Exploring the Distance from (0,0) to (1,1) with Limits
Date: 10/15/2006 at 00:28:21 From: Marc Subject: Does the distance from (0,0) to (1,1) have a discontinuity Any monotonic route from (0,0) to (1,1) along a path of only right angles will have a total distance of 2. As the number of "steps" increases to infinity, the distance is constant. Yet in the limit, the distance traveled should converge to the diagonal length of sqrt (2). How can this be? I'm confused because my intuition tells me this function has got to be continuous. I don't know what reasoning to bridge the gap between a right angled path and the diagonal route. I understand arc length derivation, but that also makes the leap from right-angled steps to diagonal in the differential [ds = sqrt (dx^2 + dy^2)]. Why can't we press onto smaller and smaller steps and still get the answer sqrt(2)? If I take n equal length steps, the distance traveled is n*(1/n)*2. The limit of this is 2*inf/inf -LHop-> 2*1/1 = 2. But as n increases, I'm getting arbitrarily close to walking the diagonal, which I know has length sqrt(2). Why doesn't my limit calculation apply here?
Date: 10/17/2006 at 15:55:07 From: Doctor Rick Subject: Re: Does the distance from (0,0) to (1,1) have a discontinuity Hi, Marc, thanks for writing to Ask Dr. Math. Your intuition goes astray not on the continuity issue, but when you assume that in the limit, the distance should converge to the length of the diagonal line. Here is how I answered a similar question a year ago: ================================================================= Question: Hi, my math teacher one day said it was ridiculous to do this. Take a triangle with both of the smaller side (a and b) equal to one. Using Pythagoras it is easy to see that the other side is (c) equal to square root of 2. Ex: | \ 1 | \ square of 2 |___\ 1 Why then can't you approximate the last line by doing a small escalator that if pushed to infinity it would approximate one doing something like this __ 1 | | | -- |_____| then split again 1 _ | | | - 1 | | | - |_____| 1 if you continue like this how can you say that adding the little lines you won't get a good approximation of the line c?? To put it into a reasonable mathematical inequality such as lim x + lim x ain't equal to ... .. I'm pretty sure it has to do with limits and infinite sums but I can't put my finger on it... perhaps there's an impossible inversion? Answer: You are assuming that, in the limit, the total length of that "zigzag" line is equal to the length of the straight line joining its endpoints. This is not true. Consider the length of a "zigzag" before you take the limit: +---+ \ | +---+ \ | +---+ \ | +---+ \ | + In each of the small triangles, the ratio of the sum of the horizontal and vertical legs to the hypotenuse is 2:sqrt(2), or sqrt(2):1. Thus the ratio of the total length of the zigzag to the length of the straight line is also sqrt(2):1. Thus if the length of the straight line is sqrt(2), the length of the zigzag is 2. This is true no matter how small you make the triangles! The sequence of lengths of the zigzags is 2, 2, 2, 2, 2, ... What is the limit of this sequence? Obviously it is 2, not sqrt(2). The zigzag, in the limit, may "look" like a straight line--but it is not a straight line! Its length is not equal to the length of the straight line. This is not intuitive. The message is: Don't judge based on the appearance of the limiting case. Examine the sequence carefully *before* you take the limit. ================================================================= Does that answer your question? - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
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