An informal definition of limits

Section 3.1 An informal definition of limits

Our brief overview of calculus convinced us that the study of limits is foundational in calculus. In this section we give an informal definition of limits, and show how the value of a limit can (sometimes, but not always) be estimated using tables of values. We also define the notion of one-sided limits.

Objectives

You should be able to:

Explain and illustrate the concept of the limit of a function \(f(x)\) as \(x\) approaches \(a\text{.}\)
Construct examples of limits that exist and examples of limits that do not exist.
Identify a given limit (or conclude that it does not exist) from the graph of a function.
Explain the difference between left- and right-sided limits.
Relate limits to one-sided limits.
Estimate the value of a limit using a table of values.

Subsection 3.1.1 Instructional video

Subsection 3.1.2 Key concepts

Concept 3.1.1. The informal definition of limits.

We write

\begin{equation*} \lim_{x \to a} f(x) = L, \end{equation*}

and say that the limit of \(f(x)\text{,}\) as \(x\) approaches \(a\text{,}\) is equal to \(L\), if we can make the values of \(f(x)\) arbitrarily close to \(L\) by taking \(x\) sufficiently close to \(a\) (on either side of \(a\)) but not equal to \(a\text{.}\)

Note that saying that \(\displaystyle \lim_{x \to a} f(x) = L\) is not the same as saying that \(f(a) = L\text{.}\) The former is about the behaviour of \(f(x)\) for \(x\) near \(a\text{,}\) while the latter is the value of the function \(f(x)\) at the point \(x=a\text{.}\)

Concept 3.1.2. Estimating the value of a limit using a table of values.

We can often estimate \(\displaystyle \lim_{x \to a} f(x)\) by writing down a table of values for the function \(f(x)\) with \(x\) closer and closer to \(a\) (from both sides).

Concept 3.1.3. One-sided limits.

We write

\begin{equation*} \lim_{x \to a^-} f(x) = L, \end{equation*}

if \(f(x)\) gets close to \(L\) when \(x\) approaches \(a\) from the left (from below), and

\begin{equation*} \lim_{x \to a^+} f(x) = L, \end{equation*}

if \(f(x)\) gets close to \(L\) when \(x\) approaches \(a\) from the right (from above).

Concept 3.1.4. Limits vs one-sided limits.

Note that

\begin{equation*} \lim_{x \to a} f(x) = L \end{equation*}

if and only if

\begin{equation*} \lim_{x \to a^-} f(x) = L \qquad \text{and} \qquad \lim_{x \to a^+} f(x) = L. \end{equation*}

MATH 144: Calculus for the Physical Sciences I