A new approach to the fundamental limits in the theory of limits
The purpose of the paper is an alternative way of obtaining of the fundamental limits in the elementary theory of limits. The approach uses two double inequalities. The method is applied to the both limits simultaneously, that makes the theory universal. Proof of the limits is quite new and origin...
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| Cite this: | A new approach to the fundamental limits in the theory of limits / L.P. Mironenko // Искусственный интеллект. — 2014. — № 2. — С. 9–14. — Бібліогр.: 6 назв. — англ. |
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| citation_txt | A new approach to the fundamental limits in the theory of limits / L.P. Mironenko // Искусственный интеллект. — 2014. — № 2. — С. 9–14. — Бібліогр.: 6 назв. — англ. |
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| description | The purpose of the paper is an alternative way of obtaining of the fundamental limits in the elementary theory
of limits. The approach uses two double inequalities. The method is applied to the both limits simultaneously,
that makes the theory universal. Proof of the limits is quite new and original. Apart from that our theory gives
many new representations of the limits (total 37).
Метою статті є підхід, який забезпечує єдиний спосіб отримання фундаментальних границь у теорії границь.
Підхід заснований на використанні нерівностей і граничному переході в них. Підхід достатньо простий у
використанні. Єдина система нерівностей забезпечує одночасне отримання формул першого та другого
границь і, практично, всіх наслідків з них. Більш того, теорія призводить до великої кількості нових наслідків.
В работе предложен единый подход к фундаментальным пределам в теории пределов. Подход
основан на использовании двойных неравенств и предельном переходе в них. Метод применим к
обоим фундаментальным пределам одновременно, что значительно упрощает общепринятые подходы. Доказательства обоих пределов дано в одном стиле и носит оригинальный характер. Кроме
того, получены новые следствия из фундаментальных пределов.
|
| first_indexed | 2025-12-01T10:36:57Z |
| format | Article |
| fulltext |
ISSN 1561-5359 «Штучний інтелект» 2014 № 2 9
1M
UDK 514.116
L.P. Mironenko
Donetsk National Technical University
Ukraine, 83000, Donetsk, Аrtema st., 58, mironenko.leon@yandex.ru
A New Approach to the Fundamental Limits
in the Theory of Limits
Л.П. Мироненко
Донецкий национальный технический университет, Украина
Украина, 83000, г. Донецк, ул. Артема, 58, mironenko.leon@yandex.ru
Новый подход для фундаментальных пределов
в теории пределов
Л.П. Мироненко
Донецький національний технічний університет, Україна
Україна, 83000, м. Донецьк, вул. Артема, 58
Новий підхід до фундаментальних границь
у теорії границь
The purpose of the paper is an alternative way of obtaining of the fundamental limits in the elementary theory
of limits. The approach uses two double inequalities. The method is applied to the both limits simultaneously,
that makes the theory universal. Proof of the limits is quite new and original. Apart from that our theory gives
many new representations of the limits (total 37).
Keywords: methodic, theory of limits, fundamental limits, function, sine, hyperbolic sine, method,
inequalities, standard limits.
В работе предложен единый подход к фундаментальным пределам в теории пределов. Подход
основан на использовании двойных неравенств и предельном переходе в них. Метод применим к
обоим фундаментальным пределам одновременно, что значительно упрощает общепринятые под-
ходы. Доказательства обоих пределов дано в одном стиле и носит оригинальный характер. Кроме
того, получены новые следствия из фундаментальных пределов.
Ключевые слова: методика, теория пределов, фундаментальные пределы, неравенство,
функция, синус, гиперболический синус, двойное неравенство.
Метою статті є підхід, який забезпечує єдиний спосіб отримання фундаментальних границь у теорії границь.
Підхід заснований на використанні нерівностей і граничному переході в них. Підхід достатньо простий у
використанні. Єдина система нерівностей забезпечує одночасне отримання формул першого та другого
границь і, практично, всіх наслідків з них. Більш того, теорія призводить до великої кількості нових наслідків.
Ключові слова: методика, теорiя границь, фундаментальні границі, функція, синус,
гіперболічний синус, нерівність, граничний перехід.
Introduction
The fundamental limits in mathematical analysis are known as the first and second
fundamental (”wonderful”) limits. They are used mainly for the determination of derivati-
ves of the elementary functions such as xx aexx ,,cos,sin and so on [1], [2]. The fundamen-
tal limits (we shall call simply limits) are used practically in all branches of mathematics,
also for a calculation of the entire class of limits.
