The Cantor Set and Fractals
Take a line segment:
Draw dots to divide the line segment into thirds:
Then erase the middle third (but keep the dots):
Repeat with the two remaining line segments. Draw dots to divide into thirds:
Then erase the middle thirds (but keep the dots):
Continue until the end of time. The limit is the Cantor Set, which will contain no line segments since any line segments have been broken up. Instead, we find infinitely many dots. We get more dots every time we break up line segments.
If we go up yet another dimension,we discover the dizzying idea of the Menger Sponge.
Why do we care about these weird objects? Not only is their construction a cool usage of infinite limits, these objects have properties you probably didn't think were actually possible. The Sierpinski Carpet, for one, has finite area, but an infinite perimeter. We sure as heck didn't think that was possible until infinite limits opened our eyes to the possibility.