One-Sided and Infinite Limits

Just as we first gained an intuitive understanding of limits and then moved on to a more rigorous definition of a limit, we now revisit one-sided limits. To do this, we modify the epsilon-delta definition of a limit to give formal epsilon-delta definitions for limits from the right and left at a point. These definitions only require slight modifications from the definition of the limit. In the definition of the limit from the right, the inequality \(0 < x - a < \delta\) replaces \(0 < \left|\right. x - a \left|\right. < \delta ,\) which ensures that we only consider values of x that are greater than (to the right of) a. Similarly, in the definition of the limit from the left, the inequality \(- \delta < x - a < 0\) replaces \(0 < \left|\right. x - a \left|\right. < \delta ,\) which ensures that we only consider values of x that are less than (to the left of) a.

Example 2.44

Proving a Statement about a Limit From the Right

Prove that \(\underset{x \rightarrow 4^{+}}{\text{lim}} \sqrt{x - 4} = 0 .\)

Solution

Let \(\epsilon > 0 .\)

Choose \(\delta = \epsilon^{2} .\) Since we ultimately want \(\left|\right. \sqrt{x - 4} - 0 \left|\right. < \epsilon ,\) we manipulate this inequality to get \(\sqrt{x - 4} < \epsilon\) or, equivalently, \(0 < x - 4 < \epsilon^{2} ,\) making \(\delta = \epsilon^{2}\) a clear choice. We may also determine \(\delta\) geometrically, as shown in Figure 2.42.

Figure 2.42 This graph shows how we find δ for the proof in Example 2.44.

Assume \(0 < x - 4 < \delta .\) Thus, \(0 < x - 4 < \epsilon^{2} .\) Hence, \(0 < \sqrt{x - 4} < \epsilon .\) Finally, \(\left|\right. \sqrt{x - 4} - 0 \left|\right. < \epsilon .\)

Therefore, \(\underset{x \rightarrow 4^{+}}{\text{lim}} \sqrt{x - 4} = 0 .\)

We conclude the process of converting our intuitive ideas of various types of limits to rigorous formal definitions by pursuing a formal definition of infinite limits. To have \(\underset{x \rightarrow a}{\text{lim}} f \left(\right. x \left.\right) = + \infty ,\) we want the values of the function \(f \left(\right. x \left.\right)\) to get larger and larger as x approaches a. Instead of the requirement that \(\left|\right. f \left(\right. x \left.\right) - L \left|\right. < \epsilon\) for arbitrarily small ε when \(0 < \left|\right. x - a \left|\right. < \delta\) for small enough \(\delta ,\) we want \(f \left(\right. x \left.\right) > M\) for arbitrarily large positive M when \(0 < \left|\right. x - a \left|\right. < \delta\) for small enough \(\delta .\) Figure 2.43 illustrates this idea by showing the value of \(\delta\) for successively larger values of M.

Figure 2.43 These graphs plot values of \(\delta\) for M to show that \(\underset{x \rightarrow a}{\text{lim}} f \left(\right. x \left.\right) = + \infty .\)