fin118 unit 2

جامعة اإلمام محمد بن سعود اإلسالمية

كلية االقتصاد والعلوم اإلدارية

قسم التمويل واالستثمار

Al-Imam Muhammad Ibn Saud Islamic University College of Economics and Administration Sciences

Department of Finance and Investment

Financial Mathematics Course

FIN 118 Unit course

2 Number Unit

Limits of Function Continuity of Function

Unit Subject

Dr. Lotfi Ben Jedidia Dr. Imed Medhioub

1

2

1. Limit of function

2. Limits to the right, to the left

3. Computing limits

4. Properties of limits

5. Computing limits of indeterminate forms

6. Continuity of function

7. Essential discontinuities

8. Properties of continuous functions

9. Applications

We will see in this unit

Learning Outcomes

3

At the end of this chapter, you should be able to:

1. Understand what is meant by “limit of function”.

2. Compute these limits.

3. Check existence of these limits.

4. Understand what is meant by a “continuous function”.

5. Check if a function is continuous graphically.

6. Check if a function is continuous algebraically.

4

Limit of Function

Example 1: Deduce the value of each function when x gets

close to +/- infinity

5

Limit of Function



6

Limit of Function



7

Limit of Function

Example 4: Deduce the value of the function when x gets

close to +/- infinity and when x gets close to the left of zero (0-) and the right of zero (0+)

8

Definition1 :

The limit of f (x), as x gets close to a, equals L is

written as:

Sometimes the values of function f tend to different

limits as x approaches a number “a” from the left side

and from the right side like the example 4.

LxfLimax

Limit of Function

8

Limits to the left, to the right

Left-Hand Limit (LHL) :

Right-Hand Limit (RHL) :

9

1LxfLimax

2LxfLimax

0 axaxax

0 axaxax

Theorem

10

if and only if:

and

Lxfax

lim

Lxfax

lim Lxfax

lim

This theorem is used to verify whether a limit exists or not.

Computing limits

• To evaluate limit of a function, we substitute the

value of (a) in the function.

• When the value of (a) does not in the domain of a

function, then we must calculate the LHL and RHL.

• Sometimes we get an indeterminate form. So we

must use one of three techniques: factoring or a

property of limit or the conjugated form.

11

12

Computing limits by substitution

Example1: Find the limit of the function at x = 0,

At x = 1, x = -2, and x = -3.

Solution:

132 2 xxxf

110302132lim22

0

xx

x

411312132lim22

1

xx

x

112322132lim22

2

xx

x

813332132lim22

3

xx

x

13

Example2:

Find the Left-hand Limit and the Right-hand Limit of the function at x=0.

What we can conclude ?

Solution

left-hand limit is different to right-hand limit then the limit at x = 0, does not exist.

0 1

0

xifx

xifxxf

000

xLimxfLim

xx

1100

xLimxfLim

xx

let

Limits to the right, to the left Computing limits by substitution

14

Properties of limits

P1:If c is any real number, ,

Then,

Lxfax

lim Mxgax

lim

a) lim ( ) ( )x a

f x g x L M

b) lim ( ) ( ) x a

f x g x L M

c) lim ( ) ( )x a

f x g x L M

( )d) lim , ( 0)

( )x a

f x L Mg x M

e) lim ( )x a

c f x c L

f) lim ( ) n n

x af x L

g) lim x a

c c

h) lim x a

x a

i) lim n n

x ax a

j) lim ( ) , ( 0)x a

f x L L

15

P2:If

Then

P3:If

then

0,)( 01

1

1

n

n

n

n

n aaxaxaxaxf

n

nxx

xaxf

lim)(lim

0,0,)(01

1

1

01

1

1

nmn

n

n

n

m

m

m

m babxbxbxb

axaxaxaxf

n

n

m

m

xx xb

xaxf

lim)(lim



P4:If then and

P5:If then and

16

xexf )( 0)(lim

xfx

)(lim xfx

xxf ln)(

)(lim0

xfx

)(lim xfx

xe

xln

Computing limits by using the highest degree term

Example2: Find the limit of the function when x gets close to (+/-) infinity.

Solution:

17

132 2 xxxf

222 22lim132lim xxxxx

222 22lim132lim xxxxx


Example3: Find the limit of the function when x gets close to (-)infinity.

Solution:

18

22

1323

2

x

xxxf

01

lim2

2lim

22

132lim

3

2

3

2

xx

x

x

xx

xxx


Example4: Find the limit of the function when x gets close to (+)infinity.

