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This is more specific than this similar question on math.SE, since I'm not satisfied with the answers there.

Question: Can you provide an interesting, natural and simple example of some physical/geometrical etc. system, which has two continuous quantities $y,x$ with $y=f(x)$, such that $y$ is not defined at a finite point $x=a$, but we would still find it interesting to ask for $\lim_{x\to a}y$? Ideally the example should be such that $f(x)$ is an elementary function of $x$ whose expression can be derived from the description of the system and $\lim_{x\to a}f(x)$ is finite.

My target audience are beginning engineering students, so "interesting, natural and simple" for them.

Background: I teach an introductory calculus course to engineers. The main focus is not limits, but we have a short section on limits of functions (not sequences) before defining derivatives. We don't do $\epsilon-\delta$ but want students to be aquatinted with the notation and idea of $\lim_{x\to a} f(x)$.

Most books & resources I've found do a good job of illustrating the notion graphically, numerically and of course with formula examples like $\lim_{x\to 1}\frac{\sqrt{x}-1}{x-1}$, where one can do some clever rearranging to see what the limit is. But I haven't seen any motivating example that shows how we might arrive at an expression like $\lim_{x\to 1}\frac{\sqrt{x}-1}{x-1}$ from a real life application.

Of course I could (and currently do) tell them: we'll later see examples where such things appear, like when dealing with derivatives, integrals etc., but I'd like to give them some applied example right when the notion is introduced, without having to look ahead at derivatives. So please no examples that say "derivatives".

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    $\begingroup$ The example is the tangent line. I delay the limits section because it's so technical. $\endgroup$
    – Sue VanHattum
    Oct 10, 2023 at 15:19
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    $\begingroup$ To complement @Sue's suggestion: How to define instantaneous velocity and other rates, surely something of concern to engineers. The basic gist is that the calculus of rates and accumulation is well-founded. Whether or not an engineer ever has to calculate a limit after they finish with school, I don't know. I wouldn't be surprised if the answer is, "No." $\endgroup$
    – user1815
    Oct 10, 2023 at 19:51
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    $\begingroup$ It's worth noting that the concept of limit is best understood in terms of real-world imprecision, a foundational engineering concept. $\lim_{x \to c}f(x) = L$ means that if one desires $f(x)$ to be within any $\epsilon$ of L, one can find a precision with which to specify that $x$ is close to $c$ (ignoring the probability $0$ case where $x=c$) to achieve this. $\endgroup$
    – TomKern
    Oct 11, 2023 at 14:16

6 Answers 6

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First thing that comes to my mind is the limit $$\lim\limits_{x \to 0} \dfrac{\sin x}{x} = 1.$$

This limit justifies the small-angle approximation $\sin \theta \approx \theta$ (for $\theta \approx 0$) that is widely leveraged in engineering/physics classes to tame unwieldy equations.

(Often, the existence of a sine term prevents the existence of a closed-form solution, but if you replace that sine term with a linear term, then you can get a closed-form first-order approximation of the solution that is sufficiently accurate for practical purposes.)

A concrete and accessible example would be calculating the period of a pendulum: http://www.physicslab.org/Document.aspx?doctype=3&filename=OscillatoryMotion_PendulumSHM.xml

Edit: Note that if you want to avoid the need for derivatives and differential equations, then you can ask the question "can you model a pendulum like a spring" instead of "what is the period of a pendulum".

Can you model a pendulum like a spring?

Springs follow Hooke's law: the restoring force is proportional to the displacement. But in a pendulum, the restoring force is $mg \sin \theta.$ So, you can't model it like a spring. Right?

... actually, for small angles, $\sin \theta \approx \theta$ and $\textrm{displacement} \approx L\theta,$ so the restoring force is $\approx mg/L \cdot \textrm{displacement}.$ So, you CAN model it like a spring when the angle is small.

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  • $\begingroup$ Thanks. No doubt an interesting and important application, but it's not as direct and simple as what I'm looking for. As I see it, one is not per se interested in evaluating a limit here, but instead wants to simplify calculations. The "starting point" is the small angle approximation. So I would first need to explain why the small angle approximation amounts to evaluating that limit (important, but not trivial for students) and second I would have to motivate why we want to use that small angle approximation. (cont.) $\endgroup$ Oct 12, 2023 at 6:43
  • $\begingroup$ Again important, but the period of the pendulum is probably to high level, since I think I need derivatives and differential equations to do it properly (correct me if I'm wrong). Remember that I want to do this before derivatives. $\endgroup$ Oct 12, 2023 at 6:43
  • $\begingroup$ This was the first example I thought of also. This example comes up in spherical coordinates all of the time (replace x with r) $\endgroup$ Oct 12, 2023 at 12:37
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    $\begingroup$ @MichaelBächtold I get what you're saying about ideally wanting a more direct/simple application. That said, you could frame it in a way that doesn't require derivatives and differential equations: "Can you model a pendulum like a spring? Springs follow Hooke's law: the restoring force is proportional to the displacement. But in a pendulum, the restoring force is $mg \sin \theta.$ So, you can't. Right? ... actually, for small angles, $\sin \theta \approx \theta$ and $\textrm{displacement} \approx L\theta,$ so the restoring force is $\approx mg/L \cdot \textrm{displacement}.$ So, you CAN. $\endgroup$ Oct 12, 2023 at 15:30
  • $\begingroup$ But I wouldn't know how to derive the formula for the period of a spring without showing that a harmonic solves the second order ODE. $\endgroup$ Oct 13, 2023 at 9:30
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"interesting, natural and simple"

