Nice examples of limits to infinity in real life

Question

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.

Answer 1

"John Napier" and how he get to "e=2.7182818284..." might be a good real life limit. (https://en.wikipedia.org/wiki/E_(mathematical_constant) )

Answer 2

COMMENT.-You can consult about examples of mathematical modeling (there are many!) and then give them the appropriate tone for your purposes. For example, suppose that a severe drought has eliminated all mammals of a certain species in a region and that a board of ecologists wants to repopulate the area in question with these mammals for which they introduce $20$ of the considered animals (males and females of course) in the affected region. Suppose that the following model has been established for the corresponding reproduction giving the number $N$ of animals as a function of time $t$ $$N(t)=\frac{20+7t}{1+0.02t}$$ According to the level of your students, several questions could be asked prior to the calculation when $t$ tends to infinity, in particular to see what value $N$ has when $t = 0$ or noting that many numerical solutions are not integer in which case the fractional part must be eliminated. But the fundamental clarification has to be the fact that a mathematical model is usually built for ideal conditions (no severe floods or wars, no fires or anything like that).

Answer 3

The problem with "real-life" applications of limits at infinity is that at a fundamental level, there are none. Every place you see something relating to the infinite in the real world, it is actually not the real world, but a mathematical model of the real world. And, without fail, if you examine the model closely, either the model breaks down - that is, fails to represent the real world situation - as you get close to the infinite; or else, the infinity exists in a part of the model that cannot be tested. (Is the universe infinite? It is impossible to prove it.) So any infinities you have "in the real world" are actually just mathematical infinities that do not relate to anything actually in the real world.

Instead I suggest you introduce infinite mathematical limits that the students are already used to dealing with, but may have never realized it. And the grandaddy of them all is non-terminating decimal notation. In particular, I suggest $0.999\ldots = 1$.

For any finite number of $9$ digits, $1 - 0.9\ldots 9 > 0$. But given any $x > 0$, you can add enough $9$ digits for $1 - 0.9\ldots 9 < x$ for any $x$. (Archimedean principle: There has to be an integer $n > \frac 1x$, and $10^n > n$.) So $1 - 0.999\ldots$ is $\ge 0$, but must be smaller and any $x > 0$. This leaves $0$ as the only possibility. In other words $$\lim_{n \to \infty} \frac 9{10} + \frac 9{10^2} + \ldots + \frac 9{10^n} = 1$$

Answer 4

Application: A limit can be interpreted as a prediction of the result of an experiment that can be held (hypothetically) for eternity without having to do it for eternity.

For example, if you throw a 6-side dice the probability of getting a 1 is $\frac{1}{6}.$ If you throw the dice $k$ times, the probability that the output is $k$-times 1 is $\left(\frac{1}{6}\right)^k$.

The limit $\displaystyle\lim_{k\to\infty}\dfrac{1}{6^k}=0$ can be interpreted as: the more times we throw the dice, the probability of always getting the same output will be closer to zero. However, never will be zero.

Similarly, the limit $\displaystyle\lim_{k\to\infty}\left(1-\dfrac{1}{6^k}\right)=1$ can be interpreted as: the more times we throw the dice, the probability of not getting the same output in all the throws will be closer to one. However, never will be 1.

Here the limit assures us certain predictions, a tendency, without having to throw a dice many times.

Answer 5

Explaining the failure of Zeno's Paradox could be a cool activity. Motion can be broken down into so many very small parts, namely steps of length $1/2^i$. Show the fact that $\lim_{n \to \infty} \sum_{i=1}^n 1/(2_i) =1$. The only prerequisite of this is understanding geometric series.

I would also suggest compound interest and its limiting case of continuous compounding, as a previous poster suggested.