Mathematics SM 015
Topic 8: Limits and Continuity
-------------------------------------------------------------------------------------------------------------LECTURE 1 OF 3
TOPIC
: 1.0 LIMITS AND CONTINUITY
SUBTOPIC
: 8.1 Limits
OBJECTIVES : ​At the end of the lesson students are able to:
​a)
state limit of a function ​f(x) as ​x approaches a given value ​a​,
b)
CONTENT
​8.1.a
.
apply the basic properties of limit.
:
Definitions of Limit
A function ​f(x) is said to approach a constant ​L as a limit when ​x approaches ​a ​as
below,
Example 1
Determine
Solution
X
f(x)​=​x​2​+1
-1
2
-0.5
1.25
Hence, the limit of
-0.25
1.0625
-0.001
1.000001
0
?
0.001
1.000001
is 1 as
approaches 0 from either side, i.e.
8.1b Properties of Limits
i.
If ​f(​ ​x​) = c, where c is a constant, then
,
Example 2
ii.
If ​f(x)​ = ​x​, then
,
Example​ ​3
iii.
If ​f(x)​ = ​x​n​, where n is a positive integer (n > 0), then
1
0.25
1.0625
0.5
1.25
1
2
Mathematics SM 015
Topic 8: Limits and Continuity
-------------------------------------------------------------------------------------------------------------,
Example​ 4
iv.
If
and
exist, then
=
=
Example 5
v.
If
and
exist, then
​
=
=
=
Example 6
vi.
If c is a constant, then
​
=
​
=
E​xample 7
vii.
If
and
exist, then
,
=
=
2
0
Mathematics SM 015
Topic 8: Limits and Continuity
--------------------------------------------------------------------------------------------------------------
EXERCISES
Find the limit of the below functions:
a.
b.
c.
ANSWERS
d.
a)
9
b) 16
c)
d)
e) 6
3
Mathematics DM025
Topic 7: Limits and Continuity
------------------------------------------------------------------------------------------------------LECTURE 2 OF 3
OBJECTIVES :
​c)
find
when
i.
ii.
and
by following methods:
factorization
multiplication of conjugates
CONTENT :
Limit of the Rational Function: ​The limit of a rational function can be found by
substitution when the denominator is not equal to zero.
If ​f(x)​ and ​g(x)​ are polynomials, and c is any number, then
Example 1
If the solution for limit problem of a rational function is
, we cannot calculate the
limit of the given rational function by substitution. We need to simplify the fraction by
(i)
factorization
(ii)
multiplication of conjugates
(i) Factorization Method
Example​ ​2
Find the
Example 3
Find the
Solution
If we factor the numerator and denominator, we can simplify the fraction as
1
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Mathematics DM025
Topic 7: Limits and Continuity
------------------------------------------------------------------------------------------------------(ii) Multiplication of Conjugates Method​.
If the solution for limit problem of a rational function related to surd using
substitution method equal to
then use the multiplication of conjugates method.
Example ​ ​4
Find the limits below:
(i)
Solution :
(ii)
(i)
(ii)
EXERCISES
1. Find the limits in the functions below:
(i)
(ii)
(iii)
(iv)
ANSWERS
2
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Mathematics DM025
Topic 7: Limits and Continuity
-------------------------------------------------------------------------------------------------------
1.
(i)
(ii)
(iii) 4
(iv)
LECTURER 3 OF 3
OBJECTIVES:
d)Find one-sided limits i.
ii.
e)Determine the existence of the limit of a function
Exclude
CONTENT:
where
One ​-​ Sided Limits
Before we discuss one-sided limit we will first look at the graphs below.
Graph (a)
and
Graph (b)
. Therefore the limit in graph (a) is ​two – sided.
but
is not exist. We say that the limit in graph (b) is
one-sided. ​A one - sided limit can either be a right-hand limit or a left –hand limit.
​Right- hand limit​:
i.e. the limit of ​f​ as ​x ​approaches ​a f​ rom the right is ​L
Left- hand limit​:
i.e. the limit of ​f​ as ​x ​approaches ​a f​ rom the left is ​M
One-sided limits are useful in taking limits of functions involving radicals.
​Example 1
​Find
Solution
3
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Mathematics DM025
Topic 7: Limits and Continuity
------------------------------------------------------------------------------------------------------​f(x)
​x
The domain for this function is
based on the right-hand limit only.
. In this example, the limit must be determined
Since
then
Example 2
Find the
​f​(​x​)
From the diagram,
therefore
Test for Existence of a Limit
If
then
Example 3
Given the function,
4
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Mathematics DM025
Topic 7: Limits and Continuity
------------------------------------------------------------------------------------------------------​f​(​x​) = ​ ​ ,
2​x +
​ 3,
Determine
Solution
Example 4
if it exists.
​ ,
2
Given ​f(x)​ = ,
​x >
​ 4.
,
find
5
.
,
Solution
Example 5
Given ​f(x)=
​
. Find
Solution
if
​if ​
5
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