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Limits and Continuity of a
Function
(Module 1)
Mat051
AY 2020-2021 (1st semester)
This material is not for sale and is owned by the Department of Mathematics and Statistics,
College of Science and Mathematics of Mindanao State University-Iligan Institute of Technology.
LIMITS AND CONTINUITY OF FUNCTIONS
The limit concept is considered to be the most essential concept in calculus. It
is used to define continuity, derivatives and integrals.
Consider the function defined by
. When
, an
indeterminate. Let’s look at how the values of the function behaves when
gets closer and closer to .
1
3
⺁
2.5
4.5
1.5
3.5
2.4
4.4
1.6
3.6
2.3
4.3
1.7
3.7
2.2
4.2
1.8
3.8
2.1
4.1
1.9
3.9
2.01
4.01
1.99
3.99
2.001
4.001
1.999 3.999
Here, we can see that as
function
⺁
2.0001 4.0001
gets closer and closer to
, the value of the
⺁ gets closer and closer to 4.
A “hole” or a skip
Figure 1 Graph of the function
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If we say that
is the limit of
that the value of
⺁ as
approaches the number
⺁ can be made close to
by taking
words, we can make the absolute difference of
very small by making the absolute difference of
close to . In other
⺁ and , written
and
, it means
or
,
very small.
This is formally stated in the following definition.
Definition Let
be a function which is defined at all
containing , except possibly at
itself. The limit of
is , written
lim
if for every
, however small, there exists a
whenever
on the open interval
⺁ as
approaches
such that
.
The following example uses the definition to prove that the given function has
the indicated limit.
Example
Solution:
Prove that the lim
t.
We need to show that for every
there exists a
that
Now,
t
t
whenever
.
Hence, we must show that
whenever
or equivalently,
Choose
whenever
.
. Then we have
whenever
or, equivalently,
This proves that
lim
t
whenever
t.
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such
Limit Theorems
1. If lim
and lim
, then
.
ᔠ
ᔠ .
2. If
and
are constants, then lim
3. If
is a constant, then for any real number , lim
4. For any real number , lim
5. If lim
and lim
lim
6. If lim
and lim
.
, then
ᔠ
⺁
, then
ᔠ lim
ᔠ
lim f1 ( x)  f 2 ( x)  lim f1 ( x)  lim f 2 ( x)  L1  L2
xa
7. If lim
and
8. If lim
, then
provided
n
xa
lim
Moreover, lim
1. lim
lim
, 
,
⺁
, then
lim
lim
exists, then lim
lim
have
⺁
lim
are two functions such that
, and if lim
Example
lim
lim
and lim
and
xa
is any positive integer, then we
L R .
9. If lim
10. If
lim

⺁ for all
exists.
.
Evaluate the following limits.
ᔠ
ᔠ
ᔠ
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Solution
2. lim
t
ᔠ
ᔠ
lim
ᔠ
ᔠ
⺁ ᔠ
⺁ᔠ
⺁ ᔠ
Solution
lim
t
t
ᔠ
ᔠ
is undefined, so apply Theorem 10 above,
lim
t
ᔠ
lim
t
t
ᔠ
t
t
ᔠ
lim
t
lim
ᔠ
t
ᔠ
t
t
3. lim
ᔠ
ᔠ
ᔠ
ᔠ
ᔠ
ᔠ
ᔠ
ᔠ
t
t
t
ᔠ
t
ᔠ
t
ᔠ
ᔠ
t
ᔠ
ᔠ
t
ᔠ
ᔠ
ᔠ
ᔠ
ᔠ
Solution
lim
ᔠ
lim
lim
lim
lim
ᔠ
ᔠ
ᔠ
ᔠ
ᔠ
ᔠ
ᔠ
ᔠ
ᔠ
ᔠ
ᔠ
ᔠ
ᔠ
ᔠ
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lim
lim
tᔠ
4. lim
ᔠ
ᔠ
ᔠ
ᔠ
ᔠ
Solution
lim
t
ᔠ
ᔠ
lim
lim
ᔠ ⺁
ᔠ
ᔠ
ᔠ ᔠ
ᔠ ᔠ ⺁
⺁ᔠ
Exercises
1. lim
t
t
2. lim
3. lim
ᔠ
4. lim
5. lim
7. lim
9.
10. lim
ᔠ
t
t
lim
lim
tᔠ
ᔠ
t
ᔠ
t
6. lim
8.
t
tᔠ
t
ᔠ
ᔠ
t
t
t
t t
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ONE-SIDED LIMITS
Consider the function
if
if
ᔠ
Its graph is shown below
approaches
approaches
approaches
Figure 2 Graph of the function
Notice that as
approaches
that
approaches
from the right,
approaches
approaches
ᔠ
if
if
from the left,
⺁ gets closer to
⺁ gets closer to . In symbols,
through values greater than
through values less than
and
but when
ᔠ
means
means that
. We shall now define one-sided
limits formally.