Mironenko L.P.
«Искусственный интеллект» 2014 № 2 10
1M
Each of the limits is proved by two different ways. The first limit is based on a limit
procedure in the trigonometric circle. Newton’s binomial is used for the second limit. Each
of the approaches is effective and attractive. Nevertheless these approaches are not
universal [1], [2].
However there are universal methods for the limits. For example, the first and second
limits can be connected one with another by Euler’s formula [3].
The next universal proof method is based on the so-called undetermined coefficient
method which allows finding limits by the standard expansion of the functions xsin and xe [4].
At last there are approaches that look alike as the method in this paper. They are also
based on inequalities and application limits to them. Such methods have some
disadvantages and there is some difficulty in the proof of the inequalities. This is the main
serious disadvantage of such approaches [6].
The inequalities must be proved by elementary mathematical methods. Other words,
the proof must be performed without using of a derivative concept.
We found a simple method to prove such limits. For this we used a deep analogy
between the trigonometric and hyperbolic functions from one side and the classical proof
method of the first limit from another side. Such way has occurred effective and helpful.
1 An auxiliary theorem of the method
The theorem that we shall use in our theory is well-known theorem about a function
)(xf which is placed between two given functions )(1 xf and )(2 xf [4].
Theorem. If the function )(xf satisfies in a vicinity of the point ox to the inequalities
)()()( 21 xfxfxf (1)
and axfxf
oo xxxx
)(lim)(lim 21 then the limit )(lim xf
oxx
exists and it is equal to a .
It is clear also, if the function )(xf is continuous at oxx then )( oxfa .
The inequalities (1) have a very important property of symmetry if all functions of
the inequalities are odd. If to replace in the inequalities x by x
)()()()()()( 2121 xfxfxfxfxfxf .
The sings of the inequalities (1) have changed.
If the functions are even
)()()()()()( 2121 xfxfxfxfxfxf ,
The sings of the inequalities (1) does not change.
We shall use the theorem for a proof of the limits in the next item.
2 Inequalities for the fundamental limits
It is well-known the inequalities for the first fundamental limit follow from the
trigonometric circle (Fig. 1)
.tansin xxx (2)
The inequalities are obvious for big values of 0x (Fig.2). It follows from a
behavior of the graphs of the functions (2) at big values of x . This fact will be used later
for a proof the inequalities for any values of x , including small values of x .
Идентификация непрерывной функции в одномерном параболическом уравнении
«Штучний інтелект» 2014 № 2 11
1M
The proof will be done by the controversy method.
Figure 1 – The geometrical proof of the inequalities .tansin xxx
The first inequality xx sin will be proved if the equation xx sin has the unique
solution 0ox . This follows directly from the figure (Fig. 2). Assume that there is a root
0ox of the equation xx sin . Execute some elementary transformations to confirm our
assumption.
22
cos
2
sin
2
cos
2
sin2sin xxxxxxxx .
We assumed that the equation
22
cos
2
sin xxx has a non-zero root ox . Therefore we
may to account the equality
22
sin oo xx in the last equation. In the result we get the
equation 1
2
cos ox . From here 0ox is an unique root of the equation xx sin .
Figure 2 – The trigonometric and hyperbolic functions demonstrate the inequalities
xxx tansin and xxx sinhtanh
The second inequality xx tan is proved as well as the previous inequality. In that
case again we assume that there is a root 0ox but now of the equation xx tan ,
2
tan1
11
2
tan1
2
tan
2
2
sin
2
22
sin2sintan
2222 xx
x
x
xxсos
xсosx
x
сosx
xxxx
.
The last equation is satisfied with only the root 0x .
Mironenko L.P.
«Искусственный интеллект» 2014 № 2 12
1M
The inequality (2) has proved complitly.
The first limit follows from the inequalities (w2) and the theorem. For this the
inequality (2) has to be divided by 0sin x and limited at 0x .
.1sinlim1sinlimcoslim1sincos
cos
1
sin
1
cos
sinsin
000
x
x
x
xx
x
xx
xx
x
x
xxx
xxx
The method works for the hyperbolic functions. As a result we shall get one of a form of
the second limit. For this consider inequalities like to the inequalities (2) (Fig. 2)
.sinhtanh xxx (3)
Repeat the method as well as in the case of the first limit. Now use the formulas
22
sinh2sinh xсoshxx ,
2
sinh
2
cosh 22 xxсoshx . For example,
01
2
cosh
22
cosh
2
sinh
2
cosh
2
sinh2sinh xxxxxxxxxx .
