A compact system of inequalities for the standard limits in the theory of limits
The purpose of the paper is an alternative way of obtaining of the standard limits in the elementary theory of limits. The approach uses two double inequalities xe^-x ≤ sin x ≤ xe^x and xe^-x ≤ shx ≤ xe^x and the limit transition. The method is applied to both standard limits simultaneously, t...
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Інститут проблем штучного інтелекту МОН України та НАН України
2013
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irk-123456789-851622015-07-22T03:01:58Z A compact system of inequalities for the standard limits in the theory of limits Mironenko, L.P. Vlasenko, A.Yu. Интеллектуальные системы планирования, управления, моделирования и принятия решений The purpose of the paper is an alternative way of obtaining of the standard limits in the elementary theory of limits. The approach uses two double inequalities xe^-x ≤ sin x ≤ xe^x and xe^-x ≤ shx ≤ xe^x and the limit transition. The method is applied to both standard limits simultaneously, that makes the theory more universal. Apart from that our theory gives many new representations of the standard limits (total number is 37). У роботі запропоновано підхід, який забезпечує єдиний засіб отримання стандартних границь у теорії границь. Підхід заснований на використанні нерівностей xe^-x ≤ sin x ≤ xe^x та xe^-x ≤ shx ≤ xe^x і граничному переході в них. Підхід достатньо простий у використанні. Єдина система нерівностей забезпечує одночасне отримання формул першого, другого стандартних границь і, практично, всіх наслідків з них. Більш того, теорія приводить до великої кількості нових наслідків із стандартних границь. В работе предложен единый подход к стандартным пределам в теории пределов. Подход основан на использовании двойных неравенств xe^-x ≤ sin x ≤ xe^x и xe^-x ≤ shx ≤ xe^x и предельном переходе в них. Метод применим к обоим стандартным пределам одновременно, что значительно упрощает общепринятые подходы. Кроме того, получены новые следствия из стандартных пределов. 2013 Article A compact system of inequalities for the standard limits in the theory of limits / L.P. Mironenko, A.Yu. Vlasenko // Искусственный интеллект. — 2013. — № 2. — С. 63–70. — Бібліогр.: 6 назв. — англ. 1561-5359 http://dspace.nbuv.gov.ua/handle/123456789/85162 514.116 en Искусственный интеллект Інститут проблем штучного інтелекту МОН України та НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| topic |
Интеллектуальные системы планирования, управления, моделирования и принятия решений Интеллектуальные системы планирования, управления, моделирования и принятия решений |
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Интеллектуальные системы планирования, управления, моделирования и принятия решений Интеллектуальные системы планирования, управления, моделирования и принятия решений Mironenko, L.P. Vlasenko, A.Yu. A compact system of inequalities for the standard limits in the theory of limits Искусственный интеллект |
| description |
The purpose of the paper is an alternative way of obtaining of the standard limits in the elementary theory of
limits. The approach uses two double inequalities xe^-x ≤ sin x ≤ xe^x and xe^-x ≤ shx ≤ xe^x and the limit
transition. The method is applied to both standard limits simultaneously, that makes the theory more
universal. Apart from that our theory gives many new representations of the standard limits (total number is 37). |
| format |
Article |
| author |
Mironenko, L.P. Vlasenko, A.Yu. |
| author_facet |
Mironenko, L.P. Vlasenko, A.Yu. |
| author_sort |
Mironenko, L.P. |
| title |
A compact system of inequalities for the standard limits in the theory of limits |
| title_short |
A compact system of inequalities for the standard limits in the theory of limits |
| title_full |
A compact system of inequalities for the standard limits in the theory of limits |
| title_fullStr |
A compact system of inequalities for the standard limits in the theory of limits |
| title_full_unstemmed |
A compact system of inequalities for the standard limits in the theory of limits |
| title_sort |
compact system of inequalities for the standard limits in the theory of limits |
| publisher |
Інститут проблем штучного інтелекту МОН України та НАН України |
| publishDate |
2013 |
| topic_facet |
Интеллектуальные системы планирования, управления, моделирования и принятия решений |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/85162 |
| citation_txt |
A compact system of inequalities for the standard limits in the theory of limits / L.P. Mironenko, A.Yu. Vlasenko // Искусственный интеллект. — 2013. — № 2. — С. 63–70. — Бібліогр.: 6 назв. — англ. |
| series |
Искусственный интеллект |
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| first_indexed |
2025-07-06T12:19:27Z |
| last_indexed |
2025-07-06T12:19:27Z |
| _version_ |
1836900017346969600 |
| fulltext |
ISSN 1561-5359 «Штучний інтелект» 2013 № 2 63
3М
UDK 514.116
L.P. Mironenko, A.Yu. Vlasenko
Donetsk National Technical University, Ukraine
Ukraine, 83000, Donetsk, Аrtema st., 58
A Compact System of Inequalities for the
Standard Limits in the Theory of Limits
Л.П. Мироненко, А.Ю. Власенко
Донецкий национальный технический университет, Украина
Украина, 83000, г. Донецк, ул. Артема, 5,
Компактная система неравенств для
стандартных пределов в теории пределов
Л.П. Мироненко, А.Ю. Власенко
Донецький національний технічний університет, Україна
Україна, 83000, м. Донецьк, вул. Артема, 58
Компактна система нерівностей для
стандартних границь у теорії границь
The purpose of the paper is an alternative way of obtaining of the standard limits in the elementary theory of
limits. The approach uses two double inequalities xx
xexxe ≤≤
−
sin and xx
xeshxxe ≤≤
− and the limit
transition. The method is applied to both standard limits simultaneously, that makes the theory more
universal. Apart from that our theory gives many new representations of the standard limits (total number is 37).
