AKLAN POLYTECHNIC COLLEGE
Quezon, Avenue, Kalibo, Aklan
SENIOR HIGH SCHOOL DEPARTMENT
Second Semester School Year 2021 -2022
Prepared by:
Approved by:
CHEJAN B. ROBERTO, LPT
Subject Teacher
Subject: Math 123
Dr. ANTONIA S. ESCALONA
Principal
Grade Level: 11
Effective Date: January 10, 2022
Date of Submission: February 08, 2022
Revision No: 00
Pages : 1 - 23
LEARNING MODULE IN
BASIC CALCULUS
(SPECIALIZED SUBJECT)
Name of Student
______________________________________________________
Grade, Strand & Section
Class Adviser
1
BASIC CALCULUS
Topic:
Limits from Tables and Graphs
MOST ENHANCED LEARNING COMPETENCIES
At the end of the lesson the students will be able to:
At the end of the lesson the students will be able to:
Illustrate the limit of a function using a table of values and the graph of a function.
DISCUSSION
The concepts of limits is important in calculus.
The limit of a function ( ) is the value it approaches as the value of x approaches a certain value.
“As x approaches a, the limit of ( ) approaches L. This is written in symbols as follows:
( )
Example:
Find
Solution: The arrowing pointing at 4 indicates that x is approaching 4 from the left side and from the
right side of 4. This means that x can take on values less than 4 and the values greater than 4. It
cannot take 4 as the value because it is just approaching 4. The first thing to do to find the limit of the
given function is to construct tables of values. In table 1, let x take on some values of x that are less
than 4. These values of x get closer and closer to 4. In table 2, let x take on some values x that are
greater than 4. This values of x get closer and closer to 4.
Table 1
( )
( )
2
4
2.5
4.5
3
5
3.1
5.1
3.3
3.5
5.3
5.5
Table 2
3.7
5.7
3.9
5.9
3.99
5.99
3.999
5.999
( )
( )
6
8
5.5
7.5
5
7
4.9
6.9
4.7
6.7
4.3
6.3
4.1
6.1
4.01
6.01
4.001
6.001
4.5
6.5
The graph is shown below.
As the value of x gets closer to 4 from the left or
as x approaches 4 from the left, the value of
( ) approaches 6. As the value of x gets closer
to 4 from right or as x approaches 4 from the
right,
( ) approaches 6. In other words, the value of
( ) gets closer and closer to 6 as the value of x
gets closer and closer to 4 from either side. This
can be written as follows:
. This
) is 6 as x
mean that the limit of (
approaches 4 from either side.
Example:
Find
.
Table 1
( )
( )
2
0.5
-2.5
1
-2
BASIC CALCULUS
1.1
-1.9
1.3
-1.7
1.5
-1.5
1.7
-1.3
1.9
-1.1
1.99
-1.01
1.999
-1.001
Table 2
( )
4
3.5
( )
1
5
The graph is shown below.
3
0
2.9
-0.1
2.7
-0.3
2.5
-0.5
2.1
-0.9
2.01
-0.99
2.001
-0.999
In table 1, as the value of x gets closer and
closer to 2 from the left of 2, the value of
( ) gets closer and closer to -1. In table 2, as
the value of x gets closer and closer to 2 from
the
right
of
2,
the
value
of
( ) gets closer and closer to -1.
Hence,
Example:
.
Solution: Construct the following table of values.
Table 1
-2
2
( )
( )
0
4
0.1
4.1
0.3
4.3
0.5
4.5
0.7
4.7
0.9
4.9
0.99
4.99
0.999
4.999
1.3
5.3
1.1
5.1
1.01
5.01
1.001
5.001
Table 2
3
7
( )
( )
2
6
1.9
5.9
1.7
5.7
1.5
5.5
The graph is shown below.
In table 1, as the value of x that is less than 1
gets closer and closer to 1, the value of
( ) gets closer and closer to 5. In table 2, as the
value of x that is greater than 1 gets closer and
closer
to
1,
the
value
of
( ) gets closer and closer to 5.
