Basic Calculus
The Limit of a Function
SENIOR
HIGH
SCHOOL
Module
1
Quarter 3
Basic Calculus
Quarter 3 – Module 1: The Limit of a Function!
First Edition, 2020
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Basic Calculus
SENIOR
HIGH
SCHOOL
Module
1
Quarter 3
The Limit of a Function
Introductory Message
For the facilitator:
Welcome to the Basic Calculus (Senior High School) Module on The Limit of a
Function!
This module was collaboratively designed, developed and reviewed by educators from
Schools Division Office of Pasig City headed by its Officer-In-Charge Schools Division
Superintendent, Ma. Evalou Concepcion A. Agustin in partnership with the Local
Government of Pasig through its mayor, Honorable Victor Ma. Regis N. Sotto.
The writers utilized the standards set by the K to 12 Curriculum using the Most
Essential Learning Competencies (MELC) while overcoming their personal, social,
and economic constraints in schooling.
This learning material hopes to engage the learners into guided and independent
learning activities at their own pace and time. Further, this also aims to help learners
acquire the needed 21st century skills especially the 5 Cs namely: Communication,
Collaboration, Creativity, Critical Thinking and Character while taking into
consideration their needs and circumstances.
In addition to the material in the main text, you will also see this box in the body of
the module:
Notes to the Teacher
This contains helpful tips or strategies that
will help you in guiding the learners.
As a facilitator you are expected to orient the learners on how to use this module.
You also need to keep track of the learners' progress while allowing them to manage
their own learning. Moreover, you are expected to encourage and assist the learners
as they do the tasks included in the module.
For the learner:
Welcome to the Basic Calculus: Module on The Limit of a Function!
The hand is one of the most symbolized part of the human body. It is often used to
depict skill, action and purpose. Through our hands we may learn, create and
accomplish. Hence, the hand in this learning resource signifies that you as a learner
is capable and empowered to successfully achieve the relevant competencies and
skills at your own pace and time. Your academic success lies in your own hands!
This module was designed to provide you with fun and meaningful opportunities for
guided and independent learning at your own pace and time. You will be enabled to
process the contents of the learning material while being an active learner.
This module has the following parts and corresponding icons:
Expectation - These are what you will be able to know after completing the
lessons in the module
Pretest - This will measure your prior knowledge and the concepts to be
mastered throughout the lesson.
Recap - This section will measure what learnings and skills that you
understand from the previous lesson.
Lesson- This section will discuss the topic for this module.
Activities - This is a set of activities you will perform.
Wrap Up- This section summarizes the concepts and applications of the
lessons.
Valuing-this part will check the integration of values in the learning
competency.
Posttest - This will measure how much you have learned from the entire
module.
EXPECTATION
Lesson: The Limit of a Function
Learning Objectives:
At the end of the learning episode, you are expected to:
1. illustrate the limit of a function using a table of values and the graph
of the function; and
2. distinguish between lim 𝑓(𝑥) and 𝑓(𝑐).
𝑥→𝑐
PRETEST
A. For each item, complete the table of values, and use the results to
estimate the value of the limit.
1. lim (𝑥 2 + 2𝑥 − 3)
𝑥→4
x
f(x)
2. lim
𝑥→1
3.9
3.99
3.999
4
?
4.001
4.01
4.1
0.9
0.99
0.999
1
?
1.001
1.01
1.1
𝑥 2 +𝑥−2
𝑥−1
x
f(x)
B. Determine if lim 𝑓(𝑥) = 𝑓(𝑐).
𝑥→𝑐
1. 𝑓(𝑥) = 𝑥 + 2; 𝑐 = −1
2. 𝑓(𝑥) = 𝑥 2 − 1; 𝑐 = −1
3. 𝑓(𝑥) =
𝑥 3 −𝑥
𝑥
;𝑐 = 0
C. Evaluate each indicated limit using the given graph.
1. lim 𝑓(𝑥)
2. lim 𝑓(𝑥)
𝑥→2
𝑥→0
3.. lim 𝑓(𝑥)
4. lim 𝑓(𝑥)
𝑥→0
𝑥→1
RECAP
It is important to first recall the important notions about functions. By
a function f with domain D, we mean a way of assigning to each object (usually
a real number) 𝑥 ∈ 𝐷 a unique object (usually a real number), which we denote
𝑓(𝑥). So, a function is always defined by specifying what is assigned to each
number in its domain. The numbers assigned, i.e., the numbers 𝑓(𝑥) where
𝑥 ∈ 𝐷, form what is called the range of the function.
