What are infinite small functions

However, a function can only be infinitesimal at a specific point. As shown in Figure 1, the function is infinitesimal only at point 0.

Figure 1. Infinitesimal function

If the limit of the quotient of two functions results in 1, the functions are said to be equivalent infinitesimals as x tends to point a.

\[\mathop(\lim )\limits_(x\to a) \frac(f(x))(g(x)) =1\]

Definition

If the functions f(x), g(x) are infinitesimal for $x > a$, then:

• A function f(x) is called infinitesimal of higher order with respect to g(x) if the following condition is satisfied:
\[\mathop(\lim )\limits_(x\to a) \frac(f(x))(g(x)) =0\]
• A function f(x) is called infinitesimal of order n with respect to g(x) if it is different from 0 and the limit is finite:
\[\mathop(\lim )\limits_(x\to a) \frac(f(x))(g^(n) (x)) =A\]

Example 1

The function $y=x^3$ is infinitesimal of higher order for x>0, in comparison with the function y=5x, since the limit of their ratio is 0, this is explained by the fact that the function $y=x^3$ tends to zero value faster:

\[\mathop(\lim )\limits_(x\to 0) \frac(x^(2) )(5x) =\frac(1)(5) \mathop(\lim )\limits_(x\to 0 ) x=0\]

Example 2

The functions y=x2-4 and y=x2-5x+6 are infinitesimals of the same order for x>2, since the limit of their ratio is not equal to 0:

\[\mathop(\lim )\limits_(x\to 2) \frac(x^(2) -4)(x^(2) -5x+6) =\mathop(\lim )\limits_(x\ to 2) \frac((x-2)(x+2))((x-2)(x-3)) =\mathop(\lim )\limits_(x\to 2) \frac((x+ 2))((x-3)) =\frac(4)(-1) =-4\ne 0\]

Properties of equivalent infinitesimals

1. The difference between two equivalent infinitesimals is an infinitesimal of higher order relative to each of them.
2. If from the sum of several infinitesimals of different orders we discard infinitesimals of higher orders, then the remaining part, called the main part, is equivalent to the entire sum.

From the first property it follows that equivalent infinitesimals can become approximately equal with an arbitrarily small relative error. Therefore, the sign ≈ is used both to denote the equivalence of infinitesimals and to write the approximate equality of their sufficiently small values.

When finding limits, it is very often necessary to use the replacement of equivalent functions for speed and convenience of calculations. The table of equivalent infinitesimals is presented below (Table 1).

The equivalence of the infinitesimals given in the table can be proven based on the equality:

\[\mathop(\lim )\limits_(x\to a) \frac(f(x))(g(x)) =1\]

Table 1

Example 3

Let us prove the equivalence of infinitesimal ln(1+x) and x.

Proof:

1. Let's find the limit of the ratio of quantities
\[\mathop(\lim )\limits_(x\to a) \frac(\ln (1+x))(x) \]
2. To do this, we apply the property of the logarithm:
\[\frac(\ln (1+x))(x) =\frac(1)(x) \ln (1+x)=\ln (1+x)^(\frac(1)(x) ) \] \[\mathop(\lim )\limits_(x\to a) \frac(\ln (1+x))(x) =\mathop(\lim )\limits_(x\to a) \ln (1+x)^(\frac(1)(x) ) \]
3. Knowing that the logarithmic function is continuous in its domain of definition, we can swap the sign of the limit and the logarithmic function:
\[\mathop(\lim )\limits_(x\to a) \frac(\ln (1+x))(x) =\mathop(\lim )\limits_(x\to a) \ln (1+ x)^(\frac(1)(x) ) =\ln \left(\mathop(\lim )\limits_(x\to a) (1+x)^(\frac(1)(x) ) \ right)\]
4. Since x is an infinitesimal quantity, the limit tends to 0. This means:
\[\mathop(\lim )\limits_(x\to a) \frac(\ln (1+x))(x) =\mathop(\lim )\limits_(x\to a) \ln (1+ x)^(\frac(1)(x) ) =\ln \left(\mathop(\lim )\limits_(x\to 0) (1+x)^(\frac(1)(x) ) \ right)=\ln e=1\]

(applied the second wonderful limit)

Infinitesimal functions.

We continue the educational series “Limits for Dummies”, which opened with articles Limits. Examples of solutions And Wonderful Limits. If this is your first time on the site, I recommend that you also read the lesson Methods for solving limits, which will significantly improve your student karma. In the third manual we looked at infinitely large functions, their comparison, and now it’s time to arm yourself with a magnifying glass so that after the Land of Giants you look into the Land of Lilliputians. I spent the New Year holidays in the cultural capital and returned in a very good mood, so the reading promises to be especially interesting.

This article will discuss in detail infinitesimal functions, which you have actually already encountered many times, and their comparison. Many events are closely related to invisible events near zero. wonderful limits, wonderful equivalences, and the practical part of the lesson is mainly devoted to calculating limits using remarkable equivalences.

