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Basic Calculus
Quarter 3 – Module 1:
Limits of Algebraic Functions
using Tables and Graphs
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Basic Calculus – Grade 11
Alternative Delivery Mode
Quarter 3 – Module 1: Limits of Algebraic Functions using Tables and Graphs
First Edition, 2020
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Basic Calculus
Quarter 3 – Module 1:
Limits of Algebraic Functions
using Tables and Graphs
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Introductory Message
This Self-Learning Module (SLM) is prepared so that you, our dear learners,
can continue your studies and learn while at home. Activities, questions, directions,
exercises, and discussions are carefully stated for you to understand each lesson.
Each SLM is composed of different parts. Each part shall guide you step-bystep as you discover and understand the lesson prepared for you.
Pre-tests are provided to measure your prior knowledge on lessons in each
SLM. This will tell you if you need to proceed on completing this module or if you
need to ask your facilitator or your teacher’s assistance for better understanding of
the lesson. At the end of each module, you need to answer the post-test to self-check
your learning. Answer keys are provided for each activity and test. We trust that you
will be honest in using these.
In addition to the material in the main text, Notes to the Teacher are also
provided to our facilitators and parents for strategies and reminders on how they can
best help you on your home-based learning.
Please use this module with care. Do not put unnecessary marks on any part
of this SLM. Use a separate sheet of paper in answering the exercises and tests. And
read the instructions carefully before performing each task.
If you have any questions in using this SLM or any difficulty in answering the
tasks in this module, do not hesitate to consult your teacher or facilitator.
Thank you.
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What I Need to Know
One of the main reasons why this module was created is to ensure that it will assist
you to understand the concept and know the process of solving limits of a function.
When you finish this module, you will be able to:
1. illustrate the limit of a function using table of values and graph of the function;
and STEM_BC11LC-IIIa-1
2. distinguish between lim 𝑓(𝑥) and 𝑓(𝑐). STEM_BC11LC-IIIa-2
𝑥→𝑐
What I Know
I.
Answer the following questions. Write your answer on a separate sheet of
paper.
1. Evaluate lim (𝑥 + 2)
𝑥→3
A. 1
2. Calculate lim (
𝑥→7
𝑥+1
2
B. 2
C. 7
D. 5
B. 5
C. 4
D. 2
B. 2
C. 4
D. 6
B. 2
C. 7
D. 3
C. 17
D. 16
)
A. 3
3. Determine lim (√𝑥 − 2)
𝑥→11
A. 3
𝑥+5
𝑥→1 𝑥+2
4. Find lim (
)
A. 1
5. Provide the value of lim(𝑥 2 + 5𝑥 − 7 )
𝑥→3
A. 13
B. 12
1
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II.
Graph the following set of ordered pairs using one Cartesian plane. (item
numbers from 6-10)
x
y
III.
0
1
1
2
2
3
3
4
4
5
Answer the following questions. Write your answer on a separate sheet of
paper.
11.
Solve lim (𝑥 2 + 5𝑥 − 1)
𝑥→2
A. 11
12.
B. 12
C. 4
D. 5
B. -2
C. – 3
D. 2
C. –3
D. 2
C.
D. 6
Find lim (𝑥 − 8)
𝑥→5
Determine lim (𝑥 2 − 𝑥 − 3)
𝑥→3
A. 3
15.
B. 3
𝑥→1
A. 3
14.
D. 14
Evaluate lim ( 2𝑥 2 + 3𝑥 − 2)
A. 2
13.
C. 13
B. –2
Provide the value of lim (
𝑥→3
A. 3
𝑥 2 −9
𝑥−3
)
B.4
2
5
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Lesson
1
Limits of Algebraic
Expressions using Tables
and Graphs
Everyone has their own limitation. Knowing your own limit helps you understand
why some things are favorable to you and some are not. In this branch of
Mathematics called Calculus, Limit is one of the important lessons that you need to
understand because it plays a vital role in the application of differentiation towards
a function.
What’s In
Complete the table of values and graph the ordered pairs using one Cartesian plane.
Write your answer on a separate sheet of paper.
𝒙
𝒚=𝒙−𝟏
1
?
2
?
3
3
?
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What’s New
Read the situation and answer the question briefly.
Situation: Have you experienced walking on a street and noticed an image of a small
billboard from afar? Because of the distance, the message on that billboard is not
readable. What action would you take to be able to see clearly and understand the
message on that billboard?