Solution:

19

454

1332

2

xx

xxxf

4

3

4

3lim

454

133lim

2

2

2

2

x

x

xx

xx

xx

Computing limits of Indeterminate

forms • The usual Indeterminate forms come in the

following types:

• We can add the three other indeterminate forms :

• So we must use one of three techniques: factoring or highest degree term or the conjugated form.

20

,0 , ,

0

0

00 ,0 ,1

21

Computing limits of Indeterminate forms

Example 1: (factoring)

•

•

0

0

22

42

2

4lim

22

2

x

x

x

42lim2

22lim

2

4lim

22

2

2

x

x

xx

x

x

xxx

0

0

18333

33

183

3lim

223

xx

x

x

9

1

6

1lim

63

3lim

183

3lim

3323

xxx

x

xx

x

xxx

22

Example 2: (highest degree term)

•

•

?34

13lim

2

2

x

x

x

?143lim 2

xxx

22 3lim143lim xxxxx

4

3

4

3lim

34

13lim

2

2

2

2

x

x

x

x

xx


23

Example 3: (conjugated form)

2

1

1

1lim

1 1

1lim

1 1

1 1lim

1

1lim

?0

0

1

1lim

1

1 1 1

1

x

xx

x

xx

xx

x

x

x

x

x

xxx

x

bababa 22 !!!


24

Find the domain and limits of the function

defined as

Solution:

1

12

2

x

xxf

Computing limits Example 4:

25

Example 4: (continued)

Time to Review !

26

1. Remember well that the search for endings or limits

is to know the behavior of the function when x gets

close to a certain point or to infinity.

2. Before calculating limits, we search the domain of

the function.

3. Computing limits can be achieved by substitution, by

factoring, by using the highest degree term, by using

conjugate forms.

4. The limit exist if and only if LHL = RHL.

Continuity of Function

27

A function f is continuous if you can draw it in one motion without picking up your pencil.

Example:

A function f is continuous at a point if the limit is the same as the value of the function.

Definition1:

Definition2:

A function f is continuous at the point x=a if the following are true:

1.

2.

3.

EX. The function is continuous at x=0.

1. ☺

2. ☺

3. ☺

definedisaf

existxfax

lim

afxfax

lim

223 2 xxxf

20 f

220203lim2

0

xf

x

20lim0

fxfx

Definition3:


28

Essential Discontinuities

f(a)=LHL

RHL

LHL=RHL

f(a)

f(a)=RHL

LHL

f(a) is not defined

a

29

Example 2:

• This function has discontinuities at x=1 and x=2. • It is continuous at x=3, because the two-sided limits match the value of the function. • It is continuous at right of x=0 and left of x=4, because the one-sided limits match the value of the function.


30

Find discontinuities of the function

Solution:

1/ the domain of the function is the real line.

2/ and

LHL = RHL=0 then f is continuous at x = 0.

3/ and

LHL is different from RHL, then f is not continuous at x = 2.

2 8

20

0

2

xifx

xifx

xifx

xf

0limlim00

xxf

xx 0limlim 2

00

xxf

xx

4limlim 2

22

xxf

xx 68limlim

22

xxf

xx

Example 3:


31

Properties of continuous functions:

P1: If f and g are continuous functions at x = a then: , , and ( with ) are

continuous functions at x = a.

P2: A polynomial function is continuous at every point.

P3: A rational function is continuous at every point in its domain.

32

gf gf g

f 0ag

0,)( 01

1

1

n

n

n

n

n aaxaxaxaxf

0,0,)(01

1

1

01

1

1

nmn

n

n

n

m

m

m

m babxbxbxb

axaxaxaxf

Examples:

1/ is a polynomial function,

then it is continuous at every point in R.

2/ is a rational function, then

it is continuous at every point in R\{2}

3/ is a continuous function in its

domain

33

10523 34 xxxxf

2

1052 3

x

xxxf

12 xxf

;

2

1fD


1. A function is continuous if you can draw it in one motion without picking up your pencil.

2. A function is continuous at a point if the limit is the same as the value of the function. 3. All constant functions are continuous. 4. The following types of functions are continuous at every member in their domain: polynomial, rational, power, root, trigonometric, exponential, and logarithmic.

Time to Review !

34

We will see in the next unit

35

1. Derivative of function

2. How to compute derivative?

3. Some rules of differentiation

4. Second and higher derivatives

5. Interpretation of the derivative

6. Critical points

7. Inflection point

35

fin118 unit 2

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