Keywords: theory of limits, standard limits, function, sine, hyperbolic sine, method, inequalities,
standard limits, limit transition.
В работе предложен единый подход к стандартным пределам в теории пределов. Подход основан на
использовании двойных неравенств xx
xexxe ≤≤
−
sin и xx
xeshxxe ≤≤
−
и предельном переходе в
них. Метод применим к обоим стандартным пределам одновременно, что значительно упрощает
общепринятые подходы. Кроме того, получены новые следствия из стандартных пределов.
Ключевые слова: теория пределов, стандартные пределы, неравенство, функция, синус,
гиперболический синус, двойное неравенство.
У роботі запропоновано підхід, який забезпечує єдиний засіб отримання стандартних границь у теорії
границь. Підхід заснований на використанні нерівностей xx
xexxe ≤≤
−
sin та xx
xeshxxe ≤≤
− і
граничному переході в них. Підхід достатньо простий у використанні. Єдина система нерівностей
забезпечує одночасне отримання формул першого, другого стандартних границь і, практично, всіх
наслідків з них. Більш того, теорія приводить до великої кількості нових наслідків із стандартних
границь.
Ключові слова: теорiя границь, стандартні границі, функція, синус, гіперболічний синус,
нерівність, граничний перехід.
Mironenko L.P., Vlasenko A.Yu.
«Искусственный интеллект» 2013 № 2 64
3М
А
Introduction
The standards limits in mathematical analysis are known as the first and second
fundamental (standard) limits. They are used mainly for the determination of derivatives of
the elementary functions xx
aexx ,,cos,sin . Thus it is built the standard table of derivatives
in differential calculus [1], [2]. The standard limits are used practically in all branches of
mathematics, also for calculation of an entire class of limits.
Each of the fundamental limits is proved by two different ways. The first standard
limit is based on a limit procedure in the trigonometric circle. For the second limit it is
used Newton’s binomial. Each of the approaches is effective and attractive. Nevertheless
they are not universal [1], [2].
Meanwhile there are universal methods for an obtaining of the limits. For example,
the first and second limits can be connected one with another by Euler’s formula [3].
The next universal method of proof is based on the undetermined coefficient method
which allows find limits by the standard expansion of the functions xsin and x
e [4].
At last there are the approaches that like to the proposal method in the paper. They
are based also on inequalities and the limiting in the inequalities. Such methods have some
disadvantages that are connected with difficulties of the proof of the inequalities. This is
one of the serious disadvantages of the theory.
According to the logic of our consideration 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 proof the standard limits. This method is based on a
specified inequality system and a limiting in the system. The approach may be applied to
both limits simultaneously. The proof of each inequality is simple and clear. Besides from
the inequalities we have got new important corollaries from the second standard limit.
1 The basic theorem of the method
The main theorem that we use in our theory is well-known theorem about a function
)(xf which is enclosed between two given functions )(
1
xf and )(
2
xf [5], [6].
Theorem. If in any vicinity of the point
o
x a function )(xf is satisfied to the inequalities
)()()(
21
xfxfxf ≤≤ (1)
and axfxf
oo
xxxx
==
→→
)(lim)(lim
21
then the limit )(lim xf
o
xx→
exists and it is equal toa .
It is clear, if the function )(xf is continuous at
o
xx = , then )(
o
xfa = .
