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

Question

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'.

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.

Answer 1

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 $\infty$".

Answer 2

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 $x$?".

Answer 3

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 $t$.

Answer 4

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.

Answer 5

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.