Investigating a Limit

Investigate \(\lim\limits_{x\rightarrow 0}\sin \dfrac{\pi}{x^{2}}\).

Solution The domain of the function \(f(x)=\sin \dfrac{\pi }{x^{2}}\) is \(\{x|x\neq 0\}\). Suppose we let \(x\) approach zero in the following way:

\(\underrightarrow{x~\hbox{approaches 0 from the left}}\) \(\underleftarrow{x~\hbox{approaches 0 from the right}}\)
\(x\) \(-\dfrac{1}{10}\) \(-\dfrac{1}{100}\) \(-\dfrac{1}{1000}\) \(-\dfrac{1}{10{,}000}\) \(\rightarrow\) 0 \(\leftarrow\) \(\dfrac{1}{10,000}\) \(\dfrac{1}{1000}\) \(\dfrac{1}{100}\) \(\dfrac{1}{10}\)
\(f(x) =\sin \dfrac{\pi }{x^{2}}\) 0 0 0 0 \(f(x)\) approaches 0 0 0 0 0

The table suggests that \(\lim\limits_{x\rightarrow 0}\sin \dfrac{\pi }{x^{2}}=0\). Now suppose we let \(x\) approach zero as follows:

\(\underrightarrow{x~\hbox{approaches 0 from the left}}\) \(\underleftarrow{x~\hbox{approaches 0 from the right}}\)
\(x\) \(-\dfrac{2}{3}\) \(-\dfrac{2}{5}\) \(-\dfrac{2}{7}\) \(-\dfrac{2}{9}\) \(-\dfrac{2}{11}\) \(\rightarrow\) 0 \(\leftarrow\) \(\dfrac{2}{11}\) \(\dfrac{2}{9}\) \(\dfrac{2}{7}\) \(\dfrac{2}{5}\) \(\dfrac{2}{3}\)
\(f(x) =\sin\dfrac{\pi}{x^{2}}\) 0.707 0.707 0.707 0.707 0.707 \(f(x)\) approaches 0.707 0.707 0.707 0.707 0.707 0.707

This table suggests that \(\lim\limits_{x\rightarrow 0}\sin \dfrac{\pi }{x^{2}}=\dfrac{\sqrt{2}}{2}\approx 0.707\).

In fact, by carefully selecting \(x\), we can make \(f\) appear to approach any number in the interval \([-1, 1]\).

Now look at the graphs of \(f(x) =\sin \dfrac{\pi}{x^{2}}\) shown in Figure 15. In Figure 15(a), the choice of \(\lim\limits_{x\rightarrow0}\sin \dfrac{\pi }{x^{2}}=0\) seems reasonable. But in Figure 15(b), it appears that \(\lim\limits_{x\rightarrow 0}\sin \dfrac{\pi }{x^{2}}=-\dfrac{1 }{2}\). Figure 15(c) illustrates that the graph of \(f\) oscillates rapidly as \(x\) approaches 0. This suggests that the value of \(f\) does not approach a single number, and that \(\lim\limits_{x\rightarrow 0}\sin \dfrac{\pi }{x^{2}}\) does not exist.