Sometimes, it doesn’t matter as much where things are right now, but where they are going. For example, I am writing this roughly six months into the COVID-19 quarantine in the United States, with the 2020 presidential election on the horizon. With an election nearing, there are a lot of polls out there about who is ahead of who. Although those could be interesting, what I think all of us care most about is how those polling results change over time. For instance, will the recent death of Supreme Court justice Ruth Bader Ginsberg alter public opinion on the election towards Republicans, towards Democrats, or neither? I imagine most people who read this will know the answer to this question – but as I write this we don’t know yet, and in any case that isn’t my point. My point is that we don’t merely case about what polls say right now – we also care about where they are going. To use fewer words, we care about the trend over time of the polls.
Polls, of course, use percentages. This makes polls very mathematical in nature. It is quite reasonable, therefore, to ask what mathematics has to say about polls. As has just been described, we as humans tend to care quite a bit not only about how things are at the moment, but how things change over time. There are numerous examples of our tendency to care about change over time, including climate change, political elections and our physical health, just to name a few. Mathematics has always been used to count things and quantify various values, but can we use mathematics to quantify how things change over time?
Thanks to calculus, the answer is yes. In fact, you could actually define calculus as the mathematical study of how things change over time. I of course am leaving out a lot of detail here, but for now this definition will suffice. When you hear calculus, think about the various problems we face in trying to understand how the world around us changes over time. How do the planets move around us? How do we move around the Sun? How does my bank statement change as I acquire interest? How does the location of a plane change as that plane flies? All of these questions – and so much more – lie in the realm of calculus.
A Foundational Concept: “Zooming In”
How do we study change over time? One way is through averages. For instance, if one year ago I had $1000 dollars in my bank account, and without ever depositing any extra money I ended up with $1010 in my account – then how much did I earn per day in interest? If I use averages, then I arrive at
This would tell me that, on average, my bank account gained 83.3 cents per month. But is that actually true? Well, not necessarily. Because of how interest works, it is actually more likely that I earned something a little bit closer to 83 cents in the early months and something a little closer to 84 cents in the later months. In this situation, the ‘error’ is pretty small. But if I tried to do the same averaging over much longer periods of time – say 10 or 20 years – then the average would become further and further from actually true.
Since averages tend to get less and less accurate as you expand the timeline, wouldn’t it make sense that narrower timelines are always better? Certainly that must be true. And in mathematics, it is true. The concept of “narrowing the timeline” is at the heart of what calculus is. In more visual terms, we can view this process as “zooming in.” The concept of “zooming in” is at the absolute heart of what calculus is. Our goal is to use the “smallest possible timeframe” for zooming in. Ideally, we’d like to use a “single instant” of time as our timeframe… but what does that mean?
The Concept of the Limit
The limit is the mathematician’s way of making sense of ‘taking an average’ at just a single point in time. This actually took a long time to figure out. For aficionados of mathematical history, you might know that Archimedes came very close to coming up with this idea in ancient Greece, about two thousand years ago. But he didn’t quite get all the way there. Nonetheless, the basic idea Archimedes used is the same core concept used in limits. The idea is to look at the trend of a process over time. Let’s use an example.
Consider the list of numbers: , a list that keeps going and going. There is a pattern in this list – each number has an extra 3 added on to the end of it. One natural question to ask about a thing like this is “does this pattern take us anywhere?” In others words: Are we moving towards something, ever closer without ever quite getting there? The answer here is yes – intuitively you’d want to say with 3’s going on forever. This actually is a number – this turns out to be exactly the value of .
Continuing a process like this all the way to the ‘end’ of a process is basically what we mean by a limit. As we get closer and closer to some ‘final point’ – is there a pattern? If so, what is the pattern?
Framing the Limit Quantitatively
This concept of a trend towards some ‘final result’ is a fairly intuitive idea. But how can we frame this into actual mathematics? To show how we do this, let me define how I would write this down as a mathematician, and then explain how a mathematician can make more exact how to calculate these things (although a bit abstractly).
The easiest way to go about this is to use function notation. The role of the ‘pattern’ is played by a function . If we want to, we could be more specific about what sort of pattern is, but we don’t have to. Imagine now that instead of wanting to what value represents, we instead want to know what is close to when is very close to 2. You might think of this as the ‘trend’ of as approaches 2. If there turns out to be an actual answer to this question, we call that answer
and we read this out loud as “the limit as approaches 2 of “. Now, what would it actually mean for there to be an answer to the question? To explain how this works, let me use a more explicit example. Let us say that the ‘pattern’ we actually care about happens to be described by the function . What would it mean for a limit, say to be equal to some number, say ? One way to think about it is you can “make zoom in on ” by “making zoom in on 2″ close enough. In more visual terms, you can zoom in on the height of by zooming in on 2 on the -axis.
A Numerical Example
For a more ‘numerical’ approach, I’ll briefly explain that . Let’s say that we are in a scenario where we are building something that will collapse if we don’t get our measurement close enough to 7 – to be more specific, we have to be within of 7. We know that if we were to use an exact value of 2 for , we would get exactly 7. But this is the real world – measuring exactly isn’t possible. So we want to make sure that we are using a measurement device that gets ‘close enough’ to 2 for our building to be safe. How close will that need to be?
Well, it would help to have an equation that represents, given the measurement we end up taking, how far off we are the value we need it to be – 7. We can express this equation as
We have stated earlier that our error has to be within in order for the building to be safe. Using the equation for error we just came up with, we can now say that we need
We can add 7 to everything, since this is an equation. When we do this, and remember that is really just , we end up with the equation
As with all equation-solving, our goal is now to get on its own. We can subtract the 3 from both sides to get closer to that goal. Once we do this, our equation looks like
Finally, we can divide both sides by 2. If we do this, we conclude that
So we now know the actual values of we require. If we instead what to know how far away from two needs to be, we can subtract 2 from both sides and conclude that . This means that in order to get our measurement close enough to 7 for our purposes, we need our input measurement to be $\pm 0.005$ away from what we want it to be.
There wasn’t really anything special here about . We could have made that number much smaller, but as long as was within plus-or-minus half of the value, we would have ended up close enough to 7. This is what we mean by a limit.
A “Fancy” Definition of the Limit
For the sake of completeness, I should define the limit in the exactly language that a mathematician would use. If your only goal is to learn what calculus is about conceptually, you probably won’t actually need this part. The intuitive descriptions given previously will serve you well in understanding what we mean by a “limit”. But if you want to know more about how calculus is “done”, then this may well be of use.
In the previous discussion, we have made frequent use of very small numbers. In order to make math easier to read, mathematicians try to use certain variables only for these very small numbers. The two most popular choices are the Greek letters epsilon and delta – which are written and . In the definition you are about to see, you should imagine both and to be some very small numbers.
Definition: is considered to be true if, for whatever choice of small number we want, we could choose another small number so that whenever is within of (also written ), the value of is within of (also written ).
There is of course nothing special about the 2. We could have put any number there. In fact, there is also a way to define limits “going to infinity”. If we replace being “close to 2” with “being extremely large”, then you basically end up with what a limit going to infinity represents. For a formal definition,
Definition: is true provided that, if we choose any number we want, we can choose a large number so that whenever is chosen to be larger than , it will always be true that .
We are now done discussing the big-picture concept of what a limit is all about, worked out a numerical example to build on that intuition, and even given a “fancy” definition of what we actually mean by limits. These examples should be very helpful in building up our intuition for understanding more topics in calculus. Next time, I will walk through some important examples of how to actually calculate limits, examples that should point out some of the special things you can calculate using limits that you can’t do any other way.