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This morning, I read Wikipedia’s informal definition of a limit:

Informally, a function f assigns an output $f(x)$ to every input $x$. The

function has a limit $L$ at an input $p$ if $f(x)$ is “close” to $L$ whenever

$x$ is “close” to $p$. In other words, $f(x)$ becomes closer and closer to $L$

as $x$ moves closer and closer to $p$.

To me that sounds like something that might be better described as a ‘target’.

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If I take a simple function, say one that only multiplies the input by $2$; and if my limit is $10$ at an input $5$: then I’ve described something that seems to match the elements contained in Wikipedia’s definition. I don’t believe that that’s right. To me it looks like an elementary-algebra problem ($2p = 10$). To make it more calculusy, I could graph the function’s output when I use inputs other than $p$, but that really wouldn’t give me anything but an illustration of the fact that one’s answer moves farther from the right answer as it becomes more wrong (go figure).

So limits are important; what I’ve just described is trivial. I do not understand them. I know calculus is often used for solving real-world challenges, and that limits are an important element of calculus, so I assume there must be some simple real-world examples of what it is that limits describe.

What is a simple example of a limit in the real world?

Thank you

-Hal.

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Your example of a limit is of a limit which is easy to evaluate, but it’s still a perfectly reasonable example!

Here’s another fairly easy to grasp example of a limit which avoids triviality.

If I keep tossing a coin as long as it takes, how likely am I to never toss a head?

Rephrased as a limit problem, we might say

If I toss a coin $N$ times, what is the probability $p(N)$ that I have not yet tossed a head? Now what is the limit as $N\to\infty$ of $p(N)$?

The mathematical answer to this is $p(N)=\left(\frac{1}{2}\right)^N$. Then $$\lim_{N\to\infty}p(N) = 0$$

because $p=\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots$ gets closer and closer to zero as $N$ gets “closer to $\infty$”.

The reading of your speedometer (e.g., 85 km/h) is a limit in the real world. Maybe you think speed is speed, why not 85 km/h. But in fact your speed is changing continuously during time, and the only “solid”, i.e., “limitless” data you have is that it took you exactly 2 hours to drive the 150 km from A to B. The figure your speedometer gives you is at each instant $t_0$ of your travel the limit

$$v(t_0):=\lim_{\Delta t\to0}{s(t_0)-s(t_0-\Delta t)\over\Delta t}\ ,$$

where $s(t)$ denotes the distance travelled up to time $t$.

It is hard for me to stray from the confines of mathematics to the ‘real world’, so let me give you this “example”:

Limits are super-important in that they serve as the basis for the definitions of the ‘derivative’ and ‘integral’, the two fundamental structures in Calculus! In that context, limits help us understand what it means to “get arbitrarily close to a point”, or “go to infinity”. Those ideas are not trivial, and it is hard to place them in a rigorous context without the notion of the limit. So more generally, the limit helps us move from the study of discrete quantity to continuous quantity, and that is of prime importance in Calculus, and applications of Calculus.

To apply this notion to physics (yes, I’m moving away from math now), it is possible to apply a *continuous* analysis to motion. We’d like to be able to measure *instantaneous* speed, which requires the notion of an *instantaneous* value. Now this is dependent on the concept of the limit. That is to say, we want to measure a quantity in an *instant*, and we define this “instant” by a limit, i.e., as an approach towards some infinitesimal time. This is how we would answer, e.g., the commonplace question “how fast was he going at time $x$?”.

A good example is continuous compounding of interest. Suppose that the money in your bank account has an annual interest rate of $r$ and it is compounded $n$ times per year. If you initially had $M_0$ dollars in your account then after $t$ years your money has grown to

$$

M_0\left(1+\frac{r}{n} \right)^{nt}.

$$

In continuous compounding your money is compounded every infinitesimal time step. This is a little non-rigorous but you can think about it as taking the number of times per year your account is compounded to infinity:

$$

\lim_{N\to\infty} M_0\left(1+\frac{r}{n} \right)^{nt} = M_0e^{rt}

$$

the well known formula for continuous compounding.

