PHILIPPINE STATE COLLEGE OF AERONAUTICS
INSTITUTE OF ENGINEERING AND TECHNOLOGY
AERONAUTICAL ENGINEERING DEPARTMENT
Learning Module 02: Limits
LEARNING
MODULE 02:
Limits
CALCULUS 1 – DIFERENTIAL
CALCULUS
Prepared by:
AERO FACULTY
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PHILIPPINE STATE COLLEGE OF AERONAUTICS
INSTITUTE OF ENGINEERING AND TECHNOLOGY
AERONAUTICAL ENGINEERING DEPARTMENT
Learning Module 02: Limits
TABLE OF CONTENTS
Time
allotment
Title
Introduction
Limits
Properties of limits
Example on limits
Activity 1
One-sided limits
Limits at infinity
Vertical asymptote
Activity 2
Tangent line
Derivatives
Examples on derivatives
Activity 3
Assignment
Online quiz
TOTAL
25 mins
20 mins
15 mins
40 mins
40 mins
20 mins
20 mins
20 mins
30 mins
90 mins
90 mins
30 mins
90 mins
107 mins
90 mins
707 mins
Page
4
6
7
8
9
9
10
11
12
12
14
16
17
17
Na
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PHILIPPINE STATE COLLEGE OF AERONAUTICS
INSTITUTE OF ENGINEERING AND TECHNOLOGY
AERONAUTICAL ENGINEERING DEPARTMENT
Learning Module 02: Limits
TABLE OF REFERENCES
References
No.
Ron Larson and Bruce Edwards, 2018 Calculus 11e
Ron Larson and David C. Falvo, 2009 Calculus An applied approach 8e
James stewart, 2016 CALCULUS 8e
John C. Sparks, 2005 Calculus without limits-Almost
Elliot Mendelson, PhD., 1998 Schaum’s outline of calculus
Ron Larson and Bruce Edwards, 2018 Calculus 11e
1
2
3
4
5
6
Table of figures
Figure no.
1
2
3
4
5
6
7
8
9
Title
Graph of a function
Limits approaching on both sides
Graph of x^3 – 1 / x - 1
Theorem of limits
Limits approaching on 1 side
Limits approaching to infinity
Tangent line
Secant lines to tangent line
General graph on derivatives
Sources
Google graphs
James stewart, 2016 CALCULUS 8e
James stewart, 2016 CALCULUS 8e
James stewart, 2016 CALCULUS 8e
James stewart, 2016 CALCULUS 8e
James stewart, 2016 CALCULUS 8e
James stewart, 2016 CALCULUS 8e
James stewart, 2016 CALCULUS 8e
James stewart, 2016 CALCULUS 8e
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PHILIPPINE STATE COLLEGE OF AERONAUTICS
INSTITUTE OF ENGINEERING AND TECHNOLOGY
AERONAUTICAL ENGINEERING DEPARTMENT
Learning Module 02: Limits
This Module aims the understanding of limits and a wider scope with Continuity and
theorems
LEARNING OUTCOMES
Course Learning Outcomes [CLO]
CLO 1 Analyze the function of
continuity by applying limit theorem.
CLO 4 Evaluate algebraic and/or
transcendental functions by applying
differentiation formulas.
CLO 7 Formulate a systematic solution
by determining the appropriate steps
and strategies in solving a problem
CLO 8 Exhibit analytical skills through
the calculation of mathematical
problems.
CLO 10 Display a professional
commitment to ethical practice through
the submission of classwork,
requirements, and activities
CLO 11 Display acquired knowledge
and skills by incorporating it into future
aeronautical engineering duties.
Module Learning Outcomes [MLO]
Topic Learning Outcomes [TLO]
MLO 1. Evaluate limits as it approach
a continuity or infinity by applying
strategies such as substitution, using TLO 3: Compute limits using
tables, simplification and/or one sided substitution, properties and continuity.
limits.
TLO 4: Compute limits as a function
approaches infinity using substitution
MLO 2. Solve derivative of algebraic and other approach such one sided
limits.
functions through application of basic
TLO 5: Compute the derivatives of
differentiation rules.
functions by using constant, power,
product, and quotient rules.
HONESTY CLAUSE
“As members of the academic community, students are expected to recognize and
uphold standards of intellectual and academic integrity. The state college assumes that
as a basic and minimum standard of conduct in academic matters, the students should
be honest and that they submit for credit the products only of their own efforts.”
