I’ve often joked about the least important thing I learned in school. The list of candidates could fill out a hypothetical March Madness style bracket (!). One day I would think it was the name of Columbus’s three ships, the next I would be sure it was penmanship. I can still do a pitch-perfect whistling rendition of ‘Hot Cross Buns’. In my most sober moments, I reflect deeply on the Lanthanide Series, and ask existential questions about the periodic table.
It was only recently that I gave the reverse question any thought. What was the most important thing? I think it has to be limits, a concept I first understood in high school calculus. A limit is a property of a mathematical formula. It states the formula’s behavior as it approaches certain numbers. (The limit of 1/x as it approaches infinity is 0.) I remember thinking at the time that this whole business with limits was a little silly. It always seemed like the answer was 0, infinity, or undefined. The mechanics of the limit I understood, the point of it not so much. Limits were the first topic we covered in that course and we soon moved on to bigger topics in the syllabus.
I look back on limits as critical because it was the first time I thought seriously about how the smallest details in the present might dictate large effects in the future. The multiplicative effect is the easiest example. A constant changes forever when X is introduced. When 2 becomes 2X, the limits are forever altered and what once remained stuck in place can now reliably be expected to approach something as powerful as infinity. This idea works both ways. A simple formula of 2X is far less powerful than X-squared. Their limits are the same, however, for as each formula approaches infinity the limit is infinity. The difference in the two is the rate of change, a difference ably examined by the rest of the calculus class, but this difference had no relevance to limits. They would both get there, eventually.
The key in the idea is the movement itself, always represented in X. There is a huge difference between stagnant and barely growing. It doesn’t matter if the stagnant is a huge idea, if the number is one thousand or one million or one billion, because in the long term anything with X involved has the power to overtake it. Studying limits was a rare moment for me in high school, one of a truly mind expanding nature, but in that very understanding is the power of the limit itself – the best ideas are the ones that expand our horizons.
