Limit of Sin(x) at Infinity

Shown is the graph of This sketch demonstrates why the limit of this function does not exist at . The function oscillates between and as . Since the function will never stop moving between those two values, there is not a single value that the function is approaching. If you pick any -value that has an absolute value of less than one, then you can find an arbitrarily large -value whose -value exceeds the one you chose. Click the "Zoom Out" button to see what happens as we change the scale on the -axis. You can see the oscillating behavior never changes and the graph becomes so dense it seems to fill the entire space. For this reason, the limit does not exist as there is no single value that the function approaches.

does not exist. The same thing happens on the left end of the graph as well, so:  does not exist either.