A function is also defined as a set of pairs (𝑥, 𝑦) with the condition that
no two distinct pairs share the same first element. With this notation, we
usually write the “rule” of the assignment as the equation 𝑦 = 𝑓(𝑥). We are,
thus prompted to call x the independent variable as opposed to the
dependent variable y whose value depends on x.
It will also be helpful to us if we can recall the different kinds of
functions, the piecewise function, and how the graph of these functions are
sketched in a Cartesian Plane.
LESSON
Consider the function f defined by 𝑓(𝑥 ) =
𝑥 2 −9
𝑥−3
. We will investigate the
function values (i.e., y’s or f(x)’s) as x gets closer and closer to 3. Here we are
not concerned with the value of the function when 𝑥 = 3. In fact, for the given
function f, 𝑓(3) is undefined in which case, it is customary to indicate this
with an open circle when sketching the graph of the function f, at 𝑥 = 3.
If x is in the domain of f, then 𝑥 ≠ 3 so that
𝑓 (𝑥 ) =
𝑥 2 − 9 (𝑥 − 3)(𝑥 + 3)
=
= 𝑥 + 3.
𝑥−3
𝑥−3
So, we can say that 𝑓 (𝑥 ) = 𝑥 + 3, 𝑥 ≠ 3. The
graph of 𝑓(𝑥) is shown in Figure 1 at the right.
Figure 1
Table 1:
x
f(x)
0
3
1
4
2
5
2.5
5.5
2.9
5.9
2.99
5.99
2.999 2.9999 2.99999
5.999 5.9999 5.99999
6
9
5
8
4
7
3.5
6.5
3.1
6.1
3.01
6.01
3.001 3.0001 3.00001
6.001 6.0001 6.00001
Table 2:
x
f(x)
In Table 1, we let x approach 3 by starting from 0 and moving toward 3
from the left of 3. In Table 2, we let x approach 3 by starting from 6 and
moving toward 3 from the right of 3.
Notice that from Table 1 and 2, as x gets closer and closer to 3, 𝑓(𝑥)
gets closer and closer to 6. The number 6, which 𝑓(𝑥) gets close to when x
gets closer to, but not equal to, 3 is said to be the limit of 𝑓(𝑥) as x approaches
3. In symbols, we write lim 𝑓 (𝑥 ) = 6. We can make 𝑓(𝑥) as close as we like to
𝑥→3
6 by keeping x close enough to 3.
Definition of the Limit of a Function
Let f be a function at every number in some open interval
containing c, except possibly at the number c itself. If the value of f is
arbitrarily close to the number L for all the values of x sufficiently close to
c, then the limit of 𝑓(𝑥) as x approaches c is L. In symbols,
lim 𝑓(𝑥 ) = 𝐿
𝑥→𝑐
Example 1: Evaluate lim 2𝑥 2 using the table of values.
𝑥→−1
Solution: Assign the values of x that are close to 1 and evaluate the function
at those values.
From the left of -1:
x
-3
-2
2
18
8
𝑓 (𝑥 ) = 2𝑥
-1.5
4.5
-1.1
2.42
-1.01
2.0402
-1.001
2.004
-1.0001
2.0004
From the right of -1:
x
1
0
2
2
0
𝑓 (𝑥 ) = 2𝑥
-0.5
0.5
-0.9
1.62
-0.99
1.9602
-0.999
1.996
-0.9999
1.9996
By observing the two tables, we will see that
from both directions, as x assumes values closer to 1, the value of 𝑓(𝑥 ) = 2𝑥 2 becomes closer to 2. Thus,
lim 2𝑥 2 = 2.
𝑥→−1
This is clearly shown in Figure 2.