What is It
•
The limit of a function 𝑓(𝑥) is the value it approaches as the value of 𝑥
approaches a certain value. “As 𝑥 approaches 𝑎, the limit of 𝑓(𝑥) approaches
L”. (Mercado, 2016)
This is written in symbols as follows;
𝐥𝐢𝐦 𝒇(𝒙) = 𝑳
𝒙→𝒄
•
One sided limit is the value (𝐿) as the 𝒙 value gets closer and closer to a
certain value 𝒄 from one side only (either from the left or from the right side).
In symbols,
𝐥𝐢𝐦 𝒇(𝒙) = 𝑳
𝐥𝐢𝐦 𝒇(𝒙) = 𝑳
𝒙→𝒄−
𝒙→𝒄+
From the left side
•
From the right side
Always remember that if the limit value from the left side is not equal to the
limit value from the right, then the limit Does Not Exist or DNE.
In symbols, if 𝐥𝐢𝐦− 𝒇(𝒙) = 𝑳
𝒙→𝒄
•
≠
𝐥𝐢𝐦 𝒇(𝒙) = 𝑳 , then 𝐥𝐢𝐦 𝒇(𝒙) 𝑫𝑵𝑬
𝒙→𝒄+
𝒙→𝒄
The limit of a function 𝐥𝐢𝐦 𝒇(𝒙) = 𝑳 is not the same as evaluating a function
𝒙→𝒄
𝒇(𝒄) because they are different in terms of concept. The limit of a function
gets its value by providing inputs that approaches the particular number while
evaluating a function is more like direct substitution process.
4
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How to Illustrate the Limit of a Function
Example:
Express in mathematical symbol: limit of the function (𝑥 + 3 ) as 𝑥 approaches 2 is
equivalent to 5.
Solution:
Write down your given 𝑓(𝑥) equivalent to (𝑥 + 3). Your 𝑐 is equal to 2 and your
limit 𝐿 is 5. Then substitute into the limit expression lim 𝑓(𝑥) = 𝐿.
𝑥→𝑐
Answer: lim(𝑥 + 3) = 5
𝑥→2
How to Solve for the Limit of a Function
Example:
1. With the given function 𝑓(𝑥) = 𝑥 + 3, solve for its limit when 𝑥 approaches 2
and graph the function.
Solution:
Step 1: Create two tables of value, one for the inputs that approaches 2 from the left
and the other is for the inputs that gets closer to 2 from the right side. (See figure
below).
𝒙<𝟐
𝒇(𝒙) or 𝒚
𝒙>𝟐
𝒇(𝒙) or y
Step 2: Choose 𝑥 − 𝑣𝑎𝑙𝑢𝑒𝑠 that approach 2 from the left side and also from the right
side. Remember that we cannot choose 2 because we are dealing with limits. (See the
number line figure below).
X – Values or
(inputs) that
approach 2
from the right
side
X – Values or
(inputs) that
approach 2 from
the left side
5
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After choosing 𝑥 − 𝑣𝑎𝑙𝑢𝑒𝑠 that approach 2 from the left and from the right side,
evaluate each input to its corresponding function and solve for its corresponding
output. (Refer to the table of values below).
𝒙<𝟐
0
1
1.5
1.9
1.99
1.999
1.9999
𝒙>𝟐
4
3
2.5
2.1
2.01
2.001
2.0001
𝒇(𝒙) or 𝒚
3
4
4.5
4.9
4.99
4.999
4.9999
𝒇(𝒙) or 𝒚
7
6
5.5
5.1
5.01
5.001
5.0001
Step 3: Now that the output values on both tables are solved, notice that it approach
5 as the inputs gets closer and closer to 2. We can illustrate both tables as one sided
limits from the left lim−(𝑥 + 3) = 5 and lim+(𝑥 + 3) = 5 from the right.
𝑥→2
𝑥→2
Step 4: Since both one sided limits from the left and right side is equivalent to 5,
therefore the limit of the function 𝑥 + 3 as 𝑥 gets closer and closer to 2 is 5.
In symbols,
lim (𝑥 + 3) = 5
𝑥→2
How to Graph the Limit of a Function
Use the coordinates from the table of values and plot them into a Cartesian plane.
Graph of
𝑓(𝑥) = 𝑥 + 3
On this area, it is
evident that as the
x-values approach 2
from the left and
from the right, the
y-values approach 5
from the left and
from the right as
well.
6
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Example:
𝑥 2 −1
)
𝑥→1 𝑥−1
2. Calculate lim (
and graph the function.
