We can’t assume one exists, and there may not be an answer to every question. Then ask the “impossible question”: what’s your predicted speed when the gap to the neighboring instant is zero? Limits help answer this conundrum: predict your speed when traveling to a neighboring instant. Speed is funny: it needs a before-and-after measurement (distance traveled / time taken), but can’t we have a speed at individual instants? Hrm. If we could use an infinite number of rectangles to simulate curved area, can we get a result that withstands infinite scrutiny? (Maybe we can find the area of a circle.) But, if we could make a prediction, is there a single rate that is ever-accurate? It seems to be around 2.71828…Ĭan we use simple shapes to measure complex ones?Ĭircles and curves are tough to measure, but rectangles are easy. We can’t easily measure the result of infinitely-compounded growth. What does perfectly continuous growth look like?Į, one of my favorite numbers, can be defined like this:
Nowadays, we have modern limit definitions of pi. It was the precursor to calculus: he determined that pi was a number that stayed between his ever-shrinking boundaries. We can’t measure a shape with seemingly infinite sides, but we can wonder “Is there a predicted value for pi that is always accurate as we keep increasing the sides?”Īrchimedes figured out that pi had a range of What things, in the real world, do we want an accurate prediction for but can’t easily measure?įinding pi “experimentally” is tough: bust out a string and a ruler? Let’s not bring out the math definitions just yet. Limits are a strategy for making confident predictions. This estimate is confirmed by our initial zoom (3:59-4:01, which estimates 9.9 to 10.1 meters) and the following one (3:59.999-4:00.001, which estimates 9.999 to 10.001 meters)”. With these requirements in place, we might say “At 4:00, the ball was at 10 meters. Which one had the ball at 4:00? This ambiguity shatters our ability to make a confident prediction. What happened? We had a sudden jump (a camera change?) and now we can’t pin down the ball’s position. Imagine at 3:59 the ball was at 10 meters, rolling right, and at 4:01 it was at 50 meters, rolling left. Uh oh! Zooming should narrow our estimate, not make it worse! Not every zoom level needs to be accurate (imagine seeing the game every 5 minutes), but to feel confident, there must be some threshold where subsequent zooms only strengthen our range estimate. The predictions agree at increasing zoom levels. With a slow-motion camera, we might even say “At 4:00, the ball was between its positions at 3:59.999 and 4:00.001”.
Our prediction is “At 4:00, the ball was between its position at 3:59 and 4:01”. Just grab the neighboring instants (3:59 and 4:01) and predict the ball to be somewhere in-between.Īnd… it works! Real-world objects don’t teleport they move through intermediate positions along their path from A to B. Even so, what’s your prediction for the ball’s position?Įasy. Unfortunately, the connection is choppy:Īck! We missed what happened at 4:00. And if the function behaves smoothly, like most real-world functions do, the limit is where the missing point must be. When our prediction is consistent and improves the closer we look, we feel confident in it. The limit wonders, “If you can see everything except a single value, what do you think is there?”. If we can directly observe a function at a value (like x=0, or x growing infinitely), we don’t need a prediction. But for most natural phenomena, it sure seems to. Our prediction, the limit, isn’t required to match reality. Why do we need limits? Math has “black hole” scenarios (dividing by zero, going to infinity), and limits give us an estimate when we can’t compute a result directly.If our prediction is always in-between neighboring points, no matter how much we zoom, that’s our estimate. How do we make a prediction? Zoom into the neighboring points.What is a limit? Our best prediction of a point we didn’t observe.Limits, the Foundations Of Calculus, seem so artificial and weasely: “Let x approach 0, but not get there, yet we’ll act like it’s there… ” Ugh.
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