The Intuitive Approach to The Limit of a Function

The concept of limit of a function is the most important of all calculus. It is used to define derivation and integration, which are the main ideas of calculus.

To make it simple, the limit of a function is what the function "approaches" when the input (the variable "x" in most cases) approaches a specific value.

Let's analyze this graph:

As the variable x approaches a, the function f(x) approaches L.

The limit of an actual function may look like this:

This is the function f(x) equals x squared. As "x" approaches 1, f(x) approaches 1. To express this we write:

Is the limit of a function useful? Very good question. It is indeed useful. Many times we find functions that are undefined at certain values. This means that the function equals something like 0/0, or infinity over infinity for specific values of "x".

We don't know what those expressions mean!

Using limits, however, we can know what the function is approaching when the variable "x" approaches that value. We don't need to care whether or not the function is defined at that point.

You'll understand this better with more examples on the page about solving limits.

Solving Simple Limits

Many limits are very easy to solve. Let's start with some:

Let's think. What will happen to the function when x approaches 1 more and more? Let's take our calculator and make a table:

The function clearly approaches 3, right?

Let's see what happens if x approaches 1, but takes values greater than one.

It also approaches 3...

So, when this happens, we write:

Limit of the Sum of Functions

Now, let's suppose we have two functions:

g(x) doesn't have an "x", so it is constant. This means its value is six, no matter what the "x" is.

What will happen if we add these functions and try to find the limit?

This problem is the same as the previous one. We don't need to make a table to know that when x approaches 2, x squared will approach 4. Six always will be six. So:

Here we can note two important properties of the limit of a function:

The first one is that the limit of the sum of two or more functions equals the sum of the limits of each function.

The second one is that the limit of a constant equals the same constant. By a "constant" we mean any number.

Limit of a Product

In our first example:

We used another important property of the limit of a function. Can you see which one?

This is similar to the property about sums, but with products:

The limit of the product of two or more functions equals the product of the limits of each function.

This also means that whenever you have a function multiplied by any number you can do this:

That is, you can take the number out of the limit sign. Another example:

Limit of a Quotient

As you probably expect by now, the limit of the quotient of two functions equals the quotient of the limits. For example:

In this example we used all the properties we learned. Using these you can solve many simple limits.

At first, you should think what properties you are using to solve your limits. But as you practice more, you'll see that you can simply replace "x" for the value it is approaching.

Let's see another example:

Conclusion

• The concept of limit is central in calculus.
• The limit of a function is what the function approaches when "x" approaches a specific value.
• The limit of a sum equals the sum of the limits.
• The limit of a product equals the product of the limits.
• The limit of a quotient equals the quotient of the limits.
• Many times you can simply replace "x" by the value it is approaching to solve limits.

Related Pages

• Day 1 of the Intuitive Online Calculus Course: Limits Intuition and Notation
• Solving Limits: Intuition and Examples of All Types of Limits
• Solving Limits at Infinity
• The Formal Limit Definition: Getting the Intuition Behind It
• Continuous Functions: An Intuitive Introduction

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