Till here, you have a good concept of what is a limit and why do we use it? When you are applying a limit at a certain point, the limit is approached from the left-hand side to that value. Suppose you have a function, at , the limit will be approached from negative infinity to the point where x is equal to 2. However, this isn't the only direction of the limit, it can be approached from infinity to the point where x is equal to 2. Both cases are correct and can be used. In this resource, you will learn the direction of limits and where it is used.

There are two types of directions of approaching and they are:

  • Left-sided limit or left-hand limit,
  • Right-sided limit or right-hand limit.

Normally, we use one side limit. It means that limit is approached from one direction only. It could be any either left-hand limit or right-hand limit. Both we give you the same answer but in some cases, we also use both sides. The case we are talking about is finding the continuity of the equation. To find whether the equation is continuous or not, we use both side limit and then compare it with each other. If the answers are the same, we call it continuous otherwise it is discontinuous. This is a very long topic and we will cover it in another resource, lets just stick to the limit sides.

The best Maths tutors available
Paolo
5
5 (63 reviews)
Paolo
£30
/h
Gift icon
1st lesson free!
Hiren
5
5 (23 reviews)
Hiren
£150
/h
Gift icon
1st lesson free!
Akash
5
5 (58 reviews)
Akash
£45
/h
Gift icon
1st lesson free!
Sehaj
4.9
4.9 (47 reviews)
Sehaj
£40
/h
Gift icon
1st lesson free!
Luke
5
5 (76 reviews)
Luke
£125
/h
Gift icon
1st lesson free!
Johann
5
5 (35 reviews)
Johann
£35
/h
Gift icon
1st lesson free!
Intasar
5
5 (48 reviews)
Intasar
£79
/h
Gift icon
1st lesson free!
Harinder
5
5 (36 reviews)
Harinder
£20
/h
Gift icon
1st lesson free!
Paolo
5
5 (63 reviews)
Paolo
£30
/h
Gift icon
1st lesson free!
Hiren
5
5 (23 reviews)
Hiren
£150
/h
Gift icon
1st lesson free!
Akash
5
5 (58 reviews)
Akash
£45
/h
Gift icon
1st lesson free!
Sehaj
4.9
4.9 (47 reviews)
Sehaj
£40
/h
Gift icon
1st lesson free!
Luke
5
5 (76 reviews)
Luke
£125
/h
Gift icon
1st lesson free!
Johann
5
5 (35 reviews)
Johann
£35
/h
Gift icon
1st lesson free!
Intasar
5
5 (48 reviews)
Intasar
£79
/h
Gift icon
1st lesson free!
Harinder
5
5 (36 reviews)
Harinder
£20
/h
Gift icon
1st lesson free!
Let's go

Left-Sided Limit or Left-Hand Limit

If the direction of the approaching is from negative infinity to the point, that means it is a left-side limit. Consider the graph below:

A continuous graph of sin x from minus infinity to positive infinity

The above graph is a simple curve. Let's say the limit is approaching the point, . For the case of the left-hand side limit, it will be written as . It means that the limit is been approached from negative infinity (which is on the left side of the graph) to the point. Hence we are travelling from the left side of the graph to the point where x is equal to . We call this direction the left-hand limit. Below is the syntax of the left-hand limit and its condition.

Right Sided Limit or Right Hand Limit

If the direction of the approaching is from positive infinity to the point, that means it is a right-side limit. Consider the graph again. Let's say this time you are approaching the point from positive infinity to . This means that you are travelling from the left side of the graph to the point where x is equal to . We call this direction the right-hand limit. Below is the syntax of the right-hand limit and its condition.

Example

The limit of a function at a point if it exists, is unique.

In this case, it can be seen that the limit from both the left and right-sided as x tends to is .

The limit of the function is as x tends to even though the function has no value at .

To calculate the limit of a function at a point, it is not of importance what happens at that particular point, but what happens around it.

Given the function:

Calculate .

The function has no limit at .

Did you like this article? Rate it!

1 Star2 Stars3 Stars4 Stars5 Stars 4.00 (4 rating(s))
Loading...
Emma

Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.