Consider the right-angled triangle with vertices (0, 0),
(1, 0), and (0, 1), and suppose we wanted to compute the length of the hypotenuse
of this triangle. Pythagoras’ theorem tells us that this hypotenuse has length √2,
but suppose for some reason that we did not know about Pythagoras’ theorem, and
wanted to compute the length using calculus methods. Well, one way to do so is to
approximate the hypotenuse by horizontal and vertical edges. Pick a large number N,
and approximate the hypotenuse by a “staircase” consisting of N horizontal edges of
equal length, alternating with N vertical edges of equal length. Clearly these edges
all have length 1/N, so the total length of the staircase is 2N/N = 2. If one takes
limits as N goes to infinity, the staircase clearly approaches the hypotenuse, and so in
the limit we should get the length of the hypotenuse. However, as N → ∞, the limit
of 2N/N is 2, not √2, so we have an incorrect value for the length of the hypotenuse.
How did this happen?