On this particular given, evaluating the function 𝑓(1) = (
(1)2 −1
) will
(1)−1
result into
0
0
or
indeterminate answer. But in the application of limits, there is a possibility that you’ll
get a defined value. For this reason, we can say that solving for the limit of a function
is different from evaluating a function.
Solution:
Step 1: Make two table of values, one for the inputs that approaches 1 from the left
and the other is for the inputs that gets closer to 1 from the right side. (See figure
below).
𝑥<1
𝑓(𝑥) or 𝑦
𝑥>1
𝑓(𝑥) or y
Step 2: Choose 𝑥 − 𝑣𝑎𝑙𝑢𝑒𝑠 that approaches 1 from the left side and also from the right
side. Remember that we cannot choose 1 because we are dealing with limits.
After choosing 𝑥 − 𝑣𝑎𝑙𝑢𝑒𝑠 that approaches 1 from the left and from the right side,
evaluate each input to its corresponding function and solve for its corresponding
output. (Refer to the table of values below).
𝑥<1
𝑓(𝑥) or 𝑦
0.5
0.9
0.99
0.999
1.5
1.9
1.99
1.999
𝑥>1
𝑓(𝑥) or y
1.5
1.1
1.01
1.0001
2.5
2.1
2.01
2.0001
Step 3: Once the output values on both tables were solved, notice that it approaches
2 as the inputs gets closer and closer to 1. We can illustrate both tables as one sided
limits from the left lim− (
𝑥→1
𝑥 2 −1
)
𝑥−1
= 2 and lim+ (
𝑥→1
𝑥 2 −1
)
𝑥−1
= 2 from the right.
Step 4: Since both one sided limits from the left and right side is equivalent to 2,
therefore the limit of the function (
In symbols,
𝑥 2 −1
)
𝑥−1
as 𝑥 gets closer and closer to 1 is 2.
lim (
𝑥→1
𝑥 2 −1
𝑥−1
)=2
Graph of the function:
In a single Cartesian plane, plot the coordinates from the table of values.
7
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Notice that a “hole’’ is
visible on the graph
because of the
Graph of 𝑓 (𝑥 ) =
0
0
or
indeterminate result
when x=1.
Nevertheless, the
limit is obvious
because it is evident
here that as the xvalues approach 1
from the left and
from the right, the yvalues approach 2
from the left and
from the right as
well.
𝑥 2 −1
𝑥−1
What’s More
Read and answer the following items. Write your answers on a separate sheet of
paper.
1. Evaluate lim(4 + 𝑥).
𝑥→2
𝑥<2
Input
from
the left
side
𝑓(𝑥) or 𝑦
Output
from the
left side
𝑥>2
Input
from the
right
side
𝑓(𝑥) or y
Output
from the
right side
0
1.5
1.99
1.9999
?
?
?
?
3
2.5
2.01
2.0001
?
?
?
?
8
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One sided limits from the right
One sided limits from the left
lim (4 + 𝑥) = ____
lim (4 + 𝑥) = ____
𝑥→2+
𝑥→2−
𝑙𝑖𝑚 (4 + 𝑥) = ____
𝑥→2
Graph of the function:
Use the coordinates from the table of values and plot them on the Cartesian plane
below.
𝑥 2 −2𝑥−3
).
𝑥−3
𝑥→3
2. Solve lim (
𝒙<𝟑
𝒇(𝒙) or 𝒚
𝒙>𝟑
𝒇(𝒙) or y
2.5
2.9
2.999
2.99999
?
?
?
?
3.5
3.1
3.001
3.00001
?
?
?
?
One sided limits from the left
𝑙𝑖𝑚− (
𝑥→3
One sided limits from the right
𝑥 2 − 2𝑥 − 3
) = ___
𝑥−3
𝑙𝑖𝑚 (
𝑥→3
𝑙𝑖𝑚+ (
𝑥→3
𝑥 2 − 2𝑥 − 3
) = ___
𝑥−3
9
𝑥 2 − 2𝑥 − 3
) = ___
𝑥−3
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Graph of the function:
Use the coordinates from the table of values and plot them on the Cartesian plane.
What I Have Learned
Express what you have learned in this lesson by completing the sentences below.
Use a separate paper for your answers.
1. The limit of a function is ___________________________________________.
2. One sided limit of a function is _____________________________________.
3. Is the limit of a function similar as evaluating a function? Explain briefly.
____________________________________________________________________
4. How can you solve the limit of a function? Explain briefly based on your own
understanding. ____________________________________________________
What I Can Do
Read and answer the given question below. Use a separate paper for your answers.
