The Limit Paradox
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.
Return to MathPages Main Menu