**A bit of History:** Already in 500 b.c. there were some discussion about the possible existence of an “infinitesimal”, i.e. is there a smallest particule of time, matter, etc. Around -450 Zeno proposed some paradoxes for both assumptions. Eudoxe proposed little later that in mathematics at least, it should be allowed to consider infinitesimals.

Wikipedia: The Eudoxian definition of proportionality uses the quantifier, “for every …” to harness the infinite and the infinitesimal, just as do the modern epsilon-delta definitions of limit and continuity.

**An example:** I propose you here to have a little look at one of the Zeno’s paradox and show (using limits) that it is in fact not a paradox. There was a guy called Achilles and a Tortoise. The turtle is at a distance $d$ from Achilles and runs with speed $v_b$ in one direction. Achilles wants to catch the turtle and runs at a speed $v_A > v_b$ in the same direction. In order to catch the turtle Achilles must first run through a distance $d_1=d$ in $t_1 = d/v_A$, meanwhile the turtle runs $d_2 = dv_b/v_A$. So Achilles needs to run the distance $d_2$ in $t_2 = d_2/v_A = dv_b/v_A^2$ before catching the turtle. But, the turtle can run $d_3 = dv_b^2/v_A^2$ during this time. Etc. At the end, Achilles never reaches the Tortoise. However thanks to limits we can show that Achilles has to run only a finite distance before he catches effectively the turtle.

At each step Achilles has to run $d_n = d(v_b/v_A)^{n-1}$ hence the total distance between the starting point of Achilles and the turtle is given by

$$D_n = \sum_{k=0}^{n-1}d\left(\frac{v_b}{v_A}\right)^{k} = d\frac{1-\left(\frac{v_b}{v_A}\right)^{n}}{1-\frac{v_b}{v_A}} \overset{n \to \infty}{\longrightarrow} d\frac{v_A}{v_A-v_b}< \infty $$

And so Achilles has to run only a finite amount of distance before catching the Turtle.

To move in a straight line from A to B, you will have to

reach the 1/2 point C between A and B. To get from C to

B, you will have to reach the midpoint of line CB.

As you continue moving 1/2 the remaining distance

you will always have a little part left between you and

point B. B is called the limit. You will get infinitely

close to it, but never really arrive at point B.

When it comes to the real world, I find that limits inform us that we need to adjust our “**rise**” and “**run**“(slope) to stay within a boundary as we approach it. And for further practical purposes, limits help us put a **finite value** (the asymptote) on a seemingly **infinite journey of precision**.

For example, as you drive your car up to a stop sign. You begin to press the brake and your acceleration decreases over time, and you notice this happening because you can see your speedometer going down. As you get closer to the stop sign, you work to adjust the rate at which your speed is falling to ensure you will stop at the right spot. You begin to notice that the changes in speed become a lot less dramatic (but the length of time it takes you to advance your position starts getting a lot longer). You are slowing down to a point where it is becoming very difficult to see if your speed is still going down.

In theory, you could keep approaching the stop sign infinitely. For each unit of time, you could be half the distance closer then you were before. However, at some point you say to yourself “this is good enough, I consider myself to have arrived at the stop sign”. **You don’t want to get a ticket for running through a stop sign, because blaming it on an experiment in Calculus probably won’t help you.** You press the brake to the floor for a full stop. The limit zero, pragmatically speaking, is zero for you.

Here is an example:

Alex joins a $100$-mile sprint competition, we denote time as $t$, distance as $F$,

we can construct $F(t)=t\cdot V$ (assuming Alex’s speed is constants like $10\ m / s$.)

so what is limit of $F$ as $t$ is approaching $20$, easily we can see $F(20)=200m$,

this is a process of limit.

how to describe this:

when t get close to $20$, F is close to $200$.

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