_______________________________
Student’s Signature over Printed Name
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PHILIPPINE STATE COLLEGE OF AERONAUTICS
INSTITUTE OF ENGINEERING AND TECHNOLOGY
AERONAUTICAL ENGINEERING DEPARTMENT
Learning Module 02: Limits
INTRODUCTION:
Over the past years, we somehow see equations like this
, where
What does it imply? As you can see, if we directly equate x as negative one, y will be
equal to undefined (0/0).
(
(
)
)
Can we just simplify the equation? The answer is no! Why? Simplifying the equation
above gives you
. The original and simplified equation might be the same
but it differs at some point. Literally, it differs at a point. Shown below are the graphs
of 2 equations.
Figure 1
𝑦
𝑦
𝑥
𝑥
𝑥
Let me show you a solution that will prove my argument. Let’s say a=b
(
)(
)
(
)
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PHILIPPINE STATE COLLEGE OF AERONAUTICS
INSTITUTE OF ENGINEERING AND TECHNOLOGY
AERONAUTICAL ENGINEERING DEPARTMENT
Learning Module 02: Limits
(
)(
(
)
)
(
(
)
)
Definitely, two will not be equal to one. Our mistake is that we divide the equation by
(
) which is equal to 0. Don’t get me wrong; even if you apply limits on this
equation, it will never be right. Limits may exist or not, but in this case it doesn’t exist.
By definition:
If ( ) becomes arbitrarily close to a single number as
side, the limit of ( ) as approaches is . we write
approaches
( )
from either
The concept of a limit plays a fundamental role in the development of calculus. In
this lesson, we define limits and show how to evaluate them. We will also discuss
properties of limits and look at different ways that a limit can fail to exist.
PART 1 LIMITS
LESSON:
To sketch the graph of the function
( )
for values other than
, you can use standard curve-sketching techniques. At
, however, it is not clear what to expect. To get an idea of the behaviour of the
graph of near
, you can use two sets of x-values—one set that approaches 1
from the left and one set that approaches 1 from the right, as shown in the table.
Figure 2
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PHILIPPINE STATE COLLEGE OF AERONAUTICS
INSTITUTE OF ENGINEERING AND TECHNOLOGY
AERONAUTICAL ENGINEERING DEPARTMENT
Learning Module 02: Limits
Figure 3
The graph of is a parabola that has a hole at the point ( ), as shown in the
Figure. Although cannot equal , you can move arbitrarily close to , and as a
result ( ) moves arbitrarily close to . Using limit notation, you can write
( )
There are many ways to find a limit. The first is the table method, which is shown
above, second is simplifying method and let me show it to you.
From the above problem
(
, we know that
(
(
)(
)(
)
)
)
PROPERTIES OF LIMITS
We learned that the limit of ( ) as approaches does not depend on the value of
at
. It may happen, however, that the limit is precisely ( ). In such cases,
you can evaluate the limit by direct substitution.
Let
and
be real numbers, and let
be a positive integer.
1.
2.
3.
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PHILIPPINE STATE COLLEGE OF AERONAUTICS
INSTITUTE OF ENGINEERING AND TECHNOLOGY
AERONAUTICAL ENGINEERING DEPARTMENT
Learning Module 02: Limits
Figure 4
IMPORTANT CONSIDERATIONS
1. Learn to recognize which limits can be evaluated by direct substitution
2. When the limit of f(x) as x approaches c cannot be evaluated by direct
substitution, try to find a function g that agrees with f for all x other than x = c.
[Choose g such that the limit of g(x) can be evaluated by direct substitution.]
Then
apply
Theorems
to
conclude
analytically
that
( )
( )
( )
3. Use a graph or table to reinforce your conclusion.
EXAMPLE:
1.
( )
2.
( )
1.9
1.99
1.999
0.3448 0.3344 0.3334
√
2
?
2.001
2.01
2.1
0.3332 0.3322 0.3226
0
?
0.001
0.01
0.1
0.2887 0.2884 0.2863
√
-0.1
-0.01 -0.001
0.2911 0.2889 0.2887
√
√
√
3.
( )
2.9
2.99
2.999
-0.0641 -0.0627 -0.0625
3
?
3.001
3.01
3.1
-0.0625 -0.0623 -0.0610
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PHILIPPINE STATE COLLEGE OF AERONAUTICS
INSTITUTE OF ENGINEERING AND TECHNOLOGY
AERONAUTICAL ENGINEERING DEPARTMENT
Learning Module 02: Limits
4.
( )
-0.1
-0.01
0.9983 0.99998
-0.001
1.000
0
?
