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I have to teach limits to infinity of real functions of one variable. I would like to start my course with a beautiful example, not simply a basic function like $1/x$. For instance, I thought of using the functions linked to the propagation of covid-19 and show that, under the basic model, the number of contaminations will go to $0$ when time goes to $+\infty$. However, this is a bad idea because the model is not so easy to explain and moreover students are sick of covid-subjects.

Hence, I ask you some help to find interesting examples from physics, geography, etc ... I suppose that an example with "time" going to $+\infty$ would be nice.

NB : This a crossed-question with MSE : https://math.stackexchange.com/questions/4340560/nice-examples-of-limits-to-infinity-in-real-life

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    $\begingroup$ Do you mean a mathematician’s notion of real-life, a physicist’s notion of real-life, or actual real life (aside from what an actual physicist, etc. actually do)? In actual real life, time does not go to $+\infty$, though physicists and mathematicians actually find limits at infinity every day. So might an engineer, but an engineer’s transients disappear in finite time, in practice. As a student, I found the real-life examples in math and physics bogus, oversimplified for the sake of solvability. I was always wondering after seeing such examples, what do you actually do in real life? $\endgroup$
    – user1815
    Dec 23, 2021 at 23:36
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    $\begingroup$ If you mean limits as the independent variable, the argument or input to the function, approaches infinity, then I would call those "limits at infinity" rather than "limits to infinity". $\endgroup$ Dec 25, 2021 at 20:21
  • $\begingroup$ @Raciquel : The fact that they's over-simplified is not the main problem; the principal problem is that they are not respectful of reality. Contempt for non-mathematical applications is among the two or three worst mistakes ever made by mathematicians. $\endgroup$ Dec 25, 2021 at 20:23
  • $\begingroup$ Could you look back at what and did for you and your class-mates, when you were studying? I'm assuming no facts and few theories have changed since then… $\endgroup$ Dec 26, 2021 at 3:00

8 Answers 8

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If you are willing to take some time to explain the model and do some simulation, I really like to show students a logistic growth model (in discrete time). The basic setup is something like the following:

Assume that there is a population of rabbits living in some area. We are going to make a few assumptions about this population:

  1. the rabbits breed once every month (say, at the end of the month);
  2. the number of baby rabbits born at the beginning of each month is proportional to the total number of rabbits (essentially, we can assume that every rabbit has some number of babies during the month); and
  3. there are limited resources in this area, which means that if there are too many rabbits, rabbits will starve at a faster rate, and fewer rabbits will survive to the beginning of the next month in order to breed.

After some work, we can get a discrete time model, which looks something like the following:

$$ P(t_{n+1}) = P(t_n)\left( 1 + r\left( 1- \frac{P(t_n)}{K}\right) \right),$$

where $P(t_n)$ is the rabbit population at the $n$-th timestep (i.e. at the beginning of month $n$), $r$ is some measure of intrinsic growth (essentially, how many baby rabbits every rabbit would have if the environment did not constrain the growth of the rabbit population), and $K$ is the carrying capacity of the environment (i.e. the maximum sustainable population size).

This model can be simulated pretty quickly with a spreadsheet. I typically don't introduce this example until the second semester of calculus, when we talk (briefly) about differential equations. But I do think that this would be doable in a first lecture on limits, and I think it pings some intuition (sequential limits are more intuitive than continuous limits, I think).

As an added bonus, you can tweak the parameters to see different kinds of behaviour in one model. For example, take $K=1000$ (I'll keep that constant), start with $10$ rabbits, and suppose that the intrinsic growth rate is $r=0.1$. Then the population increases monotonically to the carrying capacity:

enter image description here

On the other hand, if $r = 1.96$, we can start with an initial population of $900$ or so, and the population will ping-pong back and forth across the carrying capacity, but eventually die down to a limit:

enter image description here

In both of these cases, we can see a sequence which has a limit, but the limit is approached differently. Once sequence is monotonic, while the other oscillates, but damps down. Make $r$ a bit bigger (say, $r=2.2$), and you get something really interesting:

enter image description here

In this case, the sequence does not have a limit, but it does oscillate between two different values (more or less). So the limit does not exist, but it fails to exist in an interesting way. And then you can see something really crazy by increasing the intrinsic growth rate to something like $r = 2.8$:

enter image description here

In this case, the behaviour of the system is chaotic. Again, the limit fails to exist, but we get to see something interesting happen.

