
3. Given f ( x) = 3x − 1 , find:
a. f(2)
b. f(0)
4. Given f ( x) = 2 x 2 − 3 , evaluate
c. f(x+2)
f (5) − f (2)
6
What’s New
Limits are the backbone of calculus, and calculus is called the Mathematics of
Change. The study of limits is necessary in studying change in great detail. The
evaluation of a limit is what underlies the formulation of the derivative and the integral
of a function.
For starters, imagine that you are going to watch a basketball game. When you
choose seats, you would want to be as close to the action as possible. You would want
to be as close to the players as possible and have the best view of the game, as if you
were in the basketball court yourself. Take note that you cannot actually be in the court
and join the players, but you will be close enough to describe clearly what is happening
in the game.
This is how it is with limits of functions. We will consider functions of a single
variable and study the behavior of the function as its variable approaches a particular
value (a constant). The variable can only take values very, very close to the constant,
but it cannot equal the constant itself. However, the limit will be able to describe clearly
what is happening to the function near that constant.
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What Is It?
What is your idea of a limit? Do you think all things have limits? In your daily
activities, have you ever had to deal with limits?
RJ and his friends had an exciting adventure last summer. They hiked up to the
brow of the rocky mountain overlooking the Underground River. As they went up the
slope of the mountain, they got closer and closer to the brow but they had to be careful
not to go beyond, for they might fall on the river. That gave them an experience of a
limit.
http://palawanislandphilippines.com/puerto-princesa-tours/puerto-princesa-undergroundriver-tour/
A similar thing is involved in limits of functions. In this lesson, we shall try to
understand the limit of a function y = f(x) as the values approach a certain number.
We shall do this in two ways: first by using table of values for x and y; and second, by
looking at the graph of the function f.
Limit of a function by Using Tabular and Graphical Methods
Consider a function f of a single variable x. Consider a constant c which the
variable x will approach (c may or may not be in the domain of f). The limit, to be
denoted by L, is the unique real value that f(x) will approach as x approaches c. In
symbols, we write the process as
lim f ( x) = L ,
x→c
This is read as, “the Limit of f(x) as x approaches c is L”.
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The Tabular Method
Consider the linear function f ( x) = 2 x − 3 . Let us say that we want to determine
the limit of f(x) as the values of x approaches 1; that is, we are interested in looking at
the behavior of the function f as the values of x get closer and closer to number 1. In
the number line, this happens when x values approach the number 1 in Table 1 and
Table 2.
Table 1
X
f(x)=2x – 3
0
-3
0.50
-2
0.75
-1.5
0.80
-1.4
0.90
-1.2
0.99
-1.02
0.999
-1.002
lim− (2 x − 3) = −1
Table 2
x
f(x)=2x – 3
2
1
1.50
0
1.25
-0.5
1.20
-0.6
1.10
-0.8
1.01
-0.98
1.001
-0.998
lim+ (2 x − 3) = −1
x →1
x →1
▪
Table 1 shows the values of x approaching the number 1 from the left
( x → 1− ); that is, the values of x are getting closer to 1, but they are less than
1. The values of y=f(x) also get closer to the number – 1.
▪
Table 2 shows the values of x approaching the number 1 from the right
( x → 1+ ). This time, the values of x are getting closer and closer to 1, but they
are all greater than 1. Then, we see that the resulting values of f(x) also
approach the number – 1.
In this example, we observe that as the values of x approach 1 from both left
and right, the values of f(x) also approach – 1. Hence, we say that – 1 is the limit of
f ( x) = 2 x − 3 as x approaches 1. In symbols, we write it as
lim (2 x − 3) = −1
x →1−
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Example 1. Using table of values, evaluate
lim( x 2 + 2)
x →0
Solution
In this problem, f ( x) = x 2 + 2 and c = 0.