The inequality (3) divide by 0sinh x , account also xxx cosh/sinhtanh . We
have
x
x
x
x
x
x
coshsinh11
sinhcosh
1 .
Taking into account also that 1coshlim
0
x
x
, we have in the limit at 0x ,
.1sinhlim
0
x
x
x
(4)
This is one of the representations of the second fundamental limit.
The formulas (2) and (4) were received at 0x , but they are written for arbitrary
0x . It is cleared they do not change at 0x . In that case all functions in the
inequalities (2) and (3) are odd, therefore for negative x we have
xxxxxxxxx tanhsinhsinhtanh)sinh()tanh( .
Now show how to come from the form (4) to the standard form of the second limit.
Let us rewrite the limit (r4) according to the definition of the function
2
sinh
xx eex
.11lim1
2
1lim1lim1
2
lim1
2
lim
0
2
0000
x
e
x
e
ex
ee
e
e
x
ee x
x
x
xxx
xx
x
x
x
xx
x
The last limit is known well [1-3]. It represents one of the corollaries from the
second limit. Other forms of the second limit follow from this and they are described in
many handbooks.
The corollaries of the fist fundamental limit are well-known and they are described in
literature. We restrict ourselves only new corollaries (Tab. 1).
.
Идентификация непрерывной функции в одномерном параболическом уравнении
«Штучний інтелект» 2014 № 2 13
1M
Table 1 – The corollaries from the first and second fundamental limits.
First standard limit 1sinlim
0
x
x
x
Second standard limit 1sinhlim
0
x
x
x
Replacement or changing of
the variable
Result Replacement or changing of the
variable
Result
yx arcsin
1arcsinlim
0
x
x
x
yax sinh
1sinhlim
0
x
xa
x
x
xx
cos
sintan , 1coslim
0
x
x
1tanlim
0
x
x
x
x
xx
cosh
sinhtanh , 1coshlim
0
x
x
1tanhlim
0
x
x
x
yx arctan
1arctanlim
0
x
x
x
yax tanh
1tanhlim
0
x
xa
x
Results
1. It is developed a new effective method of the proof of the first and second
fundamental limits in the classical limit theory.
2. The approach generalizes usual theory of the fundamental limits and from our
point of view the theory is more universal.
3. The approach is generalized that leads to many equivalent forms of the limits
(from the first limit we have got 3 corollaries, from the second limits 34). The most of
them are new.
4. The proposed method essentially improves the classical limit theory.
Список литературы
1. Кудрявцев Л.Д. Математический анализ / Кудрявцев Л.Д. – Наука, 1970. – Том I. – 571 с.
2. Ильин В.А. Основы математического анализа / В.А. Ильин, Э.Г. Поздняк. – Москва : Изд. ФМЛ,
1956. – Том 1. – 472 с.
3. Мироненко Л.П. Эквивалентность стандартных пределов в теории пределов / Л.П. Мироненко //
Искусственный интеллект. – 2012. – № 2.
4. Мироненко Л.П. Стандартные пределы и метод неопределенных коэффициентов / Л.П. Мироненко,
И.В. Петренко // Искусственный интеллект. – 2012. – № 2.
5. Фихтенгольц Г.М. Курс дифференциального и интегрального исчисления / Фихтенгольц Г.М. –
Наука, «ФМЛ», 1972 – Том 1. – 795 с.
6. Mironenko L.P. A compact system of inequalities for the standard limits in the theory of limits /
L.P. Mironenko, A.Y. Vlasenko // Искусственный интеллект. – 2013 – № 2. – P. 61-70.
References
1. Kudriavtsew L.D. Маtematichesky analiz. Том I., Nаukа,1970 - 571 p.
2. Ilyin V.А., Pozdniak E.G. Оsсnowy mаtematicheskogo analiza, tом 1, Izd. FML, Моskwa, 1956. – 472 p.
3. Mironenko L.P. Ekvivalentnost standartnih predelov v teoryi predelov// Iskustvenyi intellekt, 2, 2012– p.
123-128.
4. Mironenko L.P., Petrenko I.V. Standartnie predeli i metod neopredelennih koefficientov// skustvenyi
intellekt, 3, 2012– p. 284-291.
