Friday, December 2, 2016

Notions of limits (written lesson)

In the previous lesson on limits, I introduced the lesson by assigning some videos that you have to watch. Today we get to the written part of the lesson.

I am going to describe this lesson and give you the assignments that you should do.

Before I continue I have to tell you that your learning should not be limited to what your teacher gives you. There are plenty of resources that you can use to learn. You can learn from a teacher, from someone who knows a subject well and can teach it to others, from books, from electronic resources like CDs, from electronic communications like radio and televisions. More importantly today there is plenty of resources in the internet that you can use if you know how to access them. You should make yourself comfortable to all types of resources that you can use for learning. Videos are great to learn something but you can't limit yourself to this only. If you want to learn something deeply you have to get the written materials. The written materials allow you to get an overview of what you  are going to learn and give you also the content. You can choose which parts to learn first or which parts to drop depending on your interests. The most important thing also is you can review the materials as much as you can. If your reading skills are good you can learn a lot from written materials but for math there isn't a lot to read. You have to do the reading and memorize certain things. You have to practice a lot.

Here is the link for the lesson but before you start read the following;

Description of the lesson

This lesson starts by a definition of limits and shows you the three methods of limits using examples. The lesson ends by giving you some problems to do. Below I give you the readings that  you have to do under each sub-title and the tasks you have to do for each lesson.


1. Objectives

You should start by reading the objectives again then read the definition of limits. The first video that you watched on the previous lesson with videos gave you verbally an idea of what a limit is. Now you are going to have a written idea of limit and three methods that allow you to calculate a limit.

2. The Idea.

You read the paragraph giving you an idea of what a limit is. You already have a video demonstration giving you the idea of a limit. Now you have a conceptual definition of a limit. You should try to state this definition either in your own words without compromising the concept or verbatim for more accuracy. Now that you have a definition of limit you are going to learn in written words how to find the limit of a function using three methods: graphtable, algebra.

3. Methods for determining limits

a) The graph method

Under this title you should see a problem named "Example 1". This problem asks you to find three limits using the graph on the right. This problem is already solved for you. You are going to do two things with this problem.

Task I 

Read the example and its solution. Read the explanations provided for the solution of the problem in case you don't understand it. Below is a guide to the explanations.

Explanation of the solution a) 

You should be able to understand the solution easily. I provide the explanations of the solution in case you don't understand it. I give you a method to understand the solution. It's graphic. You should read and do it.

Explanation of the solution of b) and c)

The same method is used for the solution of a) and b)

Explanation of the  solution of d)

You can use the same method for the solution of d) but this time notice that the function doesn't have a limit.

Task 2 

Do a pencil and paper to do the example yourself without referring to the solution. After you finish verify that your answers are correct. 

Task 3

Do Practice I.

b) Table method

In this method you are going to use two tables to find a limit. You start by giving x some values to the left of the given value of x and you group the values of x and f(x) in a table. You do the same thing to the right of the given value of x to have a second table. Even though I don't mention the tasks in the lesson you are going to do them in the same way you do for the previous problem.

Task I

Start by reading the problem they ask you to find the solution. Then read the solution. I didn't provided any explanation of the solution because the solution is explanatory by itself. Below is a guided explanation


You start by giving x a value less than 0 and closer to 0. This value has to be to the left of 0. Then you calculate the value of f(x). You give to x a second value and closer to zero than the previous one You calculate the second value of f(x). You continue to give some values to x closer and closer to x and calculate the corresponding values of x. You do a table grouping all the values of x and f(x) in a table. When you look at the table you notice that f(x) gets close to 1 to the left as x gets closer and closer to 0 to the left.

Now you give x values to the right of 0 but closer to 0 and you calculate the corresponding values of f(x). You do a table grouping the values of x and f(x). When you look at the table you notice that as x gets closer and closer to to 0 to the right f(x) gets close to 1 to the right.

Since f(x) gets close to 1 as x gets closer and closer to 0 both to the left and right to 0 we conclude the limit of f(x) is 1 when x gets close to 0.

Task 2

Take a pencil and a piece of paper to do the problem by yourself.

Task 3

Do the practice problem

Algebra method

This method is very simple but it involves some calculations to do. In this method you substitute x in the function

Task 1

Read the problem first. Then write its solution. Below is a guided explanation.


You substitute x in the function and you do the calculations to find f(x). The value of f(x) is the limit of the function

Task 2

Do the problem by yourself

Task 3

Do the practice problems

Review problems

Do the review problems involving the three methods.

Interested in learning more about limits get this free course Introduction to Calculus
You can also be enrolled in the complete Calculus course