Table 3
X
f ( x) = x + 2
-1
3
-0.5
2.25
-0.25
2.0625
-0.1
2.01
-0.01
2.0001
-0.001
2.000001
2
lim− ( x + 2) = 2
2
x →0
Table 4
x
f ( x) = x 2 + 2
1
3
0.5
2.25
0.25
2.0625
0.1
2.01
0.01
2.0001
0.001
2.000001
2
lim+ ( x + 2) = 2
x →0
Since lim ( x 2 + 2) = 2 and lim+ ( x 2 + 2) = 2 ,
x →0 −
x →0
then lim ( x + 2) = lim ( x 2 + 2)
2
x →0 −
x →0+
Therefore, lim( x 2 + 2) = 2
x →0
 x 2 − 1, x  2
Example 2. Let g ( x) = 
 x + 2, x  2
Evaluate lim g ( x)
x→ 2
Solution
In this problem c = 2.
For values of x from the left of 2, we use g ( x) = x 2 − 1
For values of x from the right of 2, we use g ( x) = x + 2
Table 5
x
g ( x) = x − 1
1
0
1.5
1.25
1.75
2.0625
1.9
2.61000
1.99
2.96010
1.999
2.996001
 lim− g ( x) = 3
2
x→2
Table 6
x
g ( x) = x + 2
3
5
2.5
4.5
2.25
4.25
2.1
4.10
2.01
4.01
2.001
4.001
 lim+ g ( x) = 4
x→2
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Since lim− g ( x) = 3 and lim+ g ( x) = 4 ,
x→2
x→2
then lim− g ( x)  lim+ g ( x)
x→2
x→2
Therefore, lim g ( x) does not exist or DNE.
x→ 2
The Graphical Method
We shall now examine the limits of the same functions using their graphs. Let
us consider again the function f ( x) = 2 x − 3 . Its graph is a straight line with slope 2
and y-intercept, - 3 , as shown below
Since we want to determine f ( x) = 2 x − 3 ,
we are interested in the behavior of the graph
around x =1. Let us consider the points of the
graph as the values approach 1 from the left.
These points have coordinates found in table 1:
(0, -3), (0.5, -2), (.75, -1.5), (0.8, -1.4), (0.9, -1.2),
(0.99, -1.01), and so on.
Notice that the points move along the straight
line and approach the poit (1, -1), where y = - 1.
You will observe a similar behavior using the
points determined by values in Table 2, where the
x values approach 1 from the right. The
corresponding y values also approach – 1.
Figure 2. Graph of f ( x) = 2 x − 3
With this, we confirm graphically that lim− (2 x − 3) = −1 .
x →1
Example 1. Consider the graph of f ( x) = x 2 + 2 as shown below.
Evaluate the following:
a. lim− ( x 2 + 2)
x →0
b. lim+ ( x 2 + 2)
x →0
c. lim( x 2 + 2)
x →0
2
Figure 3. Graph of f ( x ) = x + 2
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Solution:
Notice that as the values of x approach 0 from the left, the points of the
graph also approach a level where y = 2. The same pattern can be observed
on the right side of 0. Thus, we say that:
a. lim− ( x 2 + 2) = 2
x →0
b. lim+ ( x 2 + 2) = 2
x →0
c. lim( x 2 + 2) = 2 (since the limit of both sides of 0 approaches 2)
x →0
 x 2 − 1, x  2
Example 2. Consider the graph of g ( x) = 
as shown below.
x
+
2
,
x

2

Evaluate the following:
a. lim− g ( x)
x→2
b. lim+ g ( x)
x→2
c. lim g ( x)
x→ 2
 x 2 − 1, x  2
Figure 4. Graph of g ( x) = 
 x + 2, x  2
Solution:
Notice that as the values of x approach 2 from the left, the y values
approach 3. However, as the x values approach 2 from the right, the values of
y approach 4. Thus, we say that:
a. lim− g ( x) = 3
x→2
b. lim+ g ( x) = 4
x→2
c. lim g ( x) does not exist or DNE (since the limit of left and right
x→ 2
sides of 2 approaches on different y values)
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Example 3. Consider the graph of y = f (x) as shown below.
Evaluate the following:
a. lim− f ( x)
x→−2
b. lim+ f ( x)
x→−2
c. lim f ( x)
x→−2
d. lim− f ( x)
x→0
e. lim+ f ( x)
x→0
f. lim f ( x)
x→0
g. lim− f ( x)
x →2
h. lim+ f ( x)
x →2
i. lim f ( x)
x→2
Figure 5. Graph of y = f (x)
Solution:
a. lim− f ( x) = 3
x→−2
b. lim+ f ( x) = −1
x→−2
c. lim f ( x) DNE
x→−2
d. lim− f ( x) = −1
x→0
e. lim+ f ( x) = −1
x→0
f. lim f ( x) = −1
x→0
g. lim− f ( x) = − (Infinite limits)
x→2
h. lim+ f ( x) = + (Infinite limits)
x→2
i. lim f ( x) DNE
x→2
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What’s More?
