In economic, sociological and mathematical forecasts that are associated with continuous processes, when calculating financial rent, bank interest, constructing asymptotes for a variety of graphs, when studying their properties, knowledge of what a function limit is often required. This concept has been used at the level of intuitive perception, since the end of the 17th century, by many famous scientists, mathematicians, mechanics, physicists, astronomers, in their remarkable studies. An innovator in this matter was Isaac Newton. His example was soon followed by the famous Euler, Lagrange and other brilliant scientists. But the first concrete definitions in this area in the 19th century were given by the Czech scientist Bolzano and the mathematician from France Louis Cauchy (his portrait is presented below).
Number series
The concepts of the limit of sequence and the limit of function are closely related. And this is quite natural. Indeed, because if some function is built from a certain series, the values of which are found with increasing natural numbers from 1 to ∞, this is a numerical sequence.
We give a concrete example. Let some series of numbers be given by the expression:
a n = (n 2 + n + 1) / (n + 2).
Find out the first five components of this sequence. They turn out to be numbers: 1; 7/4; 13/5; 21/6; 31/7. You can continue this series. And it is easy to understand that any subsequent number will be greater than the previous one, and the value of each of them will approach infinity. She will become the limit of this sequence.
Below is a portrait of Bernard Bolzano, who made a great contribution to the study of sets and their limits, the author of the scientific work "Paradoxes of the Infinite".
Number range limit
Sequences of the type mentioned above are commonly called infinitely large. This means that their limit is ∞. But the number series given by the inverse formula, that is, 1 / a n, is called infinitely small in mathematics, because the values of each of the following numbers become ever smaller, tending to zero.
There are other types of sequences. For example, the series given by the expression a n = 105 - 7n approaches the value of the terms not to zero, but to infinity, only negative. And starting from n = 16, the components of the number series get negative values, so it is not considered to be increasing, as in the first case, and is called decreasing.
The series defined by the expression a n = 1 / 2n is infinitesimal. But in this case, the sequence given by the inverse formula, on the contrary, will be infinitely large.
Number series can tend not only to 0 and ∞, but to some specific number. You can specify any number of such sequences. For example, the limit a n = 5n / (n + 1) is 5.
It is possible that the sequence has no limit at all, then it is called diverging.
Concept of function limit
From the above definition it is easy to understand that, similarly to sequences, the question is considered, and also when it comes to functions. Although it has its own characteristics. If the argument (that is, the value of the variable x) tends to a certain specific number or to infinity, then the value of the function can also tend to some specific value. And in the case when it exists, it is considered the limit. But it is better and more convenient to consider this process graphically, thereby clarifying the concept of the limit of a function and its geometric meaning.
It is quite possible that a function is not defined at all at that very point of its limit or its value is not equivalent to it. Nevertheless, the limit is still considered to be existing and equal to A.
A fantastic example for clarification
Consider an important question: can a function have two limits at a selected point? No, this is not possible. If it exists, then it is necessarily the only one.
In fact, the limit is a certain quantity to which the function tends to approach. Its values come to it incredibly close to it as much as possible, but are not able to achieve this value, and are constantly only in some kind of "neighborhood" of the number expressing its value.
For clarity of explanation of the concept of the limit of a function, we can consider some fantastic situation. Suppose a runner, trying to reach the finish line, is constantly decreasing at the behest of the almighty wizard, so he takes smaller steps (their value tends to zero). Due to such witchcraft, more and more persistently moving towards the cherished goal, this record holder, however, will not become a winner, because he is not able to cross the finish line.
Symbols and properties
Limits are usually indicated by the letters lim. The arrow below them gives an idea of the value to which the argument seeks. To the right of the limit symbol, the function itself is written. This is usually followed by an equal sign and the numerical value of the limit value is indicated. Such designations are used throughout the world. To understand the concept of the limit of a function, the properties of the limits are simply necessary. They are listed below.