5. Fihtengolts G.М. Кurs differentsialnogo i integralnogo ischislenia, tом 1, Nauka, «FML», 1972 - 795 p.
6. Mironenko L.P., Vlasenko A.Y. A compact system of inequalities for the standard limits in the theory of
limits // Artificial intelligence -2, 2013– p. 61-70.
Mironenko L.P.
«Искусственный интеллект» 2014 № 2 14
1M
RESUME
L.P. Mironenko
A New Approach to the Fundamental Limitsin the Theory of Limits
Background: there is a deep analogy between the trigonometric and hyperbolic functions.
The fist fundamental limit is connected with trigonometry. It should expect an universal
method of a proof of the both fundamental limits. We found such way by using the method
of the proof from a controversy.
Materials and methods: it is used so-called the method of a proof from a controversy.
Besides in the paper we apply usual properties of limits such as taking a limit in
inequalities, some methods of elementary mathematics.
Results: it is developed a new effective method of a proof of the first and second
fundamental limits in the classical limit theory. The approach generalizes usual theory.
From our point of view the theory is more simple and effective. The approach is more
general then existing that we have got many equivalent forms of the limits (from the first
limit 3 corollaries, from the second limits 34). The most of them are new. The proposed
method essentially improves the classical fundamental limit theory.
Conclusion: it is found more effective method of a proof of two fundamental limits in the
theory of limits. The method is universal and very simple. The result can be used by
professors for students for a studying of the limit theory in colleges and universities.
The paper is received by the edititorial 05.04.2014.
|
| id | nasplib_isofts_kiev_ua-123456789-85257 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1561-5359 |
| language | English |
| last_indexed | 2025-12-01T10:36:57Z |
| publishDate | 2014 |
| publisher | Інститут проблем штучного інтелекту МОН України та НАН України |
| record_format | dspace |
| spelling | Mironenko, L.P. 2015-07-23T12:52:29Z 2015-07-23T12:52:29Z 2014 A new approach to the fundamental limits in the theory of limits / L.P. Mironenko // Искусственный интеллект. — 2014. — № 2. — С. 9–14. — Бібліогр.: 6 назв. — англ. 1561-5359 https://nasplib.isofts.kiev.ua/handle/123456789/85257 514.116 The purpose of the paper is an alternative way of obtaining of the fundamental limits in the elementary theory of limits. The approach uses two double inequalities. The method is applied to the both limits simultaneously, that makes the theory universal. Proof of the limits is quite new and original. Apart from that our theory gives many new representations of the limits (total 37). Метою статті є підхід, який забезпечує єдиний спосіб отримання фундаментальних границь у теорії границь. Підхід заснований на використанні нерівностей і граничному переході в них. Підхід достатньо простий у використанні. Єдина система нерівностей забезпечує одночасне отримання формул першого та другого границь і, практично, всіх наслідків з них. Більш того, теорія призводить до великої кількості нових наслідків. В работе предложен единый подход к фундаментальным пределам в теории пределов. Подход основан на использовании двойных неравенств и предельном переходе в них. Метод применим к обоим фундаментальным пределам одновременно, что значительно упрощает общепринятые подходы. Доказательства обоих пределов дано в одном стиле и носит оригинальный характер. Кроме того, получены новые следствия из фундаментальных пределов. en Інститут проблем штучного інтелекту МОН України та НАН України Искусственный интеллект Концептуальные проблемы создания систем искусственного интеллекта A new approach to the fundamental limits in the theory of limits Новий підхід до фундаментальних границь у теорії границь Новый подход для фундаментальных пределов в теории пределов Article published earlier |
| spellingShingle | A new approach to the fundamental limits in the theory of limits Mironenko, L.P. Концептуальные проблемы создания систем искусственного интеллекта |
| title | A new approach to the fundamental limits in the theory of limits |
| title_alt | Новий підхід до фундаментальних границь у теорії границь Новый подход для фундаментальных пределов в теории пределов |
| title_full | A new approach to the fundamental limits in the theory of limits |
| title_fullStr | A new approach to the fundamental limits in the theory of limits |
| title_full_unstemmed | A new approach to the fundamental limits in the theory of limits |
| title_short | A new approach to the fundamental limits in the theory of limits |
| title_sort | new approach to the fundamental limits in the theory of limits |
| topic | Концептуальные проблемы создания систем искусственного интеллекта |
| topic_facet | Концептуальные проблемы создания систем искусственного интеллекта |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/85257 |
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