And, if you really want, you can also just give them the function which models continuous logistic growth:

$$ P(t) = \frac{K P(0)}{P(0) + (K-P(0))\mathrm{e}^{-rt}}. $$

This is a limit which can be computed by hand, but one can also graph the behaviour of the function and see how it behaves, e.g. using GeoGebra.enter image description here

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    $\begingroup$ Isn't logistic growth curve still too close to the pandemic topic? Lol. $\endgroup$
    – user21820
    Dec 24, 2021 at 13:06
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    $\begingroup$ As a bonus, you can bring back the rabbits when you teach the Fibonacci Series. $\endgroup$
    – Barmar
    Dec 24, 2021 at 14:50
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Deeba and Rushkady Go to Town: A Fanciful Real-Life Story

A festival was just starting in town, and Deeba and Rushkady walked toward the square headed to the Infinite Pancake Eating Contest. The contest would end only when all participants agreed it was over. Deeba said, “Rushkady, since you always eat pancakes twice as a fast as me, no matter how fast or slow I eat them, I think you should give me a head start.“

“How much of a head start do you think would make a fair contest?” asked Rushkady.

“One hundred.”

“Not enough. I would overtake you.”

“One thousand.”

“Still not enough.”

“No way!” exclaimed Deeba.

“Look, if I eat twice as fast as you, then eventually I will eat twice as much as you, whatever head start I give you” replied Rushkady.

“Not if I had a head start of a million!”

“Yes, I would, or nearly so, if the contest went on long enough.”

Who was right?

[It also gives an intuitive explanation of one of L’Hôpital’s rules.]

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    $\begingroup$ Isn't this just a question about straight lines? "What is the intersection point of y = 2x and y = x + 1,000,000?" What does it have to do with limits at all? $\endgroup$
    – nick012000
    Dec 26, 2021 at 11:32
  • $\begingroup$ @nick012000 Let’s say Deeba eats half as much pancakes in an hour than in the previous hour. Rushkady has a similar rate but eats twice as much in the first hour. Then each eats a bounded number of pancakes. Here NickC’s question gains some force (culturally — being anchored in a cultural tradition is often an issue in real-life problems — a head start has to be a number of pancakes). It doesn’t matter when each starts if each eats a certain number of pancakes, there being no time limit. OTOH adding a number to D.’s score as a “head start” could make a difference in these circumstances. $\endgroup$
    – user1815
    Dec 26, 2021 at 15:52
  • $\begingroup$ Part of a real-life problem ought to include defining the problem (or at least it could). That is what happens when you make a model: decide what are important features; what is negligible; determine what parameters are unknown, whether they can be determined, and how to proceed if not. This is often done in the case study method. $\endgroup$
    – user1815
    Dec 26, 2021 at 15:58
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Maybe these are too easy but...

If you drop a bouncy ball, eventually you will stop hearing it bounce!

Well, we can model the ball by accounting for the predominant mode of energy loss via collision with the floor but ignoring air resistance and assuming constant gravitational acceleration. Then on each bounce with impact speed $v$, its rebound speed is $v·r$ for some constant $r∈(0,1)$, which is also called the coefficient of restitution, where each bounce cuts the ball's kinetic energy down to $r^2$ times the previous amount.

Then by energy conservation the ball's height on each bounce is $r^2$ times the height of the previous bounce. Hence after $k$ bounces the height the ball reaches would be $r^{2k}$ times the initial dropping height. Taking $k→∞$ we have $r^{2k}→0$, and so for any $ε > 0$ the ball height eventually no longer exceeds $ε$. Since the sound comes from the ball's energy, past that point the bounces will have too little energy for you to hear it.

Given any bouncy ball, there is some constant $c$ such that no matter how high you drop it, if $x$ seconds elapse between the first two bounces, then it stops bouncing within $c·x$ seconds!

We use the same assumptions as before. Then the time taken between consecutive bounces forms a geometric progression with ratio $r$. Thus the total time taken by all bounces after the first bounce is $x·\sum_{k=0}^∞ r^k$. How is that for a real-life limit to infinity?

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I have actually just finished teaching a first semester of calculus, and when I started it I was concerned about examples to start with. And as far as I can see, first two semesters of differential and integral calculus are about approximations: finite difference is an approximation of derivative (and vice versa!), Riemann sum is an approximation of Riemann integral (and vice versa), member of sequence is an approximation of the limit (and vice versa), etc. Problem is that (in Russia) school students are not trained in approximate calculations at all. They do have a hands-on experience with calculators or numerical software, so they have an IDEA of approximate calculations, but they are not formally trained.