Infinitesimal functions. Comparison of infinitesimals

What can I say... If there is a limit, then the function is called infinitesimal at a point.

The essential point of the statement is the fact that function can be infinitesimal only at a specific point .

Let's draw a familiar line:

This function infinitely small at a single point:
It should be noted that at the points “plus infinity” and “minus infinity” this same function will be narrower infinitely large: . Or in a more compact notation:

At all other points, the limit of the function will be equal to a finite number different from zero.

Thus, there is no such thing as "just an infinitesimal function" or "just an infinitely large function". A function can be infinitesimal or infinitely large only at a specific point .

! Note : For brevity, I will often say "infinitesimal function", meaning that it is infinitesimal at the point in question.

There can be several and even infinitely many such points. Let's draw some kind of non-frightening parabola:

The presented quadratic function is infinitesimal at two points - at “one” and at “two”:

As in the previous example, at infinity this function is infinitely large:

The meaning of double signs :

The notation means that when , and when .

The notation means that both at and at .
The commented principle of “deciphering” double signs is valid not only for infinities, but also for any end points, functions and a number of other mathematical objects.

And now the sine. This is an example where the function infinitely small at an infinite number of points:

Indeed, the sinusoid “stitches” the x-axis through each “pi”:

Note that the function is top/bottom bounded and there is no point at which it would be infinitely large, the sine can only lick its lips forever.

I'll answer a couple more simple questions:

Can a function be infinitesimal at infinity?

Certainly. There are a cartload of such specimens and a small cart.
An elementary example: . The geometric meaning of this limit, by the way, is illustrated in the article Graphs and properties of functions.

Can a function NOT BE infinitesimal?
(at any point domain of definition)

Yes. An obvious example is a quadratic function whose graph (parabola) does not intersect the axis. The opposite statement, by the way, is generally incorrect - the hyperbola from the previous question, although it does not intersect the x-axis, but infinitely small at infinity.

Comparison of infinitesimal functions

Let's construct a sequence that tends to zero and calculate several values ​​of the trinomial:

Obviously, as the “x” values ​​decrease, the function runs to zero faster than all others (its values ​​are circled in red). They say function than function , and higher order of smallness, how . But running fast in the Land of Lilliputians is not valor; “the tone is set” by the slowest dwarf, who, as befits a boss, goes to zero slowest of all. It depends on him how fast the amount will approach zero:

Figuratively speaking, the infinitesimal function “absorbs” everything else, which is especially clearly visible in the final result of the third line. Sometimes they say that lower order of smallness, how and their amount.

In the considered limit, all this, of course, does not matter much, because the result is still zero. However, the “heavyweight midgets” begin to play a fundamentally important role in limits with fractions. Let's start with examples that, albeit rarely, are found in real practical work:

Example 1

Calculate limit

There is uncertainty here, and from the introductory lesson about within the limits of functions Let’s remember the general principle of revealing this uncertainty: you need to factor the numerator and denominator, and then reduce something:

At the first step, we take out , in the numerator, and “x” in the denominator. In the second step, we reduce the numerator and denominator by “X,” thereby eliminating uncertainty. We indicate that the remaining “X’s” tend to zero, and we get the answer.

In the limit, the result is a steering wheel, therefore, the numerator function higher order of smallness than the denominator function. Or in short: . What does it mean? The numerator tends to zero faster, than the denominator, which is why it ended up being zero.

As is the case with infinitely large functions, the answer can be found out in advance. The technique is similar, but differs in that in the numerator and denominator you need to MENTALLY discard all terms with ELDER degrees, since, as noted above, slow dwarfs are of decisive importance:

Example 2

Calculate limit

Zero to zero... Let's find out the answer right away: MENTALLY let's discard everything elder terms (fast dwarfs) of the numerator and denominator:

The solution algorithm is exactly the same as in the previous example:

In this example denominator of higher order of smallness than numerator. As the "x" values ​​decrease, the slowest dwarf of the numerator (and of the entire limit) becomes a real monster in relation to its faster opponent. For example, if , then - already 40 times more... not yet a monster, of course, given the meaning of “X”, but already such a subject with a big beer belly.

And a very simple demonstration limit:

Example 3

Calculate limit

Let's find out the answer by MENTALLY throwing everything away elder numerator and denominator terms:

We decide:

The result is a finite number. The boss of the numerator is exactly twice as thick as the boss of the denominator. This is a situation where the numerator and denominator one order of smallness.

In fact, comparison of infinitesimal functions has long appeared in previous lessons:
(Example No. 4 of the lesson Limits. Examples of solutions);
(Example No. 17 of lesson Methods for solving limits) etc.

I remind you at the same time that “x” can tend not only to zero, but also to an arbitrary number, as well as to infinity.

What is fundamentally important in all the examples considered?