Illustrating something dynamically might make things interesting. For example, a geometry problem involving a limit (from an old calculus book):

Consider a circle of variable radius $r$, centered at the origin $(0,0)$, and also the unit circle centered at $(1,0)$. If the circles intersect in the first quadrant at point $D$, then draw the line through $D$ and the positive vertical intercept, $B$, of the first circle. This line will intersect the positive $x$-axis at a point $E$.

Question: If $r \to 0$, does the $x$-coordinate of point $E$ have a limiting value?

Students can guess the answer, or calculate it using a limit. Then show them an animation:

Animation of intersecting circles and line as described above. As
the variable radius approaches zero, the intersection point E
approaches the x value of 4.

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  • $\begingroup$ That's a cute example. Thanks $\endgroup$ Oct 11, 2023 at 7:15
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    $\begingroup$ And of the type I was imagining $\endgroup$ Oct 11, 2023 at 7:41
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    $\begingroup$ @MartinArgerami File->Export->Graphics View As Animated GIF. If you do this, you'll want to consider how many different images it will take, as the files can get large quickly. [MESE only allows up to 2Mb.] $\endgroup$
    – Nick C
    Oct 11, 2023 at 13:29
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    $\begingroup$ The asker is happy with this answer, so I’m not inclined against it at all. I’m curious about how this relates to "natural", as in a real world situation. $\endgroup$ Oct 12, 2023 at 4:08
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    $\begingroup$ @ToddWilcox On "It is just an invention and not something we may see acted out in nature." If you remove the limits from the circular rail described before, just thing of a classic combustion engine ;-) $\endgroup$
    – U. Windl
    Oct 16, 2023 at 11:41
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Here are some examples from a 2015 paper by Steven Jones on students' reasoning about limits.

If you pluck a stringed instrument (like a guitar or violin), the pitch of the note ($f$) depends on the length of the string ($L$), the tension of the string ($T$), and the thickness of the string ($\rho$). (A picture of a guitar is provided to the students.) Here is the formula that relates pitch, length, tension, and thickness: $f = \frac{1}{2L} \sqrt{\frac{T}{\rho}}$.

Determine the following limits. What does each one tell us?
$\lim_{L \to \infty} f$, $\lim_{L \to 0^+} f$, $\lim_{T \to \infty} f$, $\lim_{T \to 0^+} f$, $\lim_{\rho \to \infty} f$, $\lim_{\rho \to 0^+} f$

In physics, the theory of relativity says an object's mass, $m$, is related to the object's velocity, $v$, especially as speeds close to the speed of light, $c$. The following equation relates these quantities. ($m_0$ is the 'resting mass' of the object.)
$m = \frac{m_0}{1-v^2/c^2}$
Work out $\lim_{v \to c^-} m$. How did you work out your answer? What does this limit tell you about mass?

Unfortunately, with the exception of $\lim_{T \to 0^+} f$, all of these are either infinite limits or limits at infinity. I think the main reason that it's difficult to find interesting, physical, non-rate-related examples of finite limits at finite points is that physical quantities are, in general, continuous. This means that, unless we are approaching a limiting case, there's nothing stopping us from substituting values into a formula.

Here's another example that I'm adapting from Serway/Jewett's Physics for Scientists and Engineers. It's rate-related, but the derivative is hidden in the formula already.

An object at a distance $p$ from a spherical mirror is moving towards the mirror at speed $v_p$. The focal length of the mirror is $f$. The velocity of the mirror image of the object is given by $v_q = -\frac{f^2 v_p}{(p-f)^2}$. What is $\lim_{p \to 0^+} v_q$?