Figure 2
Example 2: Evaluate the indicated limit using the graph: lim 𝑓(𝑥)
𝑥→0
Figure 3
Solution: The given is an example of a piecewise function, a function that
comes from combining two or more different functions. In fact, the function
is defined by
𝑥 + 2 𝑖𝑓 𝑥 ≠ 0
}
𝑓 (𝑥 ) = {
1
𝑖𝑓 𝑥 = 0
Using the definition of limit of a function, the limit of this function is 2
even if it is evident that 𝑓(0) = 1 (See Figure 3). As f approaches 𝑥 = 0 from
both directions, the value of y approaches its “intended” value, which is 2.
Thus,
lim 𝑓(𝑥) = 2.
𝑥→0
Limit and Function Value
The limit of a function as it approaches 𝑥 = 𝑐 is not necessarily equal
to its value at c. Thus, lim 𝑓(𝑥) can assume a value different from 𝑓(𝑐).
𝑥→𝑐
Example 3: Evaluate the following limits: lim 𝑓(𝑥) and lim 𝑔(𝑥)
𝑥→0
𝑥→−1
Figure 4
Solution: The specified limits do not exist. In the first function f, the limit does
not exist because 𝑓(0) is undefined and as 𝑥 → 0, the graph (from the left and
from the right) moves to opposite directions. In the second function g, the limit
does not exist because the function must approach the same value as x
approaches c from both directions. From the graph, the limit of g as 𝑥 → 1
from the left is 2, while the limit of g as 𝑥 → 1 from the right is 3.
Existence of a Limit
The limit of a function as 𝑥 → 𝑐 exists if
•
•
𝑓(𝑐) is defined; or
if 𝑓(𝑐) is not defined, then f must approach the same value as x
moves closer to c from both directions
ACTIVITIES
Evaluate the given limits numerically (using table of values) and graphically:
1. lim
𝑥 2 −4
𝑥→2 𝑥−2
2. lim
|𝑥−2|
𝑥→2 𝑥−2
3. lim 𝑓(𝑥) when 𝑓 (𝑥 ) = {
𝑥→2
𝑥 𝑖𝑓 𝑥 ≠ 2
}
3 𝑖𝑓 𝑥 = 2
WRAP–UP
To wrap-up, answer the following questions:
1. What is the definition of a limit of a function?
2. How can we get the limit of a function using table of values? How about
graphically?
3. How do you differentiate the limit of a function from a function value?
4. When do we say that the limit of a function exist?
VALUING
The limit of a function at a specified value of x gives us a value to which
it is not possible to go beyond. Similarly, we have our own limitations. We are
restricted to do things beyond our human capacities.
How do you relate the idea of the existence of the limit of a function in
life as a student? as a family member? as part of the community?
POSTTEST
A. Using table of values, determine the limits of the following:
1. lim 𝑥
2. lim
𝑥 2 −1
𝑥→1 𝑥−1
𝑥→0
B. The graph of 𝑓(𝑥 ) is given in Figure 5. Determine the following limits:
1. lim 𝑓(𝑥)
4. lim 𝑓(𝑥)
2. lim 𝑓(𝑥)
5. lim 𝑓(𝑥)
𝑥→−4
𝑥→3
𝑥→−1
𝑥→5
3. lim 𝑓(𝑥)
𝑥→1
C. Determine if lim 𝑓(𝑥) = 𝑓(𝑐).
𝑥→𝑐
1. 𝑓(𝑥) = 𝑥 − 2; 𝑐 = 0
2. 𝑓(𝑥) = 𝑥 2 − 4; 𝑐 = 2
3. 𝑓(𝑥) =
𝑥 2 −1
;𝑐
𝑥−1
=1
Figure 5
KEY TO CORRECTION
REFERENCES
BOOK
Canlapan, Raymond B. Basic Calculus. Diwa Learning Systems, Inc., Makati
City. 2017
Cuaresma, Genaro A. et al. 2004. Analytic Geometry and Calculus 1: A
Worktext for Math 26. Los Baños, Laguna: Institute of Mathematical
Sciences and Physics, University of the Philippines.
Department of Education-Bureau of Learning Resources. 2016. Precalculus
Learner's Material.
Leithold, Louis. 1989. College Algebra and Trigonometry. Addison Wesley
Longman Inc., reprinted by Pearson Education Asia Pte. Ltd., 2002.