Definition (Right-Hand Limit)
Let
be a function which is defined for every
⺁. Then the limit of
if for every
⺁ as
approaches
limᔠ
, however small, there exists a
whenever
in some open interval
form the right is , written
such that
.
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Definition (Left-Hand Limit)
Let
be a function which is defined for every
⺁. Then the limit of
if for every
⺁ as
approaches
form the left is , written
lim
, however small, there exists a
whenever
in some open interval
such that
.
Remark The limit theorems discussed earlier still hold if “
ᔠ
by “
” or “
” is replaced
”.
The following theorem gives a relationship between the ordinary limit and the
one-sided limits.
Theorem
lim
exists if and only if limᔠ
and lim
Moreover,
lim
Example
limᔠ
Solution Since
approaches 0 from the right, it takes on positive
values, that is,
, so
limᔠ
Solution
lim
Evaluate the following limits.
1. limᔠ
2. lim
both exist and are equal.
limᔠ
ᔠ
lim
ᔠ
lim
lim
⺁
⺁
⺁
⺁ t
t
ᔠ ⺁
⺁
ᔠ ⺁
⺁
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lim
t
3.
ᔠ
lim
ᔠ
Solution
Note that
t
ᔠ ⺁
⺁
ᔠ ⺁
⺁
approaches -1 from the left. That is,
on negative values, so
ᔠ ⺁ and
ᔠ
lim
ᔠ
4. Given
determine if lim
ᔠ
ᔠ
ᔠ
takes
ᔠ
ᔠ
lim
,
t
exists.
Solution We need to find the limit of
from the left and from the right
and see if they are equal. Now,
limᔠ
lim
limᔠ
lim
ᔠ
t
ᔠ
t
t
Since the left-hand side and right-hand side limits are not the same, the
does not exist.
lim
Exercises
A. Evaluate the following limits.
1.
2.
3.
lim
lim
t
t
t
lim
t
tᔠ
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4.
5.
6.
7.
ᔠ
lim
ᔠ
limᔠ t
lim
t
lim
t
t
⺁
B. In problems 1 to 3, find (a) limᔠ
1.
determine if lim
exists.
C. Given
find the following
2.
3.
. In each case,
t
2.
1.
and (b) lim
lim ᔠ
4.
5.
lim
lim
if it exists⺁
lim ᔠ
ᔠ
lim
6. lim
,
if it exists⺁
INFINITE LIMITS
Consider the function
defined by
gets closer and closer to
. Its graph below shows that as
, the value of the function
bigger, it increases without bound. We use ᔠ
lim
ᔠ
⺁ gets bigger and
to denote this increase, that is,
.
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Figure 3 Graph of
In the same way, if the value of the function
approaches , then we say that the limit is
⺁ decreases without bound as
. The following are the formal
definitions of infinite limits.
Definition
ᔠ
Let
Limit⺁
be a function which is defined at all
containing
, except possibly at
bound as
approaches , written
if for every positive number
whenever
itself. We say that
lim
, there exists a
⺁ increases without
such that
.
Definition
Let
Limit⺁
be a function which is defined at all
containing
, except possibly at
bound as
approaches , written
if for every positive number
whenever
⺁ ᔠ
on the open interval
lim
itself. We say that
⺁
, there exists a
on the open interval
⺁ decreases without
such that
.
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Theorem
If
is a positive integer, then
limᔠ
ᔠ
lim
if is odd
if is even
ᔠ
Theorem
If
, and if lim
to zero, then
(1) If
and if
(2) If
and if
(3) If
and if
(4) If
and if
⺁
and lim
, where
⺁
is a constant not equal
through positive values of
⺁
lim
ᔠ
⺁, then
through positive values of
⺁
lim
⺁, then
through negative values of
⺁
lim
through negative values of
⺁
lim
ᔠ
⺁, then
⺁, then
Theorem
If lim
⺁
ᔠ
and lim
ᔠ
and lim
⺁
, where
lim
ᔠ
is a real number, then
⺁
ᔠ
Theorem
If lim
(1) If
(2) If
⺁
, then lim
, then lim
⺁
, where
⺁
⺁
is a nonzero real number, then
ᔠ
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Theorem
If lim
(1) If
⺁
(2) If
and lim
, then lim
, then lim
⺁
, where
⺁
⺁
is a nonzero real number, then
ᔠ
Example
Evaluate the following limits.