So I started with a question: "What is $\sqrt{2}$?" Of course, somebody inevitably answered that it is a length of diagonal of the square, and it's not a rational fraction. Then we considered a number of ways to approximate $\sqrt{2}$ with rationals:

  1. Initial bounds: $1< \sqrt{2} < 2$.
  2. Finding the next decimal digit by trial and error (much work, boring).
  3. Bisection (a bit less work, also boring). Connection to binary system.
  4. Continued fractions.
  5. I didn't do that, but Taylor expansion might be a good idea. Or not. It's tricky to get a sequence that can be proven to converge to $\sqrt{2}$.

Of course, little to no proofs are given at this time. The point is to acquaint students with ideas of:

  • quality of approximation (a good approximation lies near the approximated value);
  • convergence (broadly speaking, approximations get better over time);
  • speed of convergence (some types of approximations get better much faster or much slower than the others).

And then you can drop the definition of the limit (I used $\varepsilon$-$N$ variant). The point is that a converging sequence might lose some quality of approximation locally, but over the time such losses (dropping below the certain quality threshold $\varepsilon$) must become impossible. In other words, you can lose some quality of approximation, but if the sequence converges, the quality of approximation will get as good as requested (and stay that good after that).

This is actually what I did, and I think the students got a better grasp of "what in god's name is happening?" than usual.

Hope this helps :)

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  • $\begingroup$ Honestly, I would argue that the fundamental idea behind all of analysis is approximation. Secant lines are almost tangent lines; irregular areas are almost rectangular areas; every $L^p$ functions is almost continuous; and so on. Analysis is (in some meaningful sense) the mathematical art of approximation. $\endgroup$
    – Xander Henderson
    Dec 25, 2021 at 22:43
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Answer:

I feel like a lot of these suggestions are quite complicated. Not complicated for us, but complicated for new-to-the-topic students. If you must (but see below) go with a life example, I think doing something like Zeno's paradox of motion might be helpful. first half of the trip, half of the second half of the trip, etc. (1/2^x)

Too-long-for-comment comment:

I strongly disagree pedagogically with the impulse behind "I would like to start my course with a beautiful example, not simply a basic function like 1/x."

Every new crop of grain does not require a new sickle design. For the students, these topics, themselves, are new and wonderful revelations, even with simple explanations. And the theory of gradualism implies starting with the simplest examples first, not the hardest.

Furthermore, introducing topics as derivations of word problems ADDS COGNITIVE LOAD to what is already a new and moderately difficult task (for the student). "Word problems are hard." Yes, verily, they are.

One of my pet peeves on PDE texts is they want to introduce topics as derivations of engineering problems in statics or heat transfer. ODE texts are better as they have a semi-standard method of first teaching, training, practicing the algebra-manipulation of the techniques, with significant homework. Then, in next chapter, after success with the symbol manipulation, having the students do various word problems (from physics, econ, bio, etc.). These then give the students practice in translating to/from the real world and symbol world, as well as giving extra practice in the given techniques. But the cognitive load is easier as the students have already practiced the symbol manipulations, but are adding in some word problem complication on top. But this is different than having a NEW (to the student) symbol manipulation taught along with all the distractions of an engineering problem (or Covid, or econ or whatever). And I say all this as someone much more applied in orientation and career than the average college/HS math teacher. Not opposed to applied topics, just opposed to a poor pedagogical approach.

In general, there's a huge theme of questions here showing a desire of the teacher (often much smarter and more math experienced than his students, and usually with a background of grad school research) wanting to do something that INTERESTS THE TEACHER as opposed to what is BEST PEDOGICALLY for his students.

For the student, doing his first limits to infinity, a simple function is a feature, not a bug. And while it may be boring to the teacher to use a classic example, used in many different texts, the student is someone seeing the concept itself for the very first time. It's novel to him.

Perhaps there has even been some evolutionary process by which the texts converged on a particular approach, example, intro. "Tried and true." This is not to say one should never innovate or that it is impossible to improve or experiment. Or that practices in math education have never converged on wrong approaches. But one should be a bit cynically skeptical of new teachers pushing new methods, to include self skepticism.

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The stability of the solar system is a classic real-life problem about time going to $+\infty$. It’s not an easy problem in which to understand the mathematics, not even close to the introductory level of $1/x$. Its pedagogical value lies in the problem statement being easy to understand and to relate to, and it is a problem to which people have devoted a lot of energy over the centuries. In other words, it’s more inspiring than explanatory.

