LIMIT DEFINITION LIMIT
In this discussion, we confront the common understanding that the limit of 1/n as n goes to infinity equals zero. Seems straightforward, right?
Consider this: 1/2 + 1/2 equals 1, and 1/3 + 1/3 + 1/3 equals 1. In general, 1/n added n times equals 1. 0 + ... + 0 equals zero, not one leads us to the realization that the limit of 1/n as n approaches infinity, whatever it may be, is not exactly zero. Remember this perspective the next time someone claims limits like this are precisely zero.
The limit definition fundamental flaw can be visualized in this way: if we have a square with an area of -1, we can divide it into four equal parts, and repeat the process, each part having an area of -1/4 divided by four (16 pieces in total), essentially containing a scaled-down version of the original square in a fractal-like manner. However, the limit definition (an unproven statement) tells us that at some point, this exact but scaled-down version suddenly becomes zero and vanishes, no longer representing a square. The main challenge to accepting this idea is that we can put all these small pieces together and obtain the -1 unit square again. On the other hand, adding infinitely many zeros will result in zero and not -1. It is uncanny to believe that adding infinitely many zeros suddenly becomes a negative number.
