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:

Intuitive idea of limits

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

The limit of an actual function may look like this:

First example of a limit

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

Example of limit notation

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:

First example of a limit

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:

Limit represented in a table

The function clearly approaches 3, right?

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

Limit represented in a table

It also approaches 3...

So, when this happens, we write:

Answer to first example of limit

Limit of the Sum of Functions

Now, let's suppose we have two functions:

Second example of limit

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?

Limit of the sum of two functions

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:

Limit of the sum of two functions

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

Limit of the sum of two functions

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:

Limit of a product of two functions

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

Limit of a product of two functions

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:

Limit of a product of two functions

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

Another example of limit of the product of two functions

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:

Limit of a quotient

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:

Second example of limit of a quotient

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


Return from Limit of a Function to Limits and Continuity

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