0.001
1.000
0.01
1
0.99998 0.9983
*Use radian mode for all trigonometric functions
5.
(
)
(
*
)
(
( )
)+
( )
( )
6.
(
)
(
(
)
)
( )
( )
( )
( )
( )
( )
( )
Activity 1
Write it on a clean paper and answer the problems lengthwise.
Please write your name on the top left corner and your course, year level, and the
section below. Insert at the top center of the paper
before the honesty clause. Copy the problem first before you show your solutions.
Place the honesty clause and your signature at the beginning of your work.
No erasures or alterations allowed. Box all your final answers.
Submit your scanned work to the provided link in the google classroom.
Solve the following Limits. (10 points each)
1.
2.
3.
√
4.
5.
9|Page
PHILIPPINE STATE COLLEGE OF AERONAUTICS
INSTITUTE OF ENGINEERING AND TECHNOLOGY
AERONAUTICAL ENGINEERING DEPARTMENT
Learning Module 02: Limits
PART 2 CONTINUITY, ONE-SIDED LIMITS, AND LIMITS AT INFINITY
LESSON:
Let’s approach limits on only one side
We write
( )
and say the left-hand limit of ( ) as approaches a [or
the limit of ( ) as
approaches a from the left] is equal to if we can make the
values of ( ) arbitrarily close to by taking to be sufficiently close to a with less
than .
Notice that Definition of one sided limits differs from the definition of limits only in that
we require to be less than a. Similarly, if we require that be greater than , we
get “the right-hand limit of ( ) as x approaches a is equal to ” and we write
( )
Thus the notation
means that we consider only
greater
than .
Figure 5
LET’S APPROACH LIMITS TO INFINITY
Let
be a function defined on both sides of , except possibly at itself. Then
( )
means that the values of ( ) can be made arbitrarily large
negative by taking sufficiently close to , but not equal to .
Figure 6
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PHILIPPINE STATE COLLEGE OF AERONAUTICS
INSTITUTE OF ENGINEERING AND TECHNOLOGY
AERONAUTICAL ENGINEERING DEPARTMENT
Learning Module 02: Limits
The vertical line
is called vertical asymptote of the curve
one of the following statements is true:
( ) if at least
DEFINITION OF VERTICAL ASYMPTOTE
If ( ) approaches infinity (or negative infinity) as approaches from the right or
the left, then the line
is a vertical asymptote of the graph of .
EXAMPLE
1. Find
and
If is close to but larger than , then the denominator
is a small positive
number and
is close to . So the quotient
is a large positive number. [For
instance, if
then
.] Thus, intuitively, we see that
Likewise, if is close to but smaller than , then
but
is still a positive number (close to ). So
is a small negative number
is a numerically large negative
number. Thus
The line
is a vertical asymptote.
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PHILIPPINE STATE COLLEGE OF AERONAUTICS
INSTITUTE OF ENGINEERING AND TECHNOLOGY
AERONAUTICAL ENGINEERING DEPARTMENT
Learning Module 02: Limits
Activity 2
Write it on a clean paper and answer the problems lengthwise.
Please write your name on the top left corner and your course, year level, and the
section below. Insert at the top center of the paper
before the honesty clause. Copy the problem first before you show your solutions.
Place the honesty clause and your signature at the beginning of your work.
No erasures or alterations allowed. Box all your final answers.
Submit your scanned work to the provided link in the google classroom.
Solve for the vertical asymptote of each functions. (10 points each)
1.
( )
2.
( )
3.
( )
4.
( )
5.
( )
(
)
Part 3 DERIVATIVES
TANGENT LINE
A secant line is a line that intersects a graph at two points, while the tangent line
touches the graph at a single point. Here is an example of a secant line:
Figure 7
To find the slope of a secant line, we must use the formula “change in y over change
in x” or “rise over run”.
( )
( )
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PHILIPPINE STATE COLLEGE OF AERONAUTICS
INSTITUTE OF ENGINEERING AND TECHNOLOGY
AERONAUTICAL ENGINEERING DEPARTMENT
Learning Module 02: Limits
To find the tangent line at point P we let Q approach P along the curve C by letting x
approach a.
Figure 8
The tangent line to the curve
P with slope
( ) at the point Psa, (
( )
( )) is the line through
( )
provided that this limit exists.
Find a slope of the tangent line to the parabola
( )
( )
(
)(
(
at the point P(1,1).