The inequalities (1) have a very important property of symmetry, if all functions of
the inequalities are odd. We will replace in the inequalities x to x−
)()()()()()(
2121
xfxfxfxfxfxf ≥≥⇒−≤−≤− .
The sings of the inequalities (1) have changed.
If the functions are even
)()()()()()(
2121
xfxfxfxfxfxf ≤≤⇒−≤−≤− ,
The sings of the inequalities (1) have not changed.
Corollary. If in the inequalities (1) )()(
1
xxgxf = , )()(
2
xxxf ϕ= and at 0=
o
x
1)(lim)(lim
00
==
→→
xxg
xx
ϕ , i.e.
)()()( xxxfxxg ϕ≤≤ , (2)
then 1
)(
lim
0
=
→ x
xf
x
.
A Compact System of Inequalities ...
«Штучний інтелект» 2013 № 2 65
3М
We will use the corollary for proof of the standard limits. Besides we will need in the
generalization of the corollary. If 1)(lim)(lim
00
==
→→
xxg
xx
αϕα and
)()()( xxxfxxg αϕαααα ≤≤ , (3)
then there αα =
→
)(lim
0
xf
x
for any 0≠α .
2 The basic Inequalities for standard limits
The trigonometric and hyperbolic sine functions are satisfied to the inequalities (Fig.1).
≤≤
≤≤
−
−
.sinh
,sin
xx
xx
xexxe
xexxe
(4)
Figure 1– The graphs of the functions of the inequalities (4)
Dividing the inequalities at 0>x we get
≤≤
≤≤
−
−
.
sinh
,
sin
xx
xx
e
x
x
e
e
x
x
e
We take a limit at 0→x . Accounting that 1limlim
00
==
→
−
→
x
x
x
x
ee we find according to the
corollary
.1
sinh
lim ,1
sin
lim
00
==
→→ x
x
x
x
xx
(5)
The first formula is called the first standard (fundamental) limit, the second
represents one of the options of the second standard limit.
The formulas (5) are obtained by the assumption of 0+→x although they are
written for an arbitrary 0→x . Evidently the equalities (5) do not change at 0−→x . It is
clear from that all functions are between of the functions x
xe
− and x
xe . The functions
xsin and shx are odd. Therefore for the negative values of x we have
xxxx
e
x
x
ee
x
x
e ≥≥≥≥
−−
sinh
,
sin
. Taking the limit at 0−→x we have the equalities (5) again.
The figure (Fig. 1) demonstrates the inequalities (4) geometrically. You can see how
the inequalities (2) are changed when the point x changes the sign. All graphs of the
functions of the inequalities (4) are enclosed by the graphs only two functions x
xe
− and
x
xe (Fig. 1.). It means that the limits (5) do not depend on 0+→x or 0−→x .
You can see from the figure also how instead of inequalities (4) we may work with
the inequalities
Mironenko L.P., Vlasenko A.Yu.
«Искусственный интеллект» 2013 № 2 66
3М
А
,sinhsin
xx
xexxxe ≤≤≤
− (6)
from which also the limits (5) are followed. Here we have one more thing. The inequalities (4)
need to proof of four inequalities but the inequalities (6) only of three. For this reason the
inequalities (6) are more favorable. The proof of the inequalities is given in the appendix.
3 Corollaries from the fundamental limits
The corollaries of the fist fundamental limit are well-known and they are described in
many books. We represent the corollaries in the table (Tab. 1) without of a detailed proof.
Table 1 – The corollaries from the first and second fundamental limits
First standard limit 1
sin
lim
0
=
→ x
x
x
Second standard limit 1
sinh
lim
0
=
→ x
x
x
Replacement or
changing of the
variable
Result
Replacement or changing
of the variable
Result
yx arcsin= 1
arcsin
lim
0
=
→ x
x
x
yax sinh= 1
sinh
lim
0
=
→ x
xa
x
x
x
x
cos
sin
tan = ,
1coslim
0
=
→
x
x
1
tan
lim
0
=
→ x
x
x
x
x
x
cosh
sinh
tanh = ,
1coshlim
0
=
→
x
x
1
tanh
lim
0
=
→ x
x
x
yx arctan= 1
arctan
lim
0
=
→ x
x
x
yax tanh=
1
tanh
lim
0
=
→ x
xa
x
In the table the corollaries from the second limit are represented also. They look like
the corresponding corollaries from the first limit. These corollaries are new. Therefore we
will consider the proof of the corollaries in details:
1
cosh
1
lim
sinh
lim
tanh
lim
000
==
→→→ xx
x
x
x
xxx
, where 1
cosh
1
lim
0
=
→ xx
.