Hence,
Example 4:
Solution:
Let
Notice that
( )
( ) is not defined at x = 3. If x = 3. If x is 3, then
( )
which is
undefined. In the language of calculus, it indeterminate. What should be done here is to simplify the
expression by factoring its numerator. This can be done because
it is only approaching 3.
(
)(
)
Construct the following table of values.
Table 1
( )
( )
3
1
0
2
1
BASIC CALCULUS
2.1
1.1
2.3
1.3
2.5
1.5
2.7
1.7
2.9
1.9
2.99
1.99
2.999
1.999
Table 2
4.5
3.5
( )
( )
4
3
3.9
2.9
3.7
2.7
3.5
2.5
3.3
2.3
3.1
2.1
3.01
2.01
3.001
2.001
The graph is shown below.
As the value of x that is less than 3 gets closer
and
closer
to
3,
the
value
of
( ) gets closer and closer to 2. As the value of x
that is greater than 3 gets closer and closer to 3,
the
value
of
( ) gets closer and closer to 2.
(
Hence,
)(
)
Example: Find
Solution:
Let ( )
Notice that ( ) is not defined when x = 1. Factor the numerator of the expression
and cancel one of the factors with (
(
)(
)
)
Construct the following tables of values.
Table 1
( )
( )
-3
4
-2
1
0
1
0.3
1.69
0.5
2.25
0.7
2.89
0.9
3.61
0.99
3.9601
0.999
3.996
1.3
5.29
1.1
4.41
1.01
4.040
1.001
4.004
Table 2
( )
3
2
1.9
1.7
1.5
( )
16
9
8.41
7.29
6.25
The graph of the function is the graph of the parabola
As the value of x gets closer to 1 from the left, or
as x approaches 1 from the left, the value of
( ) Approaches 4. As the value of x gets closer
and closer to 1 from the right or as x
approaches 1 from the right, ( ) Approaches
4. In other words, the value of ( ) gets closer
and closer to 4 as the value of x gets closer and
closer to 1 from either side.
Hence,
4
BASIC CALCULUS
(
)
One Sided-Limit
Example 6: Let f be defined by the equation ( )
a. Evaluate
.
√
b. Evaluate
( )
( )
Solution:
a. The
( )
√
does not exist because √
is not a real number if
b. Table 1
( )
6
4
( )
2
1.414
The graph is shown below.
3
1
2.7
0.8367
2.5
0.7071
2.1
0.3162
2.01
0.1
2.001
0.031
2.0001
0.01
Hence, this is one –sided limit from the right. The
independent variable x approaches 2 from the
right.
√
Example: Let
a. Evaluate
be defined by the equation ( )
√
.
b. Evaluate
( )
( )
Solution:
a. The
( )
√
does not exist because √
is not a real number if
b. Table 1
( )
-2
-1
( )
2.23
2
The graph is shown below.
0
1.732
2.3
0.8367
2.5
0.7071
2.7
0.5477
2.9
0.3162
2.99
0.1
2.9999
0.01
Hence, this is a one sided limit from the left.
When a Limit Does Not Exist
Example: Suppose ( )
,
The graph is shown below.
Solution
( )
a.
b.
( )
( )
c.
( )
( )
Note that
does not exist.
Example: ( )
{
Solution:
5
BASIC CALCULUS
( )
( ) Hence,
( )
( )
( )
-2
3
-1
0
0
-1
1
0
( )
3
-1
4
0
5
1
6
2
( )
a.
( )
b.
( )
c.
( )
Hence,
( )
( ) does not exist.
Activity
I. Evaluate the following limits.
1.
(
2.
)
(
3.
)
(
)
II. Solve the given problem.
1.
( )
,
Evaluate a.
( ) b.
( ) c.
( )
Topic:
Limits Laws
MOST ENHANCED LEARNING COMPETENCIES
At the end of the lesson the students will be able to:
Distinguish between
( ) and ( )
Illustrate limit laws.
Apply the limit laws in evaluating the limit of algebraic functions (polynomial, rational and radical).
Discussion:
Theorems on Limits of Functions
1. The limit of a constant is that constant.
2. Limit of the identity function
3. The limit of a constant c times a function is
equal to the constant c times the limit of the
( ) exists.
function provided that
( )
Illustrations:
a.
b.
Illustration
a.
b.
