# 2.2: The Limit of a Function

- Page ID
- 43626

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Learning Objectives

- Using correct notation, describe the limit of a function.
- Use a table of values to estimate the limit of a function or to identify when the limit does not exist.
- Use a graph to estimate the limit of a function or to identify when the limit does not exist.
- Define one-sided limits and provide examples.
- Explain the relationship between one-sided and two-sided limits.

The concept of a limit or limiting process, essential to the understanding of calculus, has been around for thousands of years. In fact, early mathematicians used a limiting process to obtain better and better approximations of areas of circles. Yet, the formal definition of a limit—as we know and understand it today—did not appear until the late 19th century. We therefore begin our quest to understand limits, as our mathematical ancestors did, by using an intuitive approach. At the end of this chapter, armed with a conceptual understanding of limits, we examine the formal definition of a limit.

We begin our exploration of limits by taking a look at the graphs of the functions

- \(f(x)=\dfrac{x^2−4}{x−2}\),
- \(g(x)=\dfrac{|x−2|}{x−2}\), and
- \(h(x)=\dfrac{1}{(x−2)^2}\),

which are shown in Figure \(\PageIndex{1}\). In particular, let’s focus our attention on the behavior of each graph at and around \(x=2\).

Each of the three functions is undefined at \(x=2\), but if we make this statement and no other, we give a very incomplete picture of how each function behaves in the vicinity of \(x=2\). To express the behavior of each graph in the vicinity of \(2\) more completely, we need to introduce the concept of a limit.

Exercise \(\PageIndex{5}\)

Evaluate \(\displaystyle\lim_{x \to 1}f(x)\) for \(f(x)\) shown here:

**Hint**-
Compare the limit from the right with the limit from the left.

**Answer**-
\(\displaystyle\lim_{x \to 1}f(x)\) does not exist

## Key Concepts

- A table of values or graph may be used to estimate a limit.
- If the limit of a function at a point does not exist, it is still possible that the limits from the left and right at that point may exist.
- If the limits of a function from the left and right exist and are equal, then the limit of the function is that common value.

## Key Equations

**Intuitive Definition of the Limit**

\(\displaystyle \lim_{x \to a}f(x)=L\)

**Two Important Limits**

\(\displaystyle \lim_{x \to a}x=a \qquad \lim_{x \to a}c=c\)

**One-Sided Limits**

\(\displaystyle \lim_{x \to a^−}f(x)=L \qquad \lim_{x \to a^+}f(x)=L\)

## Glossary

**intuitive definition of the limit**- If all values of the function \(f(x)\) approach the real number \(L\) as the values of \(x(≠a)\) approach a, \(f(x)\) approaches L

**one-sided limit**- A one-sided limit of a function is a limit taken from either the left or the right

## Contributors and Attributions

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.