In the limit 1
sinh
lim
0
=
→ x
x
x
we substitute yx sinh= . Then, 0→y at 0→x and we
have 1
sinh
lim/1
sinh
lim
sinh
lim
000
===
→→→ y
y
y
y
x
xa
yyx
.
In the equality 1
tanh
lim
0
=
→ x
x
x
we replace yx tanh= . Then 0→y at 0→x , we get
1
tanh
lim/1
tanh
lim
tanh
lim
000
===
→→→ y
y
y
y
x
xa
yyx
.
To find the connection of 1
sinh
lim
0
=
→ x
x
x
with the known corollaries from the second
fundamental limit we may do it at least two ways. For example, we can use the definition
of the function
x
ee
x
xx −
−
=sinh :
x
e
x
e
x
ee
x
x
x
x
x
x
xx
xx
1
lim
2
11
lim
2
1
lim
2
1sinh
lim
0000
−
−
−
=
−
=
−
→→
−
→→
. In
the second limit we replace x by y− . As a result we have
y
e
x
e
y
y
x
x
1
lim
1
lim
00
−
−=
−
→
−
→
. From
where
A Compact System of Inequalities ...
«Штучний інтелект» 2013 № 2 67
3М
x
e
x
x
x
xx
1
lim
sinh
lim
00
−
=
→→
. (7)
The second way is based on the representation
x
x
xa
−
+
=
1
1
ln
2
1
tanh [5].
( ) ( )
x
x
x
x
x
x
xx
xa
xxxx
−
−
+
=
−
+
=
→→→→
1ln
lim
2
11ln
lim
2
1
.
1
1
ln
1
lim
2
1tanh
lim
0000
.
In the last equality we replace yx −→ , we get
( ) ( )
y
y
x
x
yx
−
−=
−
→→
1ln
lim
1ln
lim
00
. From where
( )
.
1ln
lim
tanh
lim
00 x
x
x
xa
xx
+
=
→→
(8)
New representation of second fundamental limit we will find from the representation
( )1lnsinh
2
++= xxxa [5].
( ) ( ) .11lnlim1ln
1
lim
sinh
lim
/1
2
0
2
00
=++=++=
→→→
x
xxx
xxxx
xx
xa
Where from
( ) 11ln
1
lim
2
0
=++
→
xx
xx
, ( ) exx
x
x
=++
→
/1
2
0
1lim . (9)
In the last equality we replace yx /1= , we get
e
x
x
x
x
=
++
∞→
11
lim
2
. (10)
The formulas (9) and (10) are new in mathematical analysis.
Other corollaries from the second fundamental limit follow by corresponding
substitutions in the formulas (7) and (8). The results are represented in the tab 2.
Table 2 – The corollaries from the second fundamental limit 1
sinh
lim
0
=
→ x
x
x
An option of the
second standard limit
Substitution or transformation Corollary
1
1
lim
2
1
lim
2
lim
sinh
lim
0
2
0
00
=
−
=
=
−
=
→→
−
→→
x
x
x
x
xx
xx
ex
e
x
ee
x
x
1
1
lim
0
=
−
→ x
e
x
x 1
sinh
lim
0
=
→ x
x
x
2
sinh
xx
ee
x
−
−
= 1
2
lim
0
=
−
−
→ x
ee
xx
x
1
1
lim
0
=
−
→ x
e
x
x
)1ln( xx +=
1
)1ln(
lim
0
=
+
→ x
x
x
1)1ln(lim
/1
0
=+
→
x
x
x ex
x
x
=+
→
/1
0
)1(lim
1)1ln(lim
/1
0
=+
→
x
x
x , yx /1= ex
x
x
=+
∞→
)/11(lim 1
)1ln(
lim
0
=
+
→ x
x
x
axx ln→
a
x
ax
x
ln
)ln1ln(
lim
0
=
+
→
a
x
ax
x
ln
)ln1ln(
lim
0
=
+
→
aax
x
x
ln)ln1ln(lim
/1
0
=+
→
aax
x
x
=+
→
/1
0
)ln1(lim
Mironenko L.P., Vlasenko A.Yu.