Illustration
( )
( )
4. The limit of the sum of two functions is equal to
the sum of the individual limit provided that the
limit of each function as
exists.
[ ( )
( )]
( )
( )
Illustration
5. The limit of the difference of two functions is
equal to the difference of the individual limit
provided that the limit of each function as
exists.
Illustration
6
BASIC CALCULUS
[ ( )
( )]
( )
( )
6. The limit of the product of two functions is the
product of their limits provided that the limit of
each function as
exists.
[ ( ) ( )]
( )
( )
Illustrations
(
a.
7. The limit of the quotient of two functions is the
quotient of their limits provided that the limit of
the divisor is not equal to 0 and the limit of each
function exists.
Illustrations
( )
( )
( )
(
provided
)
√
)
( )
( )
Illustration
(
)
(
)
( )
( )+
*
9. The limit of the nth root of a function is equal
to the principal nth root of the limit of that
function provided that n is a positive integer and
that the limit of the function is positive if n is
even.
√ ( )
(
( )
8. The limit of the nth power of a function is
equal to the power of the limit of that function
provided that n is a positive integer and the limit
of ( ) as
exists.
[ ( )]
)
Illustration:
√
√
√( )
√
( )
10. If is a polynomial function and c is a real
( )
( )
number , then
Illustration:
(
)
Solution:
(
) is a polynomial function and 5 is a real
number.
( )
( )
Hence,
( )
( )
Example: Find
Solution:
( )
(
)
(
)
( )
Example: Find
Solution:
(
)
(
)
( )
( )
Example:
Solution:
7
(
BASIC CALCULUS
)
( )
( )
( )
( )
( )
Example: Find
Solution: The limit law for the quotient of two functions cannot be applied here because
If
then x can be close to 7, that is,
Hence, the numerator and denominator can be
)(
)
divided by x – 7. Notice that
is a difference of two squares and is equal to (
(
)(
)
Example: Find
Solution:
But x is only approaching 1 and is not equal to 1. Hence, if
factor equal to x -1, this factor can be cancelled with x -1 in the denominator.
(
)(
(
(
)
( )
Example: Evaluate
Solution:
√
√
(
√
)
(
)
√
( )
√
( )
( )
√
8
√
BASIC CALCULUS
)
)
has a
√
√
√
√
√
Activity
Evaluate the following limits.
1.
(
)
2.
(
3.
)
(
)
Infinite Limits
(
)
Infinite limits- the function values decrease or increase without bounds as the independent variable
gets closer and closer to a certain fixed number.
Hence, the function has an infinite limit if the denominator is 0 and the numerator is not 0.
Example: Evaluate each of the following limits.
a.
b.
c.
Solution:
a.
Construct a table of values. Let x take on the values 4, 3.7, 3.1, 3.01, 3.001, 3.0001, and 3.00001. These
values of x are greater than 3 but are getting closer to 3. Let ( )
Notice that as the value of x
becomes smaller and approaches x, the value of ( ) becomes larger and larger.
4
( )
2
It appears that
3.7
2.85714
( )
3.4
5
3.1
20
3.01
200
3.001
2000
3.0001
20000
3.00001
200000
b.
Let
( )
Cconstruct a table of values. Notice that as x approaches 3 from the left, or as the
value of x gets closer to 3, the value of ( ) becomes smaller and smaller.
2
( )
-2
It appears that
c.
( )
Hence,
2.1
-2.2222
( )
2.3
-2.8571
2.6
-5
2.9
-20
2.99
-200
2.999
-2000
2.9999
-20000
( )
( ) does not exist.
The graph is shown below.
The broken lines that the graph of the function
approaches but never touches is a vertical
asymptote. To find the vertical asymptote, set
denominator to 0 and solve for x.
The vertical asymptote
x-3=0
x =3
9
BASIC CALCULUS
Example: Evaluate the following limits.
b.
a.
c.
Solution:
a.
Construct a table of values. Let ( )
Let x take on the values1,2,2.3,2.6 and so on. Then, observe
the values of ( )
1
2
2.3
2.6
2.9
2.99
2.999
2.9999
( )
1
2
2.8571
5
20
200
2000
20000
Notice that as the value of x gets closer to 3, the value of ( ) gets larger and larger. It appears that
( )
b.
