Informal Definition and Notation

The limit can be informally defined as the value that a function approaches as the input approaches a certain value. In other words, it is the value that the function gets closer and closer to as the input gets closer and closer to a certain value.

Consider the function . This is a reciprocal function, and as you increase the value of , the value of gets closer and closer to :

This can also be seen graphically by plotting the function. Let's plot :

When we use the graph to visualize the function, we can make an estimate for the limit. This is a good way to get an intuition for what the limit is.

As you can see, as gets larger and larger, gets closer and closer to . While never actually reaches , it gets arbitrarily close to it. This is what we call a limit.

In terms of words, we can say:

The limit of as approaches infinity is .

This is written with notation as:

Here, the symbol represents the limit, means that is approaching infinity, and is the function we are taking the limit of:

Next, consider the limit of as approaches . This time, the value of gets larger and larger as gets closer and closer to . As such, we can say that the limit of as approaches is unbounded.

This is written as:

Summary and Next Steps

We have introduced the concept of limits and how they can be informally defined. Here are the key points to remember:

1. The limit of a function is the value that the function approaches as the input approaches a certain value.
2. The limit of a function as the input approaches some value is written as .
3. If a limit is unbounded, it means that the function grows without bound as the input approaches the value.
4. A limit can be estimated by looking at the graph of the function.

Next, we will look at the concept of One-Sided Limits.