«Искусственный интеллект» 2013 № 2 68
3М
А
aax
x
x
=+
→
/1
0
)ln1(lim yx /1=
aax
x
x
=+
∞→
)ln/11(lim
1
tanh
lim
0
=
→ x
xa
x
x
x
xa
−
+
=
1
1
ln
2
1
tanh 2
1
1
ln
1
lim
0
=
−
+
→ x
x
xx
2
1
1
ln
1
lim
0
=
−
+
→ x
x
xx
2
1
1
lnlim
/1
0
=
−
+
→
x
x x
x
2
/1
0 1
1
lim e
x
x
x
x
=
−
+
→
yx /1= 2
1
1
lim e
x
x
x
x
=
−
+
∞→
2
1
1
log
log
1
lim
/1
0
=
−
+
→
x
a
a
x x
x
e
e
x
x
a
x
a
x
log2
1
1
loglim
/1
0
=
−
+
→
2
1
1
lnlim
/1
0
=
−
+
→
x
x x
x
2
1
1
log
log
1
lim
/1
0
=
−
+
→
x
a
a
x x
x
e
, yx /1= e
x
x
a
x
a
x
log2
1
1
loglim =
−
+
∞→
1
sinh
lim
0
=
→ x
xa
x
++= 1lnsinh 2
xxxa 11ln
1
lim 2
0
=
++
→
xx
xx
11ln
1
lim 2
0
=
++
→
xx
xx
11lnlim
/12
0
=
++
→
x
x
xx exx
x
x
=
++
→
/1
2
0
1lim
exx
x
x
=
++
→
/1
2
0
1lim yx /1= e
x
x
x
x
=
++
∞→
11
lim
2
11ln
1
lim 2
0
=
++
→
xx
xx
11log
log
1
lim
2
0
=
++
→
xx
ex
a
a
x
exx
x
aa
x
log1log
1
lim 2
0
=
++
→
1
tanh
lim
0
=
→ x
x
x
xx
xx
ee
ee
x
−
−
+
−
=tanh 1
1
lim
0
=
+
−
−
−
→
xx
xx
x ee
ee
x
3 The elementary generalization of the theory
The system of the inequalities (6) can be generalized by the replacement xx ⋅→α .
For definiteness we put 0>α , although the theory will be fair for any 0:≠α
≤≤
≤≤
−
−
.sinh
,sin
xx
xx
xexxe
xexxe
αα
αα
ααα
αα
(11)
Applying the corollary from the theorem (3) to the inequality (11), accounting that
1limlim
00
==
→
−
→
x
x
x
x
ee , we have
.
sinh
lim ,
sin
lim
00
α
α
α
α
==
→→ x
x
x
x
xx
(12)
The first formula can be called the fist general fundamental limit, the second is an
option of the second general limit. Using the results of the table 1 we get immediately
.
tanh
lim ,
tanh
lim ,
sinh
lim ,
sinh
lim
,
arctan
lim ,
tan
lim ,
arcsin
lim ,
sin
lim
0000
0000
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
====
====
→→→→
→→→→
x
xa
x
x
x
xa
x
x
x
x
x
x
x
x
x
x
xxxx
xxxx
. (13)
For the second limit we choose aln=α . Then we will get new equalities (Tab. 3).
A Compact System of Inequalities ...
«Штучний інтелект» 2013 № 2 69
3М
Table 3 – The corollaries from the limit a
x
xa
x
ln
)sinh(ln
lim
0
=
⋅
→
An option of the second
standard limit
Substitution or
transformation
Corollary
a
ax
a
x
aa
x
xash
x
x
x
x
xx
xx
ln
1
lim
2
1
lim
2
lim
)(ln
lim
0
2
0
00
=
−
=
=
−
=
⋅
→→
−
→→
a
x
a
x
x
ln
1
lim
0
=
−
→ α
α
=
→ x
xsh
x
)(
lim
0
, aln=α
2
xx
ee
shx
αα −
−
= , aln=α a
x
aa
xx
x
ln
2
lim
0
=
−
−
→
a
x
a
x
x
ln
1
lim
0
=
−
→
)1(log xx
a
+= a
x
x
a
x
ln
)1(log
lim
0
=
+
→
α
α
=
→ x
xarth
x
)(
lim
0
x
x
xarth
α
α
α
−
+
=
1
1
ln
2
1
, aln=α a
ax
ax
xx
ln2
ln1
1ln
ln
1
lim
0
=
⋅−
+⋅
→
a
ax
ax
xx
ln2
ln1
1ln
ln
1
lim
0
=
⋅−
+⋅
→
a
ax
ax
x
x
ln2
ln1
1ln
lnlim
/1
0
⋅=
⋅−
+⋅
→
2
/1
0 ln1
1ln
lim a
ax
ax
x
x
=
⋅−
+⋅
→
yx /1= 2
ln
ln
lim a
ax
ax
x
x
=
−
+
∞→
a
ax
ax
x
x
ln2
ln1
1ln
lnlim
/1
0
⋅=
⋅−
+⋅
→
a
ax
ax
e
x
a
a
x
ln2
ln1
1ln
log
log
1
lim
/1
0
⋅=
⋅−
+⋅
→
2
ln1
1ln
loglim
/1
0
=
⋅−
+⋅
→
x
a
x ax
ax
2
ln1
1ln
loglim