Construct a table of values. Let x take on the values 5,4,3.7,3.4, and so on. These are some values of x
greater than 3 but not equal to 3. Notice in the table that as the value of x gets closer and closer to
3, the value of ( ) becomes smaller and smaller.
5
-1
( )
It appears that
c.
( )
4
-2
3.7
-2.8571
3.4
-5
3.1
-20
3.01
-200
3.001
-2000
3.0001
-20000
The graph appears below.
( )
( )
Hence,
( ) does not exist.
Example: Consider the function defined by ( )
(
)
Evaluate
a.
(
b.
)
(
c.
)
(
)
Solution:
a.
(
)
Table 1
2
0.5
( )
b.
(
3
2
3.3
4.0816
3.6
12.5
3.9
200
3.99
20,000
3.999
2,000,000
3.9999
200,000,000
)
Table 2
( )
6
0.5
5
2
4.7
4.0816
4.4
12.5
4.1
200
4.01
20,000
4.001
2,000,000
4.0001
200,000,000
It can be seen in the two tables of values that as the value of x gets closer and closer to 4 from either
side, ( ) increases without bound.
It appears that:
10
BASIC CALCULUS
Hence,
(
)
(
)
(
)
Example: Consider the function defined by
( )
(
)
Evaluate the following.
a.
(
b.
)
(
c.
)
(
)
Solution:
a.
(
)
Table 1
2
-0.5
( )
(
b.
c.
3.3
-4.0816
3.6
-12.5
3.9
-200
3.99
-20,000
3.999
-2,000,000
5
-2
4.7
-4.0816
4.4
-12.5
4.1
-200
4.01
-20,000
4.001
-2,000,000
)
6
( )
-0.5
It appears that
a.
3
-2
(
)
(
)
(
)
Limits as Infinity
In the previous lesson, infinite limits were discussed. In infinite limits, the values of the function
decrease without bound as the independent variable approaches a certain fixed number. This time,
the independent variable either decreases or increases without bound.
Example: Consider the function defined by
Evaluate a.
( )
.
b.
Solution:
Construct a table of values. Let x take on the values 0.5,1,2,4,6,10,100, and so on. For each value of x,
find the corresponding value of ( )
0.5
1
( )
2
1
As the value of x increases,
approaches zero.
11
BASIC CALCULUS
2
4
6
10
100
1000
10000
0.5
0.25
0.16667 0.1
0.01
0.001
0.0001
( ) decreases. It appears that as x approaches positive infinity, ( )
b.
Construct a table of values. Let x take on the values -0.5,-1, -4, -6,-10,-100, and so on.Find the
corresponding values of ( )
-0.5
-2
-1
-1
-2
-0.5
-4
-.25
-6
-10
-100
-1000
-10000
( )
-0.1
-0.01
-0.001
-0.0001
0.16667
Notice that as x decreases, ( ) increases and is approaching zero. It appears that as x approaches
negative infinity, ( ) approaches zero.
The graph is shown below.
Example: Consider the function defined by ,
Evaluate a.
( )
b.
a.
Construct a table of values. Let x take on the values 0.5,1,4,8,10,100,1000, and so on.
0.5
1
( )
4
1
As x increases without bound,
4
8
0.0625
0.015625
( ) approaches 0.
10
0.01
100
0.0001
1000
0.000001
10000
0.00000001
b.
Construct a table of values. Let x take on the values -0.5, -1, -4, -8, -10, -100, -1000, and so on.
0.5
1
( )
4
1
As x decreases without bound,
The graph is shown below.
12
BASIC CALCULUS
4
8
0.0625
0.015625
( ) approaches 0.
10
0.01
100
0.0001
1000
0.000001
10000
0.00000001
If n is a positive integer, then
a.
b.
Example: Find
Example: Find
Solution:
Solution:
Example: Find
Example: Find
√
Solution:
Solution:
√
√
√
√
√
√
√
√
√
√
Limits at Infinity of Polynomials
The following are necessary in evaluating polynomials.
1.
for positive integer r
2.
