- Understand how to write the limit of a function using the correct symbols and estimate limits by examining tables and graphs
- Understand one-sided limits (approaching a point from only one direction) and how they relate to two-sided limits
- Understand and use the proper notation for infinite limits and define vertical asymptotes
The Definition of a Limit
We begin our exploration of limits by taking a look at the graphs of the functions
which are shown in Figure 1. In particular, let’s focus our attention on the behavior of each graph at and around .

Each of the three functions is undefined at , but if we make this statement and no other, we give a very incomplete picture of how each function behaves in the vicinity of . To express the behavior of each graph in the vicinity of more completely, we need to introduce the concept of a limit.
Intuitive Definition of a Limit
Let’s examine how the function behaves as f approachs . While isn’t defined at , as gets closer to from either side, approaches .
Mathematically, we say that the limit of as approaches is . We express this observation using limit notation as:
This initial exploration into limits leads us to a more formal definition.
Consider the limit of a function at a specific point as the value that the function’s output gets closer to, as the input values approach that point. Assuming such a value exists, we can articulate this concept more precisely with the following definition:
limit definition
For a function defined over an open interval around a point , possibly excluding itself, if all function values get arbitrarily close to some real number as approaches , then is the limit of as approaches .
A more succinct way to understand this definition: As gets closer to , gets closer and stays close to .
Estimating Limits Using Tables
We can estimate limits by constructing tables of functional values.
How to: Evaluate a Limit Using a Table of Functional Values
- To find , create a table with two sets of -values: those just less than and those just more than . The table below demonstrates what your tables might look like.
Table of Functional Values for Use additional values as necessary. Use additional values as necessary. - Analyze the values. If they get closer to a single number as approaches from both sides, that’s the limit.
- If both sides of is confirmed. If not, the limit may not exist.
- Use the graph of to verify your results. By plotting the function and zooming in around , you can observe if approaches the limit you calculated. This visual check complements the numerical approach.
Evaluate using a table of functional values.