1. limᔠ
Solution
positive
limᔠ
limᔠ
approaches 0 through positive values
of
ᔠ
2. limᔠ
Solution
Notice that limᔠ
and limᔠ
resulting to
we’ll find an equivalent function as follows,
limᔠ
limᔠ
limᔠ
limᔠ
limᔠ
ᔠ
ᔠ
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. So
ᔠ
Now, as
,
through positive values, that is
limᔠ
3. limᔠ
Solution
limᔠ
ᔠ
Notice that limᔠ
ᔠ
and
ᔠ
. But there’s no way we can subtract (or add) both infinite
values. So, transforming the function,
limᔠ
ᔠ
Now, as
,
limᔠ
limᔠ
through positive values. This results to
limᔠ
ᔠ
4. limᔠ
Solution From the first factor of the function, notice that limᔠ
and from the second factor limᔠ
limᔠ
ᔠ
. Thus,
ᔠ
Exercises
1.
lim
t
2. limᔠ
3.
lim
ᔠt
t
ᔠ
t
ᔠ
t
ᔠ
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ᔠ
4. limᔠ
t
t
5. lim
6. lim
lim
10.
11.
ᔠ
ᔠ
t
ᔠt
ᔠ
t
lim ᔠ
12. lim
t
ᔠ
t⺁ ᔠ ⺁
lim
lim
ᔠ
t
t
8. limᔠ
9.
t
t
t
7.
ᔠ
ᔠ t
ᔠ
t
LIMITS AT INFINITY
Definition Limit at ᔠ
Let
interval
ᔠ
be a function which is defined at every number in some open
⺁. The limit of
if for any
lim
ᔠ
increases without bound is ,
⺁
such that
.
Definition Limit at
Let
⺁
,
⺁
be a function which is defined at every number in some open
⺁. The limit of
if for any
whenever
⺁ as
, there exists a number
whenever
interval
⺁
⺁ as
lim
decreases without bound is ,
⺁
, there exists a number
such that
.
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⺁
,
Theorem
If
is a positive integer, then
lim
ᔠ
lim
Example
Evaluate the following limits.
1.
lim
t
Solution:
ᔠ
ᔠ
lim
t
ᔠ
ᔠ
lim
lim
lim
2.
t
lim
t
ᔠ
t
t
ᔠ
ᔠ
ᔠ
ᔠ
ᔠ
Solution:
lim
Note that
lim
since
. Thus,
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lim
lim
lim
lim
lim
3.
lim
ᔠ
Solution:
lim
ᔠ
ᔠ
ᔠ
ᔠ
ᔠ
lim
ᔠ
ᔠ
ᔠ
lim
ᔠ
lim
ᔠ
ᔠ
ᔠ
ᔠ ᔠ
ᔠ
ᔠ ᔠ
lim
ᔠ
lim
ᔠ
ᔠ ᔠ
lim
ᔠ
ᔠ
ᔠ
ᔠ ᔠ
ᔠ
ᔠ
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Exercises
1. lim
2.
3.
ᔠ
lim
6.
7.
8.
9.
t tᔠ ᔠ
ᔠ
ᔠ
ᔠ
lim
ᔠ
4. lim
5.
tᔠ
ᔠ
lim
lim
ᔠ
lim
ᔠ
t
t tᔠ
t
ᔠ
ᔠ
lim
ᔠ
10. lim
ᔠ
11. lim
⺁
t ᔠ
ᔠ
lim
ᔠ
ᔠ
tᔠ
t t
ᔠ
ᔠ
ᔠ ᔠ
ᔠt⺁
ᔠ
ᔠ
ᔠ
⺁
ᔠ t
12. lim
ᔠ ⺁
Continuity of a Function
Definition (Continuity at )
The function
is said to be continuous at the number
if the following three
conditions are satisfied
1.
2. lim
3. lim
exists;
exists
⺁
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If one or two of these conditions fail to hold, then we say that the function
discontinuous. If
either lim
is discontinuous at a number
⺁ or
but lim
is
exists, then
⺁ does not exist. If this happens, then we say that
the discontinuity is removable. If the discontinuity at
is not removable, then it
is called an essential discontinuity.
Theorem
If
and
and
are two functions which are continuous at
are continuous at . If
, then
ᔠ ,
then is also continuous at .
⺁
Example
Determine if the given function is continuous at the indicated number . If it is
discontinuous at , determine if the discontinuity is removable or essential.
if
1.
Solution:
;
if
⺁ does not exist since
lim
limᔠ
lim
limᔠ
⺁
lim
Therefore, the function
limᔠ
lim
is discontinuous and the discontinuity is
essential.