Such properties mean that when adding functions having a limit, to find the common lim it is necessary to add the limits of each of them. The same goes for the product and the quotient of two functions. The last property mentioned states that a common factor can be taken out of the limit sign, which does not violate the equality in expressions with limits.
Examples
We consider some problems of finding the limit of a function at a point and at infinity. The concepts described above will then become much clearer.
The limit in the first two examples is considered equal to infinity due to the fact that when the argument tends to the indicated value, the denominator in both cases turns into something infinitely small. And this means that the expression itself, on the contrary, becomes infinitely large. This is the secret to solving such examples.
In the third case, the limit is equal to some certain number. If we solve the problem without any tricks, simply substituting infinity instead of x, then the expression itself under the limit sign will take the form: (∞ / ∞). And this is uncertainty. In view of this, to solve such examples, they resort to a common method, dividing both parts of the fraction by x. Thus, the values 3 / x and 1 / x at the limit point become so small that their values for obtaining the values of the third expression become not important. Therefore, the answer comes out 2/5.
Types of Uncertainty
In the last example, which we examined above, finding the limit was difficult due to the uncertainty revealed during the calculation. Such cases are not particularly exceptional and rare. When solving a variety of problems, other types of uncertainties may arise. The concept of the limit of the function will help you learn to get out of such situations, only you should learn some techniques that help get rid of these problems.
Now consider a number of similar examples.
In the first case, there are no difficulties in finding the result. Here you just need to substitute the limit value of the variable x in the expression itself and get the answer.
But in the second example, the uncertainty of the type already considered by us is revealed: (∞ / ∞). To get out of this difficult situation, we will use a technique similar to the one we used earlier. Namely, we divide the numerator and denominator by x 7 in the expression. And then the fraction will take the form in which some terms of the expression turn into infinitely small quantities, and only 1/3 of the significant numbers remain. It will be the limit of the original function.
Expression Conversion Method
We continue to consider the concept of the limit of a function.
In the third and fourth examples mentioned above, if we substitute the marginal value in the upper and lower parts of the fraction, another kind of uncertainty arises (0/0). As you know, in mathematics this is unacceptable. In such situations, you can use the expression conversion method.
In task number 3 for starters, the numerator is laid out as the difference of squares. And then, after making reductions, substitute x = 1 into the expression. And so they get the answer. As you can see from the example, the limit of the function is 0.
Task number 4 is a bit more complicated. Here the fraction also needs to be reduced. But for factoring the numerator and denominator, they first solve the quadratic equations. Next are the roots, and factoring is done in the way that is usually done in square trinomials. Then only the expression remains under the limit sign: (x - 3) / (3x - 2). Now we should substitute the value of the variable and find out that the limit is 4.
Continuity
Connecting the concept of limits and the continuity of the function of one variable, we immediately explain some important points, clarifying that the second of these mathematical representations is usually determined through the first.
If all the conditions just indicated are fulfilled, then it can be asserted with full confidence that the function at a predetermined point is continuous. When at least one of the points is violated, this means that the chart line at the indicated location is interrupted, that is, the function suffers a break.
Similarly, the concept of the limit and continuity of the function of two variables is also related.
Limit idea
The concept considered by us, related to mathematical analysis, is rightfully considered one of the most subtle in this discipline. And although this science arose relatively recently, the very idea of the limit was used in his works by the great resident of ancient Syracuse - Archimedes. To calculate the areas and volumes of complex geometric shapes, he used the so-called exhaustion method. In some of his works, he outlined a certain axiom of continuity, which actually contained an intuitive idea of limits.
As you know, in the knowledge of mathematical truths, Archimedes was significantly ahead of his time, although before him there were great minds who expressed similar considerations. But this ingenious philosopher, engineer and scientist thus managed to calculate the area of a circle and various polygons, determined the volume of a cone, pyramid, cylinder, prism, a large number of other figures. After the death of Archimedes, the ideas of the great Greek were improved and developed for more than two millennia before becoming a theory for calculating differentials and integrals.