Journal of Information, Control and Management Systems, Vol. 4, (2007), No. 2
303
UNDERSTANDING AND COMPUTING LIMITS
BY MEANS OF INFINITESIMAL QUANTITIES
Galina JASEČKOVA 1 , Michal KAUKIČ 2
1
Fakulty of Natural Sciences
University of Žilina
2 Faculty of Management and Informatics
University of Žilina
e-mail: [email protected], [email protected]
Abstract
The concept of limit is the basic tool used throughout all the undergraduate
calculus course. We will show how to use the basic properties of infinitesimal and
infinite quantities for the easy computation and better understanding of limits of
functions.
Keywords: infinitesimal quantities, limit of function, order of growth
1
Introduction
We are convinced, that nearly all fundamental facts in calculus cannot be properly
understood without the deep understanding of the concept of limit. The main motivation for introducing limits are the difficult cases, where there is no way to make direct
evaluation of interesting quantities.
One such case is to compute functional values f (x) at some critical point xc (usually, f (xc ) is not defined), if we know, that f (x) is the “well behaved function” in the
neighbourhood of xc . Another difficult case is dealing with infinite computational processes like summing more and more consecutive terms from given sequence of numbers.
Here, the limits come to the rescue.
To be more concrete, let us consider the expression
L(x) =
f (x)
where f (x) and g (x) are going to zero for x approaching x0 :
g(x)
We will call this expression the indeterminate form of the type (0/0). Obviously, we
cannot directly compute the value L(x0 ). The key for understanding why this expression
is indeterminate is to see the “fighting quantities” f (x); g (x) both going to zero – but
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Understanding and computing limits by means of infinitesimal quantities
which one is quicker in tendency to annihilate itself?
Depending on the behaviour of f (x) and g (x) the value of L(x0 ) can be finite
number, if both quantities are of “comparable decrease” or, if f (x) is going to zero more
rapidly than g (x) then L(x0 ) = 0 and also L(x0 ) can be infinite or does not exists.
Example 1 What is the value of the function f (x) = sin x=x for x = 0?
On the first glance, this seems like the crazy question to ask. We cannot directly substitute the value x = 0 into functional recipe, so it is better (or, at least, easy) to say, that
the function is undefined at the point x = 0.
On the other hand, we can compute the functional value f (x) for arbitrarily small
(negative or positive) values of x and see that the values f (x) are pretty close to the number 1. This number can be seen as the reasonable candidate for the value f (0). We can
also consider the value of f (x) as the quotient of two functions n(x) = sin x; d(x) = x
and compare the behaviour of the functional values for x arbitrarily close to zero. In this
case, we can use the well-known fact that sin x x for small values of x (measured in
radians).
2
Basic definitions and notations
Definition 1 The numerical sequence a1 ; a2 ; : : : ; an ; : : : is called infinitely small (infinitely
large) iff lim an = 0 (resp. lim an = 1).
n!1
n!1
Definition 2 Function f (x) is infinitely small (infinitely large) for x ! a where a is the
real number or infinity iff lim f (x) = 0 (resp. lim f (x) = 1).
x! a
x! a
The basic principle used for computation of limits in this article can be formulated as
two-stage process:
knowledge of table of “equivalent” infinitely small quantities
comparison with new, unknown kind of infinitesimal quantities we need to compute actually.
The comparison of infinitely small quantities (x); (x) for x !
behaviour of the their quotient in the neighbourhood of the point a.
Definition 3 Let (x); (x) are infinitely small for x ! a and
x! a
lim
a is based on the
(x)
(x)
=
c.
Journal of Information, Control and Management Systems, Vol. 4, (2007), No. 2
305
If c is the nonzero finite number then we call the functions (x); (x) the infinitely
small of the same order and use the big-O notation (x) = O( (x)).
If c = 0 then the function (x) is called infinitely small of higher order than (x),
resp. (x) is infinitely small of lower order than
small-o notation (x) = o( (x)).
(x).
In this case we use the
If c = 1 then we call the functions (x); (x) equivalent or asymptotically equal
and write (x) (x).
Remark 1 If the limit of the quotient of infinitely small functions does not exist, then the
functions are incomparable.
Example 2 Make the comparison of the infinitely small functions
(x) = tan x2 and (x) = x.
We have:
x !0
lim
tan
x
x2
x!0
= lim
tan
x2
x2
x = xlim
!0
tan
x2
x2
x !0
lim
x = 0:
It means that the function tan x2 is infinitely small of higher order than x. There
are many functions which are infinitely small of higher order than x for x ! 0, e.g.
x52 ; sin7 x;
tan
x2007
.
x
But these functions have wildly different rates at which they
tends to zero and we need some fine tool for investigation of the “order of decrease” to
compare them.
