Limits (essay)
essay prompt: Is the idea of 'limit' indispensable? Neutral? A hindrance?
I contend that some concept of 'limit' is indispensable in mathematics, physics, and beyond. I will make this case inductively, beginning with an exposition of the facts of reality that gave rise to the need for such a concept, and concluding with a proper definition of 'limit'.
[TODO maybe also talk about invalid versions of the concept?]
Fact of reality 1:
The Earth is a sphere, but its radius is so large, and the circumstances of everyday life take place on a scale so small, that in most contexts, it is valid to think of the Earth as a plane.
We may say, informally, that a plane is the limit of a sphere as its radius "tends towards infinity". [TODO one has to be careful about which limit one is taking, TODO harry doesn't like the word infinity, but I don't know how else to say it]
We may also say that in the context set by many everyday situations, the difference between a plane, and a sphere with 4000mi radius, is nill.
Fact of reality 2:
[TODO write neatly] The sum from j = 0 to N of 2^{-j} is tabulated below for various values of N
N sum
----|-----
0 1
1 1.5
2 1.75
3 1.875
4 1.9375
5 1.96875
6 1.984375
...
30 1.999999999068677425384521484375
It appears that the larger N gets, the closer this sum gets to 2.
One might ask: How do we really know that if we continue for long enough, we'll get increasingly closer to 2? All we've seen is that we get very close to 2 after we sum the first 30 numbers. But maybe something unexpected will happen if we sum up to 3000, 300, or even 31.
Such a question can be answered as follows. First, note that
(1 + x + x^2 + ... + x^N)(1 - x) = 1 + x + x^2 + ... + x^N
-(x + x^2 + ... + x^N) - x^{N+1}
= 1 - x^{N+1}.
Therefore, rearranging, we find
1 + x + x^2 + ... + x^N = (1 - x^{N+1})/(1 - x)
or for x = 1/2,
1 + 1/2 + 1/2^2 + ... + 1/2^N = 2 - 2^{-N}.
Thus we have precisely quantified the difference between (the sum up to N) and 2; it is 2^{-N}. The larger our value of N gets, the smaller this difference will become.
In fact, for any nill value epsilon, there exists some N such that the difference between (the sum up to N) and 2 is nill.
--
One might ask: Who CARES about passing to the limit? What do we gain from talking about 2 instead of 1.984375 ? The answer is that the former is simpler than the latter.
I will introduce the principle of Occam's razor,