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:

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

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.