Ulearngo

The Existence of a Limit

As we consider the limit in the next example, keep in mind that for the limit of a function to exist at a point, the functional values must approach a single real-number value at that point. If the functional values do not approach a single value, then the limit does not exist.

Example 2.7

Evaluating a Limit That Fails to Exist

Evaluate \(\underset{x \rightarrow 0}{\text{lim}} \text{sin} \left(\right. 1 / \textit{x} \left.\right)\) using a table of values.

Solution

Table 2.5 lists values for the function \(\text{sin} \left(\right. 1 / x \left.\right)\) for the given values of x.

x \(\text{sin} \left(\right. \frac{1}{x} \left.\right)\) x \(\text{sin} \left(\right. \frac{1}{x} \left.\right)\)
−0.1 0.544021110889 0.1 −0.544021110889
−0.01 0.50636564111 0.01 −0.50636564111
−0.001 −0.8268795405312 0.001 0.826879540532
−0.0001 0.305614388888 0.0001 −0.305614388888
−0.00001 −0.035748797987 0.00001 0.035748797987
−0.000001 0.349993504187 0.000001 −0.349993504187
Table 2.5 Table of Functional Values for \(\underset{x \rightarrow 0}{\text{lim}} \text{sin} \left(\right. \frac{1}{x} \left.\right)\)

After examining the table of functional values, we can see that the y-values do not seem to approach any one single value. It appears the limit does not exist. Before drawing this conclusion, let’s take a more systematic approach. Take the following sequence of x-values approaching 0:

\[\frac{2}{\pi} , \frac{2}{3 \pi} , \frac{2}{5 \pi} , \frac{2}{7 \pi} , \frac{2}{9 \pi} , \frac{2}{11 \pi} ,\ldots.\]

The corresponding y-values are

\[1 , −1 , 1 , −1 , 1 , −1 ,\ldots.\]

At this point we can indeed conclude that \(\underset{x \rightarrow 0}{\text{lim}} \text{sin} \left(\right. 1 / \textit{x} \left.\right)\) does not exist. (Mathematicians frequently abbreviate “does not exist” as DNE. Thus, we would write \(\underset{x \rightarrow 0}{\text{lim}} \text{sin} \left(\right. 1 / \textit{x} \left.\right)\) DNE.) The graph of \(f \left(\right. x \left.\right) = \text{sin} \left(\right. 1 / x \left.\right)\) is shown in Figure 2.17 and it gives a clearer picture of the behavior of \(\text{sin} \left(\right. 1 / x \left.\right)\) as x approaches 0. You can see that \(\text{sin} \left(\right. 1 / \textit{x} \left.\right)\) oscillates ever more wildly between −1 and 1 as x approaches 0.

The graph of the function f(x) = sin(1/x), which oscillates rapidly between -1 and 1 as x approaches 0. The oscillations are less frequent as the function moves away from 0 on the x axis.
Figure 2.17 The graph of \(f \left(\right. x \left.\right) = \text{sin} \left(\right. 1 / x \left.\right)\) oscillates rapidly between −1 and 1 as x approaches 0.

Checkpoint 2.6

Use a table of functional values to evaluate \(\underset{x \rightarrow 2}{\text{lim}} \frac{\left|\right. x^{2} - 4 \left|\right.}{x - 2} ,\) if possible.

This lesson is part of:

Limits

View Full Tutorial

Track Your Learning Progress

Sign in to unlock unlimited practice exams, tutorial practice quizzes, personalized weak area practice, AI study assistance with Lexi, and detailed performance analytics.