In order to study a function and its behavior and properties, an important step is to find out the limits of the function on the ends of its domain of definition. In this article, we will introduce the idea of limits and the different cases that we can come across.
Introduction
We saw in a previous article, an introduction to functions and some of their properties, in this blog post we will learn about function’s limits, where we will try to the behavior of the function near infinity and near specific real values (i.e., the values for who the function isn’t defined), in other terms, we try to determine the function value when approaching the extremities of its domain of definition.
The idea of limits:
To better introduce the idea of limits, let’s take a look at some examples:
Example 1: Let’s the function be defined as follow:
The domain of the function is .
Let’s evaluate the function when the values of go bigger and bigger

1 
5 
10 


0.5 
22.5 
95 
9950 
999500 
We notice that, as the value of increase, the value of the increases as well, so we can predict that as the value of become bigger and bigger, the value of the function goes bigger and bigger too, meaning that as approaches infinity (infinitely big) the value of approaches as well. In this case, we can use the notation:
And we read the limit of the function as approaches , is .
Here is the graph of the function for a better illustration:
Example 2:
We have the function defined as:
The domain of the function is: .
Let’s evaluate the function as takes bigger and bigger values:

1 
5 
10 


0.36787944117 
0.00673794699 
0.00004539992 
3.720076e44 
(To close to zero that the calculator shows exactly 0) 
We can easily notice that as increases in value the value of goes closer and closer to 0, so we can predict that as approaches infinity the value of the function approaches 0, and we note:
So, From the two previous examples, we conclude that a limit of a function can a real number or .
Here is the graph representing the function , where we can see the graph approaching the Xaxis as goes towards infinity.
Example 3:
Let’s consider the function defined as follow:
We have the domain of the function is .
We know that the function h isn’t defined for (because we can’t divide by 0), but let’s try and see the values of as approaches 0 with both greater and less than 0, meaning we evaluate when approaches 0 from the right side (i.e., ), and when approaches 0 from the left side (i.e., ).
Let’s start with evaluating as gets closer and closer to zero with greater values:

1 
0.5 
0.1 
0.01 
0.001 
0.0001 
0.00001 

1 
2 
10 
100 
1000 
10000 
100000 
We can clearly notice that as gets closer and closer to zero with greater values, the values of the function become bigger and bigger, we can predict then that it keeps increasing in value towards , in other terms as approaches 0 with greater values, tend to , and we note:
And we read: the limit of the function as approaches 0 with greater values (or as approaches 0 from the right side) is .
Now let’s evaluate as approaches 0 with values less than zero:

1 
0.5 
0.1 
0.01 
0.001 
0.0001 
0.00001 

1 
2 
10 
100 
1000 
10000 
100000 
This time, notice that the value of the function gets smaller and smaller, as gets closer and closer to 0 with smaller values i.e., from the left side, so this way we can predict that will keep decreasing towards , meaning that as approaches 0 with smaller values, tends to , and we note:
Here is the graph of the function , notice that the graph goes up to if we get closer to 0 from the right side, and it goes down to if we get closer to 0 from the left side.
Let’s summarize what we learned:
Finding the limit of a function is to determine its value tendency when approaches infinity or a real number (usually a value at the extreme of the function’s domain for which the function is not defined).
We evaluate limits for when approaches the ends on the domain of definition of a function, otherwise, if we want to evaluate the function for a value of inside the domain, we just calculate , but for the values at the extremes of the domain, whether they are a real number or infinity, we cannot calculate them and that why we try to find the limits.
For example, we cannot just evaluate because infinity is not a number, infinity is an idea, a concept, so instead, we calculate the limit of the function when goes to infinity.
Same thing for a value of on the extreme of the domain and for which the function isn’t defined, for example, a function defined on a domain , we know that doesn’t belong in the domain of so we cannot evaluate , instead, we calculate the limit of the function when approaches the value of . In this case, we have two limits to determine, one for when approaches with greater value (i.e., where is very small) or in other terms from the right side; and the second for when approaches with smaller values (i.e., where is very small) or in other terms from the left side.
Finite and infinite limit when approaching infinity:
Finite limit
Definition 1: Finite limit on or :
Suppose a function defined on the domain and a real number.
To say that the limit of on is is to say that every open domain containing contains all the values of for big enough. We write
And we read tends to when tends to .
Explanation:
To say that the limit of as tends to is , means that becomes very close to and therefore the value lies in the interval , with a small real number, or in other terms .
Remarque: we get a similar definition and result for (i.e., for defined on and tends to ).
Examples:
Infinite limit
Definition 1: Infinite limit on or :
Suppose a function defined on the domain and a real number.
To say that the limit of when tends to is , means that for a real number with , every interval contains all the values of for big enough. We write
And we read tends to when tends to .
Definition 2:
Suppose a function defined on the domain and a real number.
To say that the limit of when tends to is , means that for a real number with , every interval contains all the values of for big enough. We write
And we read tends to when tends to .
Remarque: we get two similar definitions and result for (i.e., for defined on and tends to ).
Examples:
Finite and infinite limit when approaching a real value:
Finite limit when approaching a real value:
Definition:
Suppose a function defined on the domain and a real number.
To say that the limit of when tends to is equal to , means that every open interval containing contains all the values of for big enough. We write
And we read: tends to when tends to .
We distinguish two cases, the first is when tends to with greater values (or in other terms from the right side) and we note:
And the second case, when tends to with smaller values (or in other terms from the left side) and we note:
Let’s take a look at an example for better understanding:
We have the function defined on as follow:
If we take a look at the graph of the function , we notice that as approaches 0 the value approaches 1, and if we zoom the graph or if we use a table to calculate values as approaches 0 then we can see that we can get as close to 1 as we want if we get close enough to 0. Therefore, we have:
Infinite limit when approaching a real value:
Definition: Suppose a function defined on the domain .
To say that the limit of when tends to is , means that for a real number with , every interval contains all the values of for big enough. We write
And we read: tends to when tends to .
We distinguish two cases, the first is when tends to with greater values (or in other terms from the right side) and we note:
And the second case, when tends to with smaller values (or in other terms from the left side) and we note:
Here is an example for better understanding:
The function is defined on the interval as follow:
If we take a look at the graph we can clearly notice that as approaches 0 with greater value i.e., from the right side, the value of tends to , so we write:
Also, we can see that as approaches 0 with smaller values i.e., from the left side, the value of tends to , so we write:
Now why don’t you try it yourself, in the following graph of the function , you can slide the two points and on the Xaxis, and approach 0 as close as you want while noticing the value of goes to infinity with different signs
Conclusion:
We learned in this article the idea of limits and how it helps us understand the behavior of a function near the edges of its domain of definition. this is just an introduction we still have much more to learn and
Surely you want more fun with limits, so check out this random rational functions generator, as its name suggests it randomly generate a rational function (a quotient of two polynomial functions) and draws its graph. Take a look and enjoy graphicly seeing the different limits of these functions on their domains. Have Fun!!!!!
You want to learn more fun subjects, check the post about Functions and some of their properties, or the one about How to solve polynomial equations of first, second, and third degrees!!!!!
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