Limits

Limits are the most important thing that distinguishes the science of calculus from the other branches of mathematics. The concept of limits is very essential in calculus. One should not proceed into the subject without first being familiar with that concept and understanding it in a good way.

Let’s discuss the concept of limit to form a general idea about the matter. Consider a function . This function has a limit L as x approaches a number c, we mean that the value of that function  becomes closer and closer to the number L as x gets closer and closer to c.

Some examples may clarify this point.

Example

We have a function . We want to calculate the limit of this function as  approaches 1.

Solution

Practice

Try to calculate the limit of the following functions in the same way:

, as x approaches 2.

, as x approaches 0.

In the last example, one can notice that f(1)=4, and so the whole process of finding the limit does not seem so useful as we can simply calculate the value of the function at some value of x which will be equal the limit of the function at that point. In fact, there some functions where the value of the limit at some points is not equal to the value of the function. There are also functions that have a limit at some values of x while the function itself has no value at that point. That makes it necessary to calculate the limit of the function and not the value of the function.

Example

Find the limit of the function:

as x approaches 1.

Solution

NB

To say that the limit of the function f as x approaches 1 is 4, we write:

Practice

Calculate the limit of the following functions at the given points:

, as x approaches 3.

, as x approaches 0.