We introduce following basic infinitely small functions
(x) = x a for x ! a (often a = 0)
(x) = x1
for x ! 1.
Using the positive powers n (x) of those functions we can create the infinite scale of
infinitely small functions, which will serve for investigation of complicated functions.
f (x)
= c is the finite nonzero number then the function f (x) is
x!a n (x)
called infinitely small of n-th order compared to (x).
Definition 4 If
3
lim
Simplification of computation of limits
(x()x; ). (xIt);can
(xbe
); (x) are infinitely small functions for x ! a , (x) (x);
(Let
x) easily shown that the following properties of infinitely small
functions hold
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Understanding and computing limits by means of infinitesimal quantities
1.
2.
(x)
(x)
= lim
(principle of substitution of equivalent quantities).
x!a (x) x!a (x)
If lim f (x) = K is the finite nonzero number then f (x) (x) K (x).
x! a
lim
3. If (x) (x);
(x) (x) then (x) (x) (transitivity property).
If we can choose instead of complicated (x); (x) the simpler quantities (x); (x)
then the process of computation of limits can be substantially simplified. For this aim
the following table can be extremely useful.
2 (x)
(x) 2
ln(1 + (x)) (x)
arctan(x) (x)
(1 + (x))p
1 p (x); p > 0
(x) (x)
tan(x) (x)
arcsin(x) (x)
a(x) 1 (x) ln a; a > 0
sin
1
cos
Table 1 Table of equivalent quantities ((x) is infinitely small for x ! 0).
For (x) = x we have the following simple equivalence relations
x sin x tan x arcsin x arctan x ln(1 + x) ex
1
m
mp1 + x
1
:
Using the principle of substitution of equivalent quantities and the above-mentioned
table we can compute many complicated limits, as will be illustrated on the following
examples.
x
sin 5
Example 3
x!0 ln(1 + 4x)
Example 4
lim
x 0 4
x !0
= 4 lim
cos
lim
ln cos x
p
! 1+x
x
x2
1
2
1
x !0
4 lim
x + arcsin2 x
3x
sin 2
x
= lim
x2 =2
x!0 x2
=
5
x ! 0 4x
= lim
=
5
=
4
x 5x; ln(1 + 4x) 4x.
sin 5
,
ln(1 + (cos
x
x =4
1))
2
2
=
:
arctan2
x
2
x
2
:
x! 0
x !0 3 x
3
In the solution we used the fact, that the sum of infinitely small quantities is equivalent
to the quantity of the smallest order (i.e. we can neglect the higher order terms in the
sum).
Example 5
lim
Example 6
sin 3 x ln(1 + 3x)
lim
p x!0 (arctan px)2 e5 3 x 1
p
= lim
=
p3 x 3x
3
p
= lim p
= .
3
x! ( x) 5 x
5
0
2
307
Journal of Information, Control and Management Systems, Vol. 4, (2007), No. 2
Now we will bring two examples, which can be efficiently and with great elegance solved
using infinitesimal quantities, but the classical solution using L’Hospital’s rule is very
cumbersome.
Example 7
=
=
x x2 ) + arcsin 2x 3x3
10
sin 3x + tan2 x + (ex
1)
ln(1 +
x !0
= lim
(
x x2 ) + 2x 3x3
3x + x2 + x10
x + cos 2x + + cos nx n
=
sin x2
(1
cos x) + (1
cos 2x) + + (1
cos nx)
=
lim
2
x !0
sin x
2
2
2 2
2
x =2 + (2x) =2 + + (n x )=2
1
x (12 + 22 + + n2 )
lim
=
lim
x !0
x2
2 x !0
x2
1 n(n + 1)(2n + 1)
n(n + 1)(2n + 1)
=
:
Example 8
=
x! 0
lim
2
x! 0
lim
:
=1
cos
6
=
12
The experience gained by authors shows that the students are highly motivated, because of ease of computations of limits. The knowledge of properties of infinitely small
quantities can be used by students later in the calculus for investigating the rate of growth
of functions, the convergence and divergence of series and improper integrals. The infinitesimal and infinite functions are of great importance also in the study of Computer
Science (complexity of algorithms, asymptotic analysis).
REFERENCES
[1] G. M. Fikhtengolc: Kurs differencialnogo i integralnogo ischislenia I, Fizmatgiz,
Moscow, 1969
[2] I. A. Maron: Differencialnoiye i integralnoiye ischisleniye v primerakh i zadachakh,
Nauka, Moscow, 1970
[3] I. A. Vinogradova, S. N. Olekhnik, V. A. Sadownichiy: Zadachi i uprazhneniya po
matematicheskomu analizu, MGU Press, Moscow, 1988
Acknowledgement
This research is supported by Ministry of Education development project IKT/0771352.