Table of Contents
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 |
3.720076e-44 |
(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 X-axis
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.,
Let’s start with evaluating
|
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
And we read: the limit of the function
Now let’s evaluate
|
-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
Here is the graph of the function
Let’s summarize what we learned:
Finding the limit of a function is to determine its value tendency when
We evaluate limits for when
For example, we cannot just evaluate
Same thing for a value of
Finite and infinite limit when approaching infinity:
Finite limit
Definition 1: Finite limit on
Suppose
To say that the limit of
And we read
Explanation:
To say that the limit of
Remarque: we get a similar definition and result for
Examples:
Infinite limit
Definition 1: Infinite limit on
Suppose
To say that the limit of
And we read
Definition 2:
Suppose
To say that the limit of
And we read
Remarque: we get two similar definitions and result for
Examples:
Finite and infinite limit when approaching a real value:
Finite limit when approaching a real value:
Definition:
Suppose
To say that the limit of
And we read:
We distinguish two cases, the first is when
And the second case, when
Let’s take a look at an example for better understanding:
We have the function
If we take a look at the graph of the function
Infinite limit when approaching a real value:
Definition: Suppose
To say that the limit of
And we read:
We distinguish two cases, the first is when
And the second case, when
Here is an example for better understanding:
The function
If we take a look at the graph
Also, we can see that as
Now why don’t you try it yourself, in the following graph of the function
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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