An Intuitive and Down to Earth Introduction to Limits and Continuity

I know limits and continuity can be difficult to understand. After all, the best mathematicians of history worked a lot to reach the beautiful conclusions we have today. Don't feel overwhelmed, though. All you need is to develop an intuitive understanding, and you'll see how simple these concepts are.

In this page we'll try to understand the intuitive idea behind limits. If you're looking for how to solve limits, you may visit this page: Solving Limits.

The limit is the most important concept of all calculus. The main ideas of calculus, the derivative and the integral, are defined using limits.

I found limits and continuity the most difficult when I first learned calculus. Once you grasp them, though, they seem so simple.

Understanding Limits

When we talked about functions before, we only talked about the values of functions at specific points. For example, the value of f(x) at x=1.

Now, what we want to know now, is what the function is "approaching" when x "approaches" a specific value. Let's pay attention to this graph:

When x approaches "a", the function f(x) approaches "L".

Here's an example with an actual function:

This is the function f(x)=x2. Let's focus on the point (1,1). We can see from the graph that when x approaches 1, the function f(x) approaches 1. When this happens, we say that:

This is read "the limit as x approaches 1 of x squared equals 1".

Why Limits are Useful

You might ask what this is useful for. Very good question. Why would you need to know what the function is approaching? You already know the function equals 1 when x equals 1, right?

Well, the point is that we don't care what the function is at x equals 1!

To show this, let's consider the following function:

Don't let this notation intimidate you! This only means that this function equals x2 when x is anything other than 1, and equals 0 when x equals 1. This function is the same as the one we saw before, but in this case it has a "hole" at x=1. Let's see what the graph looks like:

What does the function approach when x approaches 1? It also approaches 1, right? It doesn't matter that the function is other than 1 at that point! So,

In calculus, the most useful limits are like this one. The value of the function at the specific point we care about is not defined, like 0/0 (which is complete junk), or useless, like zero or infinite.

In these cases we can know what the function is approaching, and that is what we really need. To learn more about what limits are and how to solve simple problems visit the limit of a function page.


Introduction to Continuous Functions

Limits and continuity are often covered in the same chapter of textbooks. This is because they are so related. The basic idea of continuity is very simple, and the "formal" definition uses limits.

Basically, we say a function is continuous when you can graph it without lifting your pencil from the paper. Here's an example of what a continuous function looks like:

There is a precise mathematical definition of continuity that uses limits, and I talk about that at continuous functions page. Intuitively, this definition says that small changes in the input of the function result in small changes in the output.

If you are confused by that, ignore it! You don't need to learn all at once. The most important is to recognize a continuous function when you see it.

Now, what would a discontinuous function look like? A function essentially is discontinuous when it has any "gap". For example:

Easy to recognize, isn't it?

To continue to learn about limits and continuity, visit some of the pages below.

Conclusion

  • The concepts of limits and continuity are important.
  • You may find them difficult at first, but they will seem simple after you form an intuition.
  • The limit of a function is what it approaches when the input x approaches something.
  • It doesn't matter what the actual value of the function is at that point it is approaching!
  • A function is continuous when you can graph it without lifting the pencil.

The Self-Study Course is a complete resource that will guide you in the process of learning calculus intuitively. It is much more than a textbook, as it is specifically designed for self-study.

If you want to study more in depth what you find on this website, following the same paradigm of forming an intuitive understanding first, this is the way to go. Click here to learn more.



Still have a question or doubt about limits? An "impossible problem"? Submit it here!


Learn more about limits and continuity:

The Intuitive Approach to The Limit of a Function

The concept of limit of a function is the most important one of all calculus. Learn how to solve limit problems and understand what you are doing!

Solving Limits Made Simple: Solve Any Limit You May Find

This page is the ultimate resource for solving limits. I prepared a list of all possible cases of limits. If you master these, there won't be a single limit you can't solve.

Intuition Behind the Squeeze Theorem and Applications

The squeeze theorem may seem hard on your book, but it is a very simple and commonsensical idea. Here you'll learn what it means and how to apply it to solve limits.

How to Solve Limits at Infinity: Intuition and Examples

Limits at infinity can be confusing. Here you'll learn the basic technique to solve them and some simple tricks to help you out...

The Formal Limit Definition: Getting the Intuition Behind It

The limit definition caughts many students of calculus. Don't be one of them. Here we'll arrive step by step to what may seem at first glance a convoluted and complex definition...

Continuous Functions: An Intuitive Introduction

All calculus is about continuous functions. Here you'll find the intuition behind this concept and we'll clear up a common misconception you may even find in your book.

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