{
Example: Evaluate
(
Evaluate
)
Solution:
Solution:
(
(
(
)
[
(
)]
)
(
)
To find the limit of a polynomial function where
factor out the variable x with the
highest exponent and apply the first rule above.
13
[
BASIC CALCULUS
(
(
(
)]
)
)
)
In the above example,
Hence, apply the
second rule.Since the exponent in
is odd
number,
Activity
Construct a table of values. Sketch the graph and find the vertical asymptote. Then find the
indicated limits.
1.
2. ( )
( )
a.
b.
a.
b.
Limits of Exponential, Logarithmic, and Trigonometric Functions
Topic:
MOST ENHANCED LEARNING COMPETENCIES
At the end of the lesson the students will be able to:
Compare the limits of exponential, logarithmic, and trigonometric functions using table of values and
graphs of the functions.
2. Evaluate limits involving the expressions
and
using table of values.
DISCUSSION
Limits of Exponential Functions
An exponential function is a function in which the exponent of the expression is a variable. It is of the
form ( )
Where: b = any real number such that
= any real number
The limits of exponential functions can be computed by using table of values and graphs.
Example: Consider the exponential function ( )
Compute
using a table of values and a
graph.
Solution:
a. Construct a table of values. Since
let The graph passes through the point ( ) The
x take on the number -3, -1,0,1,3,5, and so on.
graph increases without bound as x approaches
positive infinity. Notice that in the table of
( ) -3
values, as the value of x increases, the value of
-1
0
1
3
5
7
9
( ) increases too. This means as
0.125 0.5 1
2
8
32 128 512
approaches
Example: Find
( ) using a table of values and a graph.
Solution: Let x take on the values -2, -10,1,3,5, and so on. Then, for each x, find the corresponding
values of ( )
( )
14
-2
4
BASIC CALCULUS
-1
2
0
1
1
0.5
3
0.125
5
0.03125
7
0.00781
9
0.00195
The graph passes through the point ( ) The
graph is asymptotic to the x axis as x
approaches positive infinity. It can be seen in
the table as well as in the graph that as
the function
( )
( ) approaches 0.
( )
Example: Find
using a table of values and a graph.
Solution: Let x take on the values 2,1,0, -1, and so on. These number are decreasing in values. Then,
for each x find the corresponding value of ( )
2
4
( )
1
2
0
1
-1
0.5
-3
0.125
-5
0.03125
-7
0.00781
-9
0.00195
The graph passes through the point ( ) It is
asymptotic to the x-axis as x approaches
negative infinity. The table of values and the
graph show that as x approaches negative
infinity, the function approaches 0.
Example: Find
( ) using table of values and a graph.
Solution: Since
let x take on values that are approaching negative infinity such as 3,2,1,0,-1,3, and so on. Then, for each x compute the corresponding value of ( )
( )
3
0.125
2
0.25
1
0.5
0
1
-1
2
-2
4
-3
8
-7
128
-9
512
-11
2048
The graph passes through point ( ) The graph
increases without bound as x approaches
negative infinity. In the table of values, as the
value of x decreases, the value of ( )
increases. This means that as
the
function approaches negative infinity.
( )
Summary:
1. if
a.
b.
2. if
a.
b.
Natural Exponential Function
The exponential function with base e is frequently used in advanced mathematics. The natural
exponential function is ( )
where e = 2.718281828…
The limit of a natural exponential function can be determined by using table of values and graphs.
Example: Find
15
BASIC CALCULUS
by using a table of values and by sketching the graph of ( )
Solution: Construct a table of values similar to the table below.
( )
-2
0.13534
-1
0.367879
0
1
1
2.71828
2
7.38906
3
20.08554
4
54,5982
5
148.4132
Notice that as the value of x increases, the
corresponding value of ( ) rapidly increases.
In the graph, the value of x increases without
bound.
Example: Find
by using a table of values and by sketching the graph of
( )
Solution: Construct a table of values. Let x take on the values 2, 1,0, -2, and so on. Compute the
corresponding values of ( ).
( )
2
7.38906
1
2.71828
0
1
-1
0.36788
-2
0.13534
-3
0.04979
-4
0.018316
As the value of x decreases, the value of ( )
decreases too but rapidly. As the value of x
approaches negative infinity, the function
approaches 0.