Scott Tremaine traces the stability problem back to Isaac Newton in Is the Solar System Stable?, which calculus students should be able to read, and forward through many famous mathematicians, Lagrange, Laplace,…,Poincaré,…. I think one could argue that the problem goes back further, to Aristotle and probably earlier. Once you realize that there’s a universe and a solar system that changes, you are bound to wonder how long you or the human race will be safe. However, one could also argue that it became a problem in mathematical analysis and conceivable in terms of limits at infinity only after Newton developed a mathematical theory of gravitation.

According to Tremaine, the orbits will remain stable for as long as the sun is expected to live, about 8 billion years. That is, in our current understanding, we do not expect time to be able to approach $+\infty$ for those of us in the solar system, not as a real-life problem. Such limitations of reality afford the teacher an opportunity to discuss why people (and not just calculus students) model phenomena with $t \rightarrow +\infty$, when in practice $t$ just being large enough is sufficient.

A more detailed treatment of the problem may be found in Jacques Laskar, Is the Solar System Stable?

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  • $\begingroup$ Laskar's abstract takes a different view: "The recent simulations even show that planetary collisions or ejections are possible on a period of less than 5 billion years, before the end of the life of the Sun." $\endgroup$
    – J.G.
    Dec 24, 2021 at 22:04
  • $\begingroup$ @J.G. I noticed that but II guess I didn't think being exact was that important. It was one reason to include the ink. I rather wonder how up-to-date Laskar's result is. (I'm not an expert, if you haven't noticed.) Do you know more? $\endgroup$
    – user1815
    Dec 24, 2021 at 22:20
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A nice easy limit that can be seen in everyday life.

Suppose your student looks along a perfectly level road, with perfect visibility. They look down at an angle that allows them to see the road 10m ahead of them. Then 20m ahead. Then 30m ahead. And so on, for any amount of distance ahead.

You can easily show, draw, find an equation, even tabulate, and intuitively perceive, that the angle they look at tends ever closer to the horizontal (0 or 90, however you choose to define it), but never quiiiiiiite reaches it.

You can also show how limits work in mathematics - for any angle (0-or-90) + ε close to the horizontal you can name, however tiny, the value will get closer to 0 or 90 than that, if you look far enough ahead. (And you can demonstrate/calculate when it will, for 1/100, 1/1000000 and 10-100 of a degree if needed, in this case, to make it concrete,and show it never reaches 0-or-90 even so: infinity isn't a "number".) Classic "≤ ε" stuff, ties nicely into how and why we define a limit that way.

Its one of the easiest examples I can think of, for introducing the concepts or a limit in an everyday context.

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  • $\begingroup$ If the road is on the surface of a perfectly round planet, then there is a maximum angle (of slightly less than 90°) at which the road surface can still be seen and beyond which one will only see the sky. On a not so perfectly round planet, there are lots and lots of roads where the surface is visible when looking exactly straight ahead. $\endgroup$
    – Jan
    Dec 25, 2021 at 1:53
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    $\begingroup$ This is mostly a worthless point to make here. The context is clearly "real world" educational example ideas, suited for a classroom, and that point is true for almost all teaching examples drawing on the real world for material. If a ball rolls down a real world slope, as in many classroom examples, its never perfectly smooth either. Nor is any real world example in education perfectly ideal. The OP asked for examples from a wide range of "real world" contexts, including physics/geography. I think its clear from context that they understand the real world is not an idealised one $\endgroup$
    – Stilez
    Dec 25, 2021 at 2:26
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    $\begingroup$ Well, at least it would be good to explicitely state whether your simplified real-life road is on a simple limitless plane or on a simple sphere or on something else. To me, this is not at all clear from context, the only hint is that your reasoning does not work on a sphere. But for your students that might not be obvious and they might get some wrong impression. E.g. I think the angle you are looking for in your last-but one paragraph is about one arc minute when standing on the surface on a perfect sphere with the same radius as earth. Not really that small of an angle. $\endgroup$
    – Jan
    Dec 25, 2021 at 11:49
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    $\begingroup$ I figure an educator ("I have to teach...") would be more than capable of such detailing, and take them for granted, and was simply asking for what was stated in the OP - accessible examples as inspiration for teaching material. Its not like a textbook where you have to spell it all out because you don't know the reader's situation and competence. The OP is clearly beyond the point they need round vs flat ideal world spelt out, to make the example helpful. $\endgroup$
    – Stilez
    Dec 25, 2021 at 14:38
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As an amateur photographer, my first thought was the lens position for focus on an object. It should be easy to intuitively explain that the closer the object the further the image because the lens only bends the rays so much. As the object gets very far away, the location of the image moves toward a limit, which is the focal length of the lens. Like the 1/x example, it can show that you don't have to be "close to infinity" to be near the limit.

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