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Polynomials are everywhere, and in engineering or physics, it is extremely common to use linear approximations to them

$$ \begin{align} \lim_{x \rightarrow 0} \frac{(x + 1)^d - 1}{x} = d \end{align} $$

Granted, this is just the derivative, but it need not be presented as such and may in fact motivate derivatives later. Armed with this identity, high school knowledge is enough to

  1. Approximate square roots, e.g. $\sqrt{1.002} \approx 1.001$

  2. Compute how far the horizon is. For example, suppose you're on board a ship $10$m above the water, how far can you see? What if you're in a crow's nest that's also $10$m high? $$ \begin{align} \text{horizon distance} &= \sqrt{(\text{radius of earth + height})^2 - \text{radius of earth}^2} \\ &\approx \sqrt{2 * \text{radius of earth} * \text{height}} \end{align} $$ Even if you could just type everything into a calculator, this approximation shows that the height of the crow's nest yields diminishing returns.

  3. Show that tidal forces scales inversely with distance $r$ cubed and proportional to displacement $d$ from the center $$ \begin{align} \frac{Gm}{r^2} - \frac{Gm}{(r \pm d)^2} &= \frac{Gm}{r^2}(1 - (1 \pm \frac{d}{r})^{-2}) \\ &\approx \mp\frac{2Gmd}{r^3} \end{align} $$ Thus you can show that a) the sun's tidal force is smaller than the moon's on earth and b) tides are only obvious in large bodies of water.

  4. Derive all sorts of simple harmonic oscillators.

The list goes on.

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  • $\begingroup$ Thanks, these are nice examples, but similar to my comment to Justin Skycak's answers, not as direct as what I'm looking for. The main point here is the approximation $(1+x)^d\approx 1+dx$ when $x\approx 0$, which is important, but for students not obviously equivalent to $\lim_{x\to 0} \frac{(1+x)^d-1}{x}=d$. Establishing this equivalence plus showing the use of the approximation would take too much time and maybe distract from the main focus of the class, wich is introducing the notion of limits. $\endgroup$ Oct 12, 2023 at 7:24
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Suppose you have an event with tiny but positive probability $p$. As an example, you have $100$ ping-pong balls, all white except for $1$ golden ball, and you choose among them with equal probability, so the probability of choosing the golden ball is $p = \frac{1}{100}$.

Now suppose that you repeat the event $1/p$ times rounded down to the nearest integer, i.e. a number of times equal to $\lfloor \frac{1}{p}\rfloor$. In the ping-pong ball example, you replace the chosen ball, mix up the 100 balls, and choose again randomly, repeating this a total of $100$ times.

What's the probability of observing the event at least $1$ time?

The answer is $$1 - \bigl(1 - p\bigr)^{\lfloor 1/p \rfloor} $$ Of course, this makes no sense when $p=0$.

But as $p$ gets closer and closer to zero, this probability converges to $$\lim_{p \to 0} \left( 1 - \bigl(1 - p\bigr)^{\lfloor 1/p \rfloor}\right) = 1 - e^{-1} \approx 0.63212055882 $$

Comment: My first version of this had an integer variable and the expression $\lim_{n \to \infty} (1 - (1-1/n)^n) = 1-e^{-1}$, and then I read more closely to see that you asked for a continuous variable. So I had to fudge the example, which leaves a trace of its discrete origin in the expression $\lfloor 1/p \rfloor$.

Still, though, engineers might like this, they have to deal with probabilities. What's particularly nice is that it challenges some peoples' faulty intuition of probability, which goes something like this: "If you do $n$ repetitions of an event of probability $1/n$, you should definitely see a success"... Well, no, but you do have about a $63\%$ chance of success.

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I'll share an idea, but it requires geometric series, a topic typically covered at the end of a Calculus II course. Still, you might consider using this as a way to preview some future work in calculus for the students, while also providing another motivating context for limits, i.e. the sum of a series. (What I describe below is given to students as a problem set in my Calculus II course.)

Consider a geometric series with first term 2 and multiplicative ratio $r$, so we have $2+2r+2r^2+2r^3+\cdots$ What possible sum value $S$ could this series achieve? After deriving the formula for the sum, namely $S=\frac{2}{1-r}$, you can ask some leading questions about what $S$ could possibly be. I suggest working with the students to consider at least the cases where $S=10$ ($r=4/5$), $S=4$ ($r=1/2$), $S=2$ ($r=0$), $S=1$ ($r=-1$), and $S=0$ (impossible).

Then, look back at your derivation of the sum formula and remind students of the domain: $-1<r<1$. Graph the function $S(r)=\frac{2}{1-r}$ on the domain $-1<r<1$. Note that the point $(-1,1)$ is not on the graph, because of the domain issue (when $r=-1$, the geometric series does not converge). However, it's visually clear that the limit exists: $\lim_{r\to (-1)^+} S(r) = 1$. All of this can help students grapple with the idea that this infinite series can converge to any sum $S$ that satisfies $S>1$, but not $S=1$ itself. I hope this will also provide them with a fairly natural example of where a function clearly has a limiting value at a finite input, even though that point isn't really "achievable," in some sense.

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