Example: Find
by using a table of values and by sketching the graph of
( )
Solution: Construct a table of values. Let x take on the values -2, -1,0,1, and so on. Find the
corresponding values of ( ) for each value of x.
( )
-2
7.38906
-1
2.71828
0
1
1
0.36788
2
0.13534
3
0.04979
4
0.018316
As the value of x increases, the value of ( )
decreases rapidly. As x approaches positive
infinity, ( ) approaches 0.
Example: Find
by using a table of values and by sketching the graph of
( )
Solution: Construct a table of values. Let x take on the values 2, 1,0, -1,-2, and so on. Find the
corresponding values of ( ) for each value of x.
( )
16
2
0.13534
BASIC CALCULUS
1
0.36788
0
1
-1
2.71828
-2
2.71828
-3
20.08554
-4
54.5982
As the value of x decreases, the value of ( )
increases rapidly . As the value of x approaches
negative infinity, the function ( ) approaches
positive infinity.
Summary:
1.
2.
3.
4.
Limits of Logarithmic Functions
( )
Exponential functions of the form
which is
is also a function. For
equivalent to
where graphed in the previous lesson. The inverse of
and
the logarithmic function
is
Remember that
is equivalent to
To construct a table of values for
Choose values for y and calculate the corresponding value of x.
use
Let y take on the values -3, -1,0,1,3,5,7, and so on. Compute the corresponding value of y for each
value of x.
Table 1
0.125 0.5 1 2 8 32 128 512
( )
-3
-1 0 1 3 5
7
9
2048
11
Let y take on the values 9, 7, 5,3,1, and so on.
Compute the corresponding values of x.
512 128
( ) 9
7
Table 2
32 8 2 1 0.5 0.125
5
3 1 0 -1 -3
0.03125
-5
In table 1, the values of x are increasing
rapidly. The values of y are also increasing,
although slowly. Although slowly, y will
approach infinity. This can be seen in the
graph.
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BASIC CALCULUS
Hence,
In table 2, as the value of x approaches zero
from the right, the values of approaches
negative infinity.
Hence,
Notice that if the graphs of
and
are symmetrical about
hence the graphs
of
and the graph of
are
symmetric about
The same thing is true with the graphs of
( )
and ( )The symbol ( ) is used for
natural logarithms.
( )
Table 1 for ( )
( )
Table 2 for
( )
-2
0.13534
-1
0.367879
0
1
1
2.71828
2
7.38906
3
20.08554
4
54.5982
0.13534
-2
0.367879
-1
1
0
2.71828
1
7.38906
2
20.08554
3
54.5982
4
Notice that in table 1, as the value of x increases
the value of ( ) increases rapidly. In table, the
value of x increases rapidly. The value of y also
increases,
although
slowly.
In the graph of ln x, as x moves to the positive
( )
direction, although slowly. Hence,
If the value of x decreases and approaches
zero, the value of y will approach negative
infinity.
( )
Example: Find the limits of the following.
1.
2.
3.
Solution:
Summary:
1.
3.
2.
a. The limit of
( )
( )
4.
( )
as t approaches 0.
Construct table of values. The value of t should be in radians.
(
)
-1.5
0.664997
-0.8
0.896695
-0.5
0.958851
-0.2
0.993347
-0.08
0.998934
-0.03
0.999850
-0.01
0.999983
-0.001
0.999999
(
)
1.5
0.664997
0.8
0.896695
0.5
0.958851
-0.2
0.993347
0.08
0.998934
0.03
0.999850
0.01
0.999983
0.001
0.999999
1.
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BASIC CALCULUS
a.
(
b.
)
(
c.
)
(
)
Follow this procedure for the other values of
To sketch of the graph of
is shown below.
It can be observed from the table of values that
b. The limit of
as
can also be determined by using tables of values.
(
)
-1.5
-0.619508
-0.5
-0.244835
-0.2
-0.099667
-0.08
-0.039979
-0.03
-0.014999
0.01
-0.005
-0.001
-0.0005
(
)
1.5
-0.619508
0.5
-0.244835
0.2
0.099667
0.08
-0.039979
0.03
0.014999
0.01
0.005
0.001
0.0005
Computations:
a. *
(
)+
b. *
(
)+
c. *
(
)+
d. *
(
)+
e. *
(
)+
f. *
(
g. *
)+
(
)+
It can be observed from the table of values above that as the values of t approaches zero, the
values of
also approaches zero. This means that the limit of
is zero as
Activity
Evaluate the following limits.
