Right Hand Limit

Introduction to Course

09/19/2023 1

Calculus

Differential Calculus Integral Calculus

Limit Continuity and Differentiability

Chapter 1

Function

That defines a relationship between one variable (the

independent variable) and another variable (the dependent variable).

Function,

in mathematics

An Expression, Rule

or Law

Function

Function:

Example:

f(x) = 3x/2

It is a function because each input "x" has a single output

• x = 2 ; f(2) = 3

• x = 16 ; f(16) = 24

• x = - 10 ; • f(−10) = −15

• etc...

f(x) = 3x/2

Limit

1.9

2

2.2

2

Limit

Right Hand Limit

Left Hand Limit

Limit of a function

If the values of f(x) can be made as close as we like to ‘’

by taking values of x sufficiently close to a ,

then ‘’ is defined as the limit of f(x) as x tends to a.

 

This is symbolically written as ,

Or

 

limx®a f (x) = l

Example

The limit of  as x approaches 1 is 2

 

Now we see that x gets close to 1 then close to 2.

 

� ( � )

 

limx®1

x² - 1

x- 1 = 2 = l

We are now faced with an interesting situation:

•When x = 1 we don't know the answer (it is indeterminate)

•But we can see that it is going to be 2

We want to give the answer "2" but can’t,

So instead mathematicians say exactly what is going on by using the special word "limit"

The limit of  as x approaches 1 is 2

 

And it is written in symbols as:

So it is a special way of saying, "ignoring what happens when we get there, but as we get closer and closer the answer gets closer and closer to 2"

Types of Limit of a function

Limit of a function

Left Hand Limit

Right hand Limit

1.9

2.2 2

2

Left Hand Limit

If the values of f(x) can be made as close as we like to ‘’

by taking values of x sufficiently close to a , ( But less than a) then ‘’ is defined as the left hand limit of f(x) as x tends to a.

 

This is symbolically written as ,

L.H.L = lim

x®a^- f (x) = l

Right Hand Limit

If the values of f(x) can be made as close as we like to ‘’

by taking values of x sufficiently close to a , (But greater than a) then ‘’ is defined as the right hand limit of f(x) as x tends to a.

 

This is symbolically written as ,

R.H.L = lim

x®a⁺ f (x) = l

Left Hand Limit

Right Hand Limit

Existence of limit of a function

 

Existence of limit of a function at :

 

· L.H.L= Lim

x®a^- f(x) exists

· R.H.L= Lim

x®a⁺ f(x) exists

· L.H.L= R.H.L= l

The limit of a function f(x) at x= a i.e. Lim_x®af(x)= l exists if

Fundamental Properties of limit:

If are two functions and

 

�

� (¿ )

¿¿

� ( � ) ± ���

� → � ¿

���

� → � { ^{� ( � )} ± � ( � )}^{= ���}

� → � ¿

 

�

lim

→ �

� ( � ) ��� lim

�→ �

� ( � ) ������� h ��

 

�

� (¿ )

¿

� ( � ) ׿ ���

� → � ¿

� ¿ ���

� → � { � ( � ) × � ( � ) }= ���

� → � ¿

 

c)  

Fundamental Properties of limit:

d)

 

�

� ( ¿ )

���� → � ¿ ^}�

¿¿

{ � ( � ) }^�= ¿

� ¿ ���

� → � ¿

 

� ¿ ���

� →�

( �������� ) = ��������

 

Problem

Problem 1: A function is defined as bellow :

Does

Exist ?

 

2

2

1

2.4 1

1 1

x when x

f x when x

x when x

 

=  =

 + 



1

 

limx f x

®

Problem

Solution : Given ,

Since

So ,

Does not Exist .

 

2

2

1

2.4 1

1 1

x when x

f x when x

x when x

 

=  =

 + 



1

 

. . lim

L H L x _- f x

= ®

2

lim1

x x

= ®

12

=

1

 

. . lim

R H L x ₊ f x

= ®

= 1



²



lim1 1

x x

= ® +

12 1

= +

= 2

. . . .

L H L  R H L

1

 

lim

x

f x

®

Problem

Problem 2: A function is defined as bellow :

Find the value of

 

2 1 0

1 0

1 0

x when x

f x when x

x when x

 + 

=  =

 + 



0

 

lim

x

f x

®

C.W : problem no 2.

Problem

Practice

A function is defined as bellow :

Find the value of

lim

� → 1

� ( � )

and  

0

 

lim

x

f x

®

 

















 +

=

1 when

1 ,

1 0

when ,

0 when

,

2 1 x x

x x

x x

x f

Problem

Practice

Continuity :

A function f(x) is said to be continuous at a point x = a if

Or , If it is satisfy the following condition :

& finite and equal to the functional values of f(x)

L.H.L = R.H.L= f(a)

limx®a f (x) is exits

Problem

A function is defined as bellow :

Discuss the continuous at point x = 1

 

















 +

=

1 when

1 ,

1 0

when ,

0 when

,

2 1 x x

x x

x x

x f

Solution

Solution: Given that,

= 1

Here,

Now, the functional value at is

Since, L.H.L = R. H.L = f(1)

at

 

2 1 0

0 1

1 1

x when x

f x x when x

when x x

 + 

=   

 



1  

. . lim

L H L x _- f x

= ®

1  

limx x

= ®

= 1

1  

. . lim

R H L x ₊ f x

= ®

1

 

lim 1

x® x

=

. . . . L H L R H L=

1

x = ^f  ^{1 =1}

1 x =

Problem

Solution