𝑥 + 3, 𝑥 > 3
Suppose you are given a piecewise function 𝑓(𝑥) = {
, evaluate the
𝑥² − 3, 𝑥 ≤ 3
limit of this function as 𝑥 approaches 3. Use the steps shown to you to get the final
answer.
10
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Assessment
I.
Calculate the limits of the following functions. Write your answer on a separate
sheet of paper.
1. lim (𝑥 − 10)
𝑥→6
A. 4
B. -4
C. – 5
D. 5
B. -7
C. – 6
D. 6
B. -13
C. 12
D. -12
B. -2
C. 2
D. 3
B. 2
C. 3
D. 0
2. lim (𝑥 2 + 5𝑥 − 7)
𝑥→2
A. 7
3. lim (
𝑥 2 −36
𝑥→6
𝑥−6
)
A. 13
4. lim (3𝑥 − 10)
𝑥→4
A. – 3
5. lim (𝑥 2 + 4𝑥 − 5)
𝑥→1
A. 1
II.
Read, analyze, and write the letter of the correct answer on a separate
sheet of paper.
6. If the left side limit of a function is not equal to the right-side limit, then
the limit exists.
A. True
B. False
C. Not Sure
D. No answer
7. If the left side limit of a function is equal to the right-side limit, then the
limit does not exist.
A. True
B. False
C. Not Sure
D. No answer
8. One sided limit from the left side is illustrated as lim+ 𝑓(𝑥).
𝑥→𝑐
A. True
B. False
C. Not Sure
D. No answer
9. One sided limit from the right side is illustrated as lim− 𝑓(𝑥).
𝑥→𝑐
A. True
B. False
C. Not Sure
D. No answer
10. There is no difference between the limit of a function and evaluating a
function.
A. True
B. False
C. Not Sure
D. No answer
11
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III.
For questions 11 to 15, refer to the graph below and write the letter of the
correct answer on a separate sheet of paper.
𝐺𝑟𝑎𝑝ℎ 𝑜𝑓 𝑓(𝑥)
11. Calculate lim+ 𝑓(𝑥)
𝑥→3
A. 1
B. 2
C. 3
D. 4
B. 1
C. 3
D. 4
B. 1
C. 2
D. 4
B. 3
C. 1
D. DNE
B. 2
C. 3
D. DNE
12. Solve lim− 𝑓(𝑥)
A. 2
𝑥→3
13. Determine lim 𝑓(𝑥)
𝑥→−3
A. 3
14. Evaluate lim 𝑓(𝑥)
𝑥→3
A. 2
15. Find lim + 𝑓(𝑥)
A. 1
𝑥→−3
12
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Additional Activities
Read, understand, and solve the given problem below. Use a separate sheet of paper
for your answer.
Vic and Joey argue about the limit of the function 𝑓(𝑥) =
1
𝑥
as 𝑥 approaches 0 . Vic’s
opinion is that the limit is 0 while Joey claims that the limit does not exist. Write a
short explanation that discusses the pros and cons of Vic and Joey’s opinion.
13
14
What’s More
What’s More
1.
2.
Assessment
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
B
A
C
C
D
B
B
B
B
B
A
B
C
C
B
What's New
Answers may
vary.
What I Know
What's In
x
y
1
0
2
1
1.
2.
3.
4.
5.
3
2
D
C
A
B
C
6-10.
Additional Activities
Joey is correct and Vic is wrong base
on their own answers about the limit
of the function because their one
sided limits is different from each
other, therefore the limit does not
exist.
11.
12.
13.
14.
15.
C
B
C
A
D
Answer Key
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References
DepEd. 2013. Basic Calculus. Teachers Guide.
Lim, Yvette F., Nocon, Rizaldi C., Nocon, Ederlina G., and Ruivivar, Leonar A. 2016.
Math for Engagement Learning Grade 11 Basic Calculus. Sibs Publishing
House, Inc.
Mercado, Jesus P., and Orines, Fernando B. 2016. Next Century Mathematics 11
Basic Calculus. Phoenix Publishing House, Inc.
Geogebra. Graphing application for android. Playstore.
15
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Department of Education – Region III,
Schools Division of Bataan - Curriculum Implementation Division
Learning Resources Management and Development Section (LRMDS)
Provincial Capitol Compound, Balanga City, Bataan
Telefax: (047) 237-2102
Email Address: bataan@deped.gov.ph