/1
0
=
⋅−
+⋅
→
x
a
x ax
ax
yx /1= 2
ln
ln
loglim =
−
+
∞→
x
a
x ax
xa
++= 1ln
22
αα xxarshx ,
aln=α a
axax
xx
ln
1lnlnln
1
lim
22
0
=
=
++⋅
→
α
α
=
→ x
xarsh
x 0
lim
ααα =
++
→
x
x
xx
/1
22
0
1lnlim ,
aln=α a
axax
x
x
=
=
++⋅
→
/1
22
0
1lnlnlim
a
axax
x
x
=
=
++⋅
→
/1
22
0
1lnlnlim yx /1= a
x
axa
x
x
=
++
∞→
22
lnln
lim
a
axax
x
x
ln
1lnlnlnlim
/1
22
0
=
=
++⋅
→
ea
axax
ex
a
a
a
x
log/1ln
1lnlnlog
log
1
lim
22
0
==
=
++⋅
→
1
1lnlnlog
1
lim 22
0
=
=
++⋅
→
axax
x
a
x
α
α
=
→ x
xth
x 0
lim
xx
xx
ee
ee
thx
−
−
+
−
= , aln=α a
aa
aa
x
xx
xx
x
ln
1
lim
0
=
+
−
−
−
→
APPENDIX 1. The proof of the inequalities (4) 0≥x
≤≤
≤≤
−
−
.sinh
,sin
xx
xx
xexxe
xexxe
.
To the first inequality we apply the well-known inequality xx ≤sin . Then
x
xexx ≤≤sin . The last inequality is obvious because 1≥⇒≤
xx
exex .
Mironenko L.P., Vlasenko A.Yu.
«Искусственный интеллект» 2013 № 2 70
3М
А
For proving the inequality xxe
x
sin≤
− we replace x by x− . we have
( )xxe
x
−≤− sin . We use the property ( ) xx sinsin −=− . Multiplying the inequality
xxe
x
sin−≤− by the factor 1− we have the previous inequality x
xex ≤sin .
To the second inequality (4) we apply the inequality xe
x
≥−1 . The second inequality
(4) is proved like the first, only instead of the inequality xx ≤sin we will use the
inequality xe
x
≥−1 .
122
2
sinh
22
−≤≤⇒≤
−
=
−
−
xxx
xx
exxexe
ee
x .
we have the obvious inequality 122
22
≤⇒≤
−− xx
exxe . For the proving of the inequality
x
xex ≤sinh we replace x by x− . We have x
xex
−
−≤− )sinh( . We use the property
( ) xx sinhsinh −=− . Multiplying the inequality x
xex
−
−≤− sinh by the factor 1− we return
to the proved inequality x
xex
−
≥sinh .
APPENDIX 2. The proof of the inequalities .sinhsin xx ≤
We replace the left part of the inequality by a value x that is more then xsin , we get
xx sinh≤ or xx
eex
−
−≤2 . We rewrite the inequality in the form 012
2
≥−−
xx
xee . The
roots of the equation are 1
2
+±= xxe
x . The negative root we omit because 0>
x
e .
Geometrically we have a parabola 12
2
−−=
xx
xeey respect to x
e . The parabola is touched
the axis x only at the point 0=x , the branches of the parabola are directed up, i.е. 0>y .
Therefore the inequality 012
2
≥−−
xx
xee is fulfilled for any x .
Corollary.
2
1
2
x
xe
x
++≥ . Here it is used the inequality 2/11
22
xx +≤+ which is
checked by raising both parts of the inequality into square.
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 simple and effective.
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 improves the classical theory of the fundamental limits.
Literature
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. Gursa E. Кurs mаtematicheskogo analiza, tом 1, Gosudarstvennoe techniko-tworcheskoe iezdatelstvo –
Моskwa, 1933. – 368 p.
Статья поступила в редакцию 02.04.2013.
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