Potential Flaws in Empirical Evidence for Limits

Does $\sin(10\pi/x)$ have a limit as $x\to0$ ?

Hint: Try plugging in smaller and smaller values like $\pm0.1$ , $\pm0.01$ , $\pm0.001$ , etc. What do you find?

Here's a graph to visualize what's going on as $x$ gets smaller and smaller:

$x$

$y$

Actually, when $x$ is too small, the graph pictured above is no longer accurate---the function $\sin(10\pi/x)$ oscillates too wildly!

Give your best guess: Why is the graph inaccurate for small values of $x$ ?

Great. If you want a more accurate, high-resolution, graph of this wildly oscillating function, try http://www.wolframalpha.com/input/?i=graph+sin[10pi%2Fx]

If you're interested, the graph is closely related to something known as the topologist's sine curve.

Keep going