1.
2.
3.
CONTINUITY
Topic:
MOST ENHANCED LEARNING COMPETENCIES
At the end of the lesson the students will be able to;
Illustrate continuity of a function at a number;
Determine whether a function is continuous at a number or not;
Illustrate continuity of a function on an interval;
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BASIC CALCULUS
.
Determine whether a function is continuous on an interval or not.
Continuity at a Number
A function ( ) is said to be continuous at the number a if all the following three conditions are
satisfied.
( ) exists
( )
( )
a. ( ) exists
b.
c.
If any one or more of the above three conditions fail, the function ( ) is said to be discontinues at a
number a.
Example: Determine whether
is continuous at x = 1.
( )
Solution:
a. ( )
b.
( )
Example: Determine whether the function ( )
1. x = 0
( )
( )
c.
The three conditions are
satisfied. The function is
continuous at x = 1.
( )
( )
continuous at :
2. x = 2
3. x = 3
1. At x = 0
2. At x = 2
3. at x = 3
a. ( )
a. ( )
a. ( )
Solution:
( )
( )
( )
. The limit is indeterminate.
b.
(
b.
b.
=
(
)(
)
)
=
=3 + 3 = 6
( )
( ) Hence ( ) is
c.
( )
discontinuous at x = 3.
( )
The three conditions are
satisfied. Hence, ( )is
continuous at x = 0.
c.
( )
( )
The three conditions are
satisfied. Hence, ( ) is
continuous at x = 2.
The graph is shown below.
There is a break in the graph of the function at the point where x = 3. This indicated by the hole.
20
BASIC CALCULUS
Example: Investigate the continuity of the function.
( )
{
Solution: The graph is shown below.
( )
( )
( )
3
-2
4
-4
( )
3
2
2
1
There is a break in the graph at x = 3. Find out if the three conditions are satisfied.
At x = 3
a. ( )
( )
( )
b.
( )
(
)
( )
( )
c.
( )
( )
The second condition is not satisfied at x = 3. Hence, the function is discontinuous at x = 3.
Continuity in an open interval
A function ( ) is continuous on an open interval (
(
)
21
BASIC CALCULUS
) if it is continuous at every point on the interval
Illustration:
Continuity on a Closed Interval
A function ( ) is continuous on a closed interval [
1. it is continuous on the open interval (
] if
)
2. It is continuous from the right of a.
a. ( ) exists.
b.
( )
( )
( )
( )
( )
( )
3. It is continuous from the left of b.
a. ( ) exists.
b.
Illustration:
Continuity of a Polynomial, Rational, and Square Root Functions
1. A polynomial function is continuous at every real number, that is, it is continuous everywhere.
2. A rational function is continuous at every number in its domain.
3. A square root is continuous at every number in its domain.
Example: Determine whether the following function is continuous on the given interval.
( )
(
)
Solution:
( )
22
-3
4
BASIC CALCULUS
-2
0
-1
-2
0
-2
1
0
2
4
Since
the
polynomial
function
is
( )
continuous
(
at
)
every
real
number
then,
ACTIVITY
I. Evaluate the following. (Refer to lesson 1)
1.
2.
3.
II. Evaluate the following limits. (Refer to lesson 2)
1.
2.
3.
III. Construct table of values. Sketch the graph and find the vertical asymptote. Then find the
indicated limits. (Refer to lesson 2 –Infinite limits.)
1. ( )
2. ( )
a.
a.
b.
b.
IV. Evaluate the following limits. (Refer to Lesson 3)
2.
1.
3.
V. Determine whether or not the following are continuous functions. Sketch the graph.
1. ( )
2. ( )
3. ( )
Reference:
Orines, Fernando B. and Mercado Jesus P.Next Century Mathematics Basic Calculus, Phoenix
Publishing House, Quezon City Philippines, 2016.
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BASIC CALCULUS