)
)
There is another expression for the slope of a tangent line that is sometimes easier
to use. If
, then
and so the slope of the secant line PQ is
(
)
( )
Notice that as x approaches a, h approaches 0 (because
) and so the
expression for the slope of the tangent line in Definition becomes
(
)
( )
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PHILIPPINE STATE COLLEGE OF AERONAUTICS
INSTITUTE OF ENGINEERING AND TECHNOLOGY
AERONAUTICAL ENGINEERING DEPARTMENT
Learning Module 02: Limits
Figure 9
Find a slope of the tangent line to the parabola
(
)
( )
at the point P(1,1) using
.
(
)
(
)
( )
(
)
(
)
( )
DERIVATIVES
The shortcut on finding the slope of tangent line is to find the derivative. Let’s study
the basic derivatives.
Derivatives can be denoted as
( )
.
can be read as dy over dx or (first) derivative of y with respect to x
( ) can be read as f prime of x or (first) derivative of function of x
can be read as y prime or (first) derivative of y
14 | P a g e
PHILIPPINE STATE COLLEGE OF AERONAUTICS
INSTITUTE OF ENGINEERING AND TECHNOLOGY
AERONAUTICAL ENGINEERING DEPARTMENT
Learning Module 02: Limits
1.
( )
2.
( )
3.
(
)
4.
(
)
5.
(
6.
(
7.
( )
)
)
What are these formulas?
1. The derivative of a constant is always zero
2. The derivative of any variable with respect to its variable is 1
3. The derivative of a constant multiplied by a variable has a constant not equal
to zero.
4. Power rule which can be associated as chain rule
5. Sum of derivatives
6. Product rule
7. Quotient rule
Apply power rule on finding the derivative of
.
Let;
Then,
*𝑥’ follows rule no. 2
From the above example
(
)
As you can see, from finding the equation of sloe of tangent line using both method
are the same. Therefore, finding the derivatives is finding the slope of tangent line.
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PHILIPPINE STATE COLLEGE OF AERONAUTICS
INSTITUTE OF ENGINEERING AND TECHNOLOGY
AERONAUTICAL ENGINEERING DEPARTMENT
Learning Module 02: Limits
EXAMPLES
Derive the following functions
1.
2.
(
)
1.
( )
(
(
)
)
(
(
)
)
( )
( )
2.
( )
(
)
[
(
)
(
(
)*
( )
( )+
)(
)
]
(
(
)
(
)*
( )
)(
)
( )+
)
)
(
(
)
(
)
(
)
(
(
)
)
(
(
(
)
)
16 | P a g e
PHILIPPINE STATE COLLEGE OF AERONAUTICS
INSTITUTE OF ENGINEERING AND TECHNOLOGY
AERONAUTICAL ENGINEERING DEPARTMENT
Learning Module 02: Limits
Activity 3
Write it on a clean paper and answer the problems lengthwise.
Please write your name on the top left corner and your course, year level, and the
section below. Insert at the top center of the paper
before the honesty clause. Copy the problem first before you show your solutions.
Place the honesty clause and your signature at the beginning of your work.
No erasures or alterations allowed. Box all your final answers.
Submit your scanned work to the provided link in the google classroom.
Solve for the Derivative of each functions. (10 points each)
1.
2.
3.
4.
(
(
(
(
)
)
)
)
5.
( )
6.
7.
8.
( )
( )
( )
9.
( )
10. ( )
√
√
(
(
)
)
)(
Assignment
Write it on a clean paper and answer the problems lengthwise.
Please write your name on the top left corner and your course, year level, and the section below. Insert
“Midterm Assignment” at the top center of the paper
before the honesty clause. Copy the problem first before you show your solutions.
Place the honesty clause and your signature at the beginning of your work.
No erasures or alterations allowed. Box all your final answers.
Submit your scanned work to the provided link in the google classroom.
Solve the following differentiation problems covered in the Midterm Modules.
A. Solve the following limits
a.
b.
√
c.
d.
e.
B. Solve for the vertical Asymptote of each functions
a. ( )
b.
( )
c.
( )
17 | P a g e
PHILIPPINE STATE COLLEGE OF AERONAUTICS
INSTITUTE OF ENGINEERING AND TECHNOLOGY
AERONAUTICAL ENGINEERING DEPARTMENT
Learning Module 02: Limits
d.
( )
e.
( )
C. Solve for the derivative of each functions
1. ( )
2. ( )
3. ( )
4. ( )
5. ( )
6. ( )
7. ( )
8. ( )
√
√
(
)
9. ( )
10.
( )
(
)
)(
(
)
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