How to explain the limits in a simple way: mathematics in everyday life

When we hear “mathematical”, We often scare ourselves: it seems something of incomprehensible! But mathematics, in truth, describes the reality that surrounds us, we just have to learn to interpret it in the right way. Let’s take an example: i mathematical limits. When we studied them at school they may seem abstract and not very useful; The limits represent an abstract concept that describes the behavior of a function near a point or infinity and studying it allows to understand their trend in these conditions. In reality, the limits can help us describe concrete situations In which we are observing something that continues to change and we want to understand the fate after some time. We will deepen examples like that of Tulip growthand of city ​​that change over the centuriesuntil the Motorcycle of a boa in the seaall using the limits!

The limit is a concept that describes the behavior of a function or succession near a point or infinity. Studying the limits allows you to understand their trend in these conditions, providing a more precise analysis.

The mathematical limits in real life: the example of the tulip

What is a mathematical limit? Well, it is certainly an abstract, theoretical concept that helps us to study the functions. But … can we understand it based on real life? We think of this situation: we are at school and we are waiting for the bell to sound to be “finally free”. It happened to everyone to look at the clock with impatience, and the more we look at the clock and the more the sound of the bell seems to move away. It seems that the minute hands try to complete the tour without succeeding: the more we look at the clock and the more it seems to slow down. Situations like these can be described using the limits. We understand what it is with some examples.

Have you ever observed a seedling, for example a tulip, grow day after day? At the beginning we see her sprout from the ground and appears only as a green dot. Then day by day we see her grow up to reach its maximum height, which usually happens in April, when the flower opens in all its splendor. Using mathematical language we can say that

When the date curtains In April the tulip curtains at its maximum height, while when the date curtains In May the tulip curtains to dry.

What we have just done was describing a mathematical limit! If in fact instead of “tending” we use the concept of limit, we can say that:

The limit of the tulip when the date curtains In April it is flowering, while the limit of the tulip when the date tends in May is to dry.

Describe something that changes looking at final state to which it tends, it is one of the key ideas of the concept of limit and we can find it in many real life situations, Don’t you believe it? Here are examples: when the service timetable tends to the end, the bus tends to the terminus; When our hunger tends to the maximum, we tend to the refrigerator; When sleep tends to the maximum we tend to the bed, and whoever has more. This suggests that the limits can be seen as a approximation process More and more accurate: those who speak “Matematica” know that this concept is present within the formal definition of limit. If we resume in hand the example of the tulip, in fact, could we ask ourselves: “How much should I get closer in April so that the height is the definitive one and the plant flourishes?”.

What we have just seen is the case in which a limit tends to a precise moment, that is, to a finite quantity. But what happens when a limit Do you tend to infinity?

When the limit tends to infinity

Let’s now consider a series of maps which represent the same city over the centuries. We can verify how the shape of the city changes when the years go by and therefore the form to which it “tends” when the years “They tend to 2025”that is, a finite limit, as already described. But we can also try to imagine the shape to which it will tend in one hundred years: it will be difficult to predict, but if for example the city is on the sea we can predict that it will tend to expand on the opposite side to this. If we imagine that the city is eternal (like Rome …) we can imagine the form to which he will tend between millions of years and beyond. Said in mathematical terms, we are imagining the limit of the form of the city when time tends to Infinito.

With the word limitin fact, the mathematicians refer somehow to the arrival point of a process that “tends to something”, a point of arrival that is not necessarily reached in a finished time, but which can also be reached in a time infinite.

A geometric examplewhich visually makes the idea of ​​this concept is that of the figure at the top. Let’s take a first step dash (one segment) vertical and we attack two dresses, slightly more short that form a branch at the top, as if it were a tree. As a second step we attack two new lips, even smaller, to each of the smallest dolls. We can repeat this operation many times. Step by step the drawing begins to take on a form that looks more and more to the formthe one he tends to when the number of steps tend to infinite. This type of geometric figure is called fractal.

An interesting thing about this example is that when the number of steps tends to infiniteon the one hand you have that theimage tends to a precise figure As you can see in the image, but on the other you have that the number of donge is bigger and bigger. But how is it possible? This happens because despite the dates they continue to increase indefinitely, their length becomes smaller and smaller, so small that it seems not to make a contribution to the entertainment of the tree. In this case it is said that the length of the new limbs becomes infinitesimal and the function that describes the tree we designed has a asymptotethat is, a maximum value that will never be reached, nor exceeded.

It is not always possible to find a limit

But is it always possible to predict the fate of something we are observing? It would think yes, but it is not that simple. In fact there are phenomena that they do not tend to stabilize towards something precise.

We observe, for example, theheight to which there is one buoy Which floats in the sea: if it is a calm day of the sea the height of the buoy it will be stable, but if there is moved sea it will go up and descend following the rhythm of the waves. If we ask ourselves what height will tend to the boa in 10 years, 1000 years or infinite years, we cannot predict it why periodically It will happen that the sea will be moved and the boa will continue to swing, moving up and down. Similarly, we cannot establish the limit to which the Moon phase in the skyWhy oscillates continuously Between full and new (that is, when it is not visible) without ever stopping. In these cases we will say that the limits of the height of the buoy and the lunar phase do not exist or, said in “mathematical”, they do not converge.