There is an interesting "thought experiment" that often puzzles students when they first learn about limits in calculus. This is known as the Limit Paradox, and is sometimes presented in the form of an equalateral triangle as shown below: The path ABC is twice as long as AC. Similarly the path ADEFC is also twice as long as AC, as is the path AGHIEJKLC, and so on. Breaking down the jagged path into smaller and smaller jags, the deviation of the jaged path from the straight line AC goes to zero, so, in a sense, the line AC is the "limit" of the sequence of jagged paths. This might seem to suggest that the length of AC is twice the length of itself! Paradoxes like this were discussed extensively (and very seriously) during the early history of calculus. Another example - one that may help to illustrate the fallacy of these paradoxes - is to consider the sequence of numbers 0.9, 0.99, 0.999, etc. Clearly none of these numbers is an integer. However, these numbers approach ever more closely the number 1.0, so are we justified in concluding that the number 1.0 is not an integer? No. Similarly, we could note that the average size of the non-zero decimal digits of 0.9, 0.99, etc is 9, so we might think the average size of the non-zero digits of 1.0 must also be 9, but of course it isn't. In the words of The Encyclopedia Britanica, "The limit paradox is the result of the mistaken idea that the limiting configuration must have properties which are the limiting cases of the corresponding properties of the approximating configurations." While I wouldn't cite this as a model of didactic clarity, it does make the key point, which is that entities in any given sequence generally possess multiple properties, and an entity that possesses the limiting value of one of those properties doesn't necessarily possess the limiting value of ALL those properties (many of which may not even converge). In the case of the jagged paths, we are actually considering the boundaries of the minimal envelope surrounding the path, and noting that the limiting jagged path resides entirely within an arbitrarily small envelope around the line AC. The boundaries of this envelope approach the line AC, in position as well as length, but the length of the jagged line within this envelope does not converge on the length of the envelope containing it. In fact, we could easily construct a sequence of looping paths, with an exponentially increasing number of geometrically decreasing loops, such that the total length of the looping path from A to C goes to infinity as the envelope converges on the straight line AC. These examples illustrate that you have to be careful about which property is being taken to the limit. It's worthwhile to keep this in mind when considering things like Koch's snowflake and other fractal boundaries, which tend to be defined as the limiting cases of progressive fragmentation processes. Those constructions are more complicated and involve more subtle issues, because the minimal envelope itself becomes progressively more convoluted, in contrast to the simple "limit paradox" we've been considering, where the limiting envelope is well-behaved.

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