2.
Solution:
if
if
;
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lim
⺁
lim
⺁
Therefore,
t
3.
ᔠ
lim
lim
lim
ᔠt
ᔠt
is continuous at
;
Solution: Examining each condition for continuity, we have
is not defined
lim
lim
t
t
t
ᔠ
ᔠ
t
ᔠ
ᔠ
t
ᔠ
t
t
t
t
ᔠ
lim
Therefore, the function
lim
at
t
ᔠ
t
ᔠ
ᔠ
ᔠ
ᔠ
ᔠ
ᔠ
ᔠ
t
ᔠ
ᔠ
ᔠ
t
lim
ᔠ
t
ᔠ
ᔠ
t
ᔠ
t
ᔠ
ᔠ
t
ᔠ
ᔠ
ᔠ
ᔠ
t
ᔠ
is not continuous
continuos at
ᔠ
⺁
t
. However, since the
exists, the discontinuity is removable. To make it continuous
, the function can be redefined as,
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t
ᔠ
t
if
if
Exercises
Determine if the given function is continuous at the indicated number . If it is
discontinuous at , determine if the discontinuity is removable or essential.
ᔠ
1.
at
;
and at
ᔠ
2.
at
t and at
3.
4.
ᔠt⺁
5.
ᔠ
t ᔠt
7.
ᔠ
⺁
; at
; at
t
if
if
ᔠ
9.
at
ᔠ
10.
t
; at
if
ᔠ
8.
t
; at
if
6.
11.
t
t
t;
t
; at
if
if
; at
if
at
ᔠ
at
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12.
13. Let
at
ᔠ
be defined by
ᔠ
For what value of
is
14. For what values of
continuous at
ᔠ
if
if
and
a continuous function at
and
t and
?
is , defined by
if
t
ᔠ if t
if
?
Intermediate Value Theorem
Theorem (Intermediate Value Theorem)
Suppose that
between
is continuous on the closed interval
⺁ and
⺁. Then there is a number
e and
is any number
such that
.
The intermediate value theorem says that a function must take on every value
between its endpoints at least once provided the function is continuous on a
compact interval. It means that a continuous function
over any values between
to jump across the line
⺁ and
on
⺁. Otherwise, the graph of
e cannot skip
would need
, something that continuous functions cannot do.
A function may take on a given value
more than once. Although these
geometric representations make the IVT seem reasonable, the proof is more
complicated than one might imagine and an interested reader may refer to an
advanced calculus or real analysis text.
The following result is a special case of the Intermediate Value Theorem. It is
also the basis of a Root-Finding algorithm called the Bisection Method.
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Corollary
Suppose that
is continuous on
⺁ such that
e and
.
. Then there exists
Example
Use the Intermediate Value Theorem to verify that the following functions
have zero/s in the given interval. Then use the method of bisections to find an
interval of length t that contains the zero.
1.
t
2.
3.
4.
t
t
cos
ᔠ
Squeeze Theorem and Limits Involving Sine and Cosine
Example
1. Show that lim
sin
.
Solution: It can be seen that
t
sin
Since sin
, for all
sin
sin , we have
t
sin
, for all
and
.
, for all
By Squeeze Theorem,
lim
lim
sin
lim
lim
sin
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2. Show that lim
cos
Solution: For all
.
,
cos
Thus,
cos
cos
, for all
cos
, for all
and
.
Let
and
Hence, for
cos
Now,
lim
By Squeeze Theorem,
lim
lim
cos
Example
Evaluate the following limits.
1. lim
sin
Solution
lim
sin
ᔠ
ᔠ
This limit is of the form .
lim
sin
lim
ᔠ
sin
lim
lim
sin
sin
lim
lim
ᔠ
ᔠ
ᔠ
where
note
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2. lim
t
t
cos
Solution:
lim
cos
cos
ᔠ cos
lim
lim
lim
3. lim
cos
lim
sin
ᔠ cos
sin
lim
ᔠ
tan
ᔠ cos
ᔠ cos
ᔠ cos
Solution:
lim
tan
lim
lim
sin
cos
sin
cos
lim
cos
Exercises
Evaluate the following limits.
1. lim
2.
sin
lim
sin t
ᔠ
ᔠ ⺁
3. lim t cot t
4. lim
tan
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cos
5. lim
t
t
6. lim csc
⺁
7. lim
cot
9. lim
cos
11. lim
sin cos
sin
cos
sin
8. lim
t
tant
10. lim
t
sin
12. lim
t
13. lim ttan
14. lim
sin
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