Fun Limit Problems – Not quite infinite!

Posted on by Dinh Quan Tran

Limits. This concept, with its explicit meaning “a point or level beyond which something does not or may not extend or pass”, is probably very familiar in real life. In Mathematics, limits are also very much the same despite its formal definitions having to use quite a number of Greek letters and signs, and such a concept has been inspired by or is applied in many interesting problems and phenomena. Today I will discuss two such problems I found interesting and relevant to the concept of limits: The Achilles and the Turtle paradox and the regular polygon area calculation. But first, let’s review the formal definitions.

  1. What is a limit?

The formal definition, or the {\displaystyle (\varepsilon ,\delta )} definition of limits, goes as follows:

Let {\displaystyle f} be a real-valued function defined on a subset {\displaystyle D} of the real numbers. Let {\displaystyle c} be a limit point of {\displaystyle D} and let {\displaystyle L} be a real number. We say that:

{\displaystyle \lim _{x\to c}f(x)=L}

if for every {\displaystyle \varepsilon >0} there exists a {\displaystyle \delta >0} such that, for all {\displaystyle x\in D}, if {\displaystyle 0<|x-c|<\delta }, then {\displaystyle |f(x)-L|<\varepsilon}.

Notice that we are dealing with limits here, so the function is not necessarily defined at {\displaystyle c}, and moreover {\displaystyle c} can be positive or negative infinite. To put it in an informal way, the definition implies that the limit is {\displaystyle L} if there is always a value attainable by the function that is near {\displaystyle L} enough, i.e. its difference to {\displaystyle L} is smaller than a specified value {\displaystyle \varepsilon >0} no matter how small {\displaystyle \varepsilon} can be.

  1. Achilles and the Turtle paradox:
One of very simple but confusing paradoxes involved in this book is the paradox of Achilles and the Turtle. Achilles and the Tu… | Book projects, Paradox, This book
Running as fast as he can, will Achilles overtake the turtle?

The Achilles and the Turtle paradox, one of those created by the ancient Greek Philosopher Zeno of Elea, was a perfect demonstration of how a seemingly infinite thing can never exceed a limited value. Let’s see the paradox after I have plugged in some numbers:

Achilles starts 10 meters behind the turtle. The turtle has the speed of 1 m/s, and Achilles has the speed of 10 m/s. Achilles will reach the starting point of the turtle in 1 second, in which time the turtle has crawled 1 meter. Achilles will have run that distance in the next 0.1 seconds, but the turtle has also crawled 0.1 meters in the meantime. Therefore, as the turtle is always in front of Achilles, he will never catch up to the turtle.

Of course we know that the statement is not true. Using physics, we can deduce that Achilles will catch up to the turtle in 

{10 / (10-1) = 1.111...} (seconds)

So why does the statement sound so true? Well, if you notice, the time intervals considered by the paradox are reduced by a factor of 10 each time (1 seconds to 0.1 seconds and so on). Calculating the limit to the sum of those intervals using the formula for geometric series {\dfrac{1}{1-q}}, we get the value 10/9. Therefore, the total time that the paradox considers will never reach 10/9 seconds, which is the time needed for Achilles to catch up to the turtle. 

This is often considered one of the earliest problems concerning limits (it comes sooner than the concept of limit for probably nearly 2 thousand years). The infinity property of the paradox, therefore, meant that this is not disproven for a very long time.

  1. Regular polygon’s area:
Demonstration of the area of circumscribed polygons. Notice how the blank area gets smaller.

As can be seen in the figure above, the area of the polygon comes closer to the area of the circle the more edges it gets. In other words, the limit of the area of a regular n-gon approaches a circle as n increases to infinity. This is because the formula of the area of a regular n-gon given its circumradius r is:

{\dfrac{r^{2}nsin(\dfrac{2\pi}{n})}{2}}

which equals to:

{\pi r^{2} \dfrac{n}{2\pi} sin(\dfrac{2\pi}{n})}


Taking the limit of the right side, using the standard limit {\lim_{x \to +\infty} \dfrac{1}{n} sin(n) = 1}, we would get the formula for the area of the circle. This is one of the most commonly known fun applications of limits.

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