ACTIVITY 2
Independent Assessment 1
Using a calculator, complete the following table of values to investigate the limits of
the following functions.
t + 2, t  −2
a. lim g (t ) =___________ , where g (t ) = 
t →−2
 2, t  −2
t
g (t ) = t + 2
-3
-2.5
-2.25
-2.1
-2.01
-2.001
lim− g (t ) = _______
t →−2
t
g (t ) = 2
-1
-1.5
-1.75
-1.9
-1.99
-1.999
lim+ g (t ) = _______
t →−2
Independent Assessment 2
Consider the function f(x) whose graph is shown below.
Determine the following
a. lim− f ( x) = _______
t →1.5
lim f ( x) = _______
t →1.5+
lim f ( x) = _______
t →1.5
b. lim− f ( x) = _______
t →0
lim f ( x) = _______
t →0+
lim f ( x) = _______
t →0
c. lim− f ( x) = _______
t →2
lim f ( x) = _______
t →2+
lim f ( x) = _______
t →2
d. lim− f ( x) = _______
t →4
lim f ( x) = _______
t →4+
lim f ( x) = _______
t →4
Figure 6. Graph of y = f (x)
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What I Have Learned?
I learned that:
In order to get the limit of a function y=f(x) as x approaches to the number
c, we simply do the following:
1. Compute for the values of f(x) as x approaches c from the left. Check
whether these values also approach a certain number. We shall symbolize
the limit from the left as lim− f ( x) .
x →c
2. Do the same for the values of f(x) as x approaches c from the right. We
shall symbolize the limit from the right as lim+ f ( x) .
x →c
lim f ( x) and lim+ f ( x) are also called one-sided limits.
x →c −
x →c
3. Verify whether lim− f ( x) = lim+ f ( x)
x →c
x →c
a. If lim− f ( x) = lim+ f ( x) = L, then lim f ( x) = L
x →c
x →c
x →c
b. If lim− f ( x)  lim f ( x) , we say that lim f ( x) does not exist (DNE).
x →c
x →c +
x→ c
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Assessment
POST TEST
1
1. Using a calculator, complete the following table of values to investigate the
limits of the following functions.
lim
t →1
x2 − 2x + 1
=___________
x −1
x2 − 2x + 1
y=
x −1
x
0
0.5
0.75
0.9
0.99
0.999
x2 − 2x + 1
y=
x −1
x
2
1.5
1.25
1.1
1.01
1.001
lim−
t →1
x2 − 2x + 1
x −1
lim+
t →1
x2 − 2x + 1
x −1
2. From the graph of y = f(x) below, evaluate the following:
a. lim− f ( x) = _______
t →−3
b. lim+ f ( x) = _______
t →−3
c. lim f ( x) = _______
t →−3
d.
e.
f.
lim f ( x) = _______
t →−0−
lim f ( x) = _______
t →−0+
lim f ( x) = _______
t →0
g. lim− f ( x) = _______
t →3
h. lim+ f ( x) = _______
t →3
i.
j.
lim f ( x) = _______
t →3
lim f ( x) = _______
t →6−
k. lim+ f ( x) = _______
t →6
l.
Figure 7. Graph of y = f (x)
lim f ( x) = _______
t →6
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References
Balmaceda, J. M., Arceo, P. P., Lemence, R. S., Ortega, O. M., & Vallejo, L. J.
(2016). TG for SHS Basic Calculus. Quezon City: Commision on Higher
Education.
Rodriguez, E. M. (2017). Conceptual Math and Beyond: Basic Calculus. Quezon
City: Brilliant creations Publishing, Inc.
http://fode.education.gov.pg/courses/Mathematics/Grade%2012/Advanced/Unit3.pdf
http://palawanislandphilippines.com/puerto-princesa-tours/puerto-princesaunderground-river-tour/
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