Firstly, the limit must exist at all at a given point. For example, there is no limit. If , then the numerator function is not defined at the point “plus infinity” (under the root it turns out infinitely large a negative number). Similar, seemingly fanciful examples are found in practice: unexpectedly, there is also a comparison of infinitesimal functions and “zero to zero” uncertainty. Indeed, if , then . …Solution? We get rid of the four-story fraction, get uncertainty and reveal it using the standard method.

Perhaps those starting to study the limits are drilled by the question: “How is this possible? There is an uncertainty of 0:0, but you can’t divide by zero!” Absolutely right, it’s impossible. Let's consider the same limit. The function is not defined at point zero. But this, generally speaking, is not required. important so that the function exists ANYWHERE infinitely close to zero point (or more strictly - at any infinitesimal neighborhood zero).

THE MOST IMPORTANT FEATURE OF LIMIT AS A CONCEPT

is that "x" infinitely close is approaching a certain point, but he is not “obligated” to “go there”! That is, for the existence of a limit of a function at a point doesn't matter, whether the function itself is defined there or not. You can read more about this in the article Cauchy limits, but for now let’s return to the topic of today’s lesson:

Secondly, the numerator and denominator functions must be infinitesimal at a given point. So, for example, the limit is from a completely different command, here the numerator function does not tend to zero: .

Let’s systematize information about comparing infinitesimal functions:

Let - infinitesimal functions at a point(i.e. at ) and there is a limit to their relationship. Then:

1) If , then the function higher order of smallness, how .
The simplest example: , that is, a cubic function of a higher order of smallness than a quadratic one.

2) If , then the function higher order of smallness, how .
The simplest example: , that is, a quadratic function of a higher order of smallness than a linear one.

3) If , where is a non-zero constant, then the functions have same order of smallness.
The simplest example: , in other words, the dwarf runs towards zero exactly twice as slow as , and the “distance” between them remains constant.

The most interesting special case is when . Such functions are called infinitesimal equivalent functions.

Before giving a basic example, let's talk about the term itself. Equivalence. This word has already been encountered in class. Methods for solving limits, in other articles and will appear more than once. What is equivalence? There is a mathematical definition of equivalence, logical, physical, etc., but let’s try to understand the essence itself.

Equivalence is equivalence (or equivalence) in some respect.. It's time to stretch your muscles and take a little break from higher mathematics. Now there is a good January frost outside, so it is very important to insulate well. Please go into the hallway and open the closet with clothes. Imagine that there are two identical sheepskin coats hanging there, which differ only in color. One is orange, the other is purple. From the point of view of their warming qualities, these sheepskin coats are equivalent. Both in the first and in the second sheepskin coat you will be equally warm, that is, the choice is equivalent, whether to wear orange or purple - without winning: “one to one equals one.” But from the point of view of safety on the road, sheepskin coats are no longer equivalent - the orange color is more visible to vehicle drivers, ... and the patrol will not stop, because with the owner of such clothes everything is clear. In this regard, we can consider that sheepskin coats are “of the same order of magnitude”, relatively speaking, an “orange sheepskin coat” is twice as “safe” as a “purple sheepskin coat” (“which is worse, but also noticeable in the dark”). And if you go out into the cold in just a jacket and socks, then the difference will be colossal, so the jacket and sheepskin coat are “of different orders of magnitude.”

...you're in trouble, you need to post it on Wikipedia with a link to this lesson =) =) =)

The obvious example of infinitesimal equivalent functions is familiar to you - these are the functions first remarkable limit .

Let us give a geometric interpretation of the first remarkable limit. Let's make the drawing:

Well, the strong male friendship of the charts is visible even to the naked eye. A Even my own mother couldn’t tell them apart. Thus, if , then the functions are infinitesimal and equivalent. What if the difference is negligible? Then in the limit sine is above you can replace"X": , or “x” below with a sine: . In fact, it turned out to be a geometric proof of the first remarkable limit =)

Similarly, by the way, one can illustrate any wonderful limit, which is equal to one.

! Attention! Equivalence of objects does not imply coincidence of objects! Orange and purple sheepskin coats are equivalently warm, but they are different sheepskin coats. The functions are practically indistinguishable near zero, but these are two different functions.

Designation: Equivalence is indicated by a tilde.
For example: – “the sine of x is equivalent to x” if .

A very important conclusion follows from the above: if two infinitesimal functions are equivalent, then one can be replaced by the other. This technique is widely used in practice, and right now we will see how:

Remarkable equivalences within

To solve practical examples you will need table of remarkable equivalences. A student cannot live by a single polynomial, so the field of further activity will be very wide. First, using the theory of infinitesimal equivalent functions, let’s click through the examples of the first part of the lesson Remarkable limits. Examples of solutions, in which the following limits were found:

1) Let's solve the limit. Let us replace the infinitesimal numerator function with the equivalent infinitesimal function:

Why is such a replacement possible? Because infinitely close to zero the graph of the function practically coincides with the graph of the function.

In this example we used table equivalence where . It is convenient that the “alpha” parameter can be not only “x”, but also a complex function, which tends to zero.

2) Let's find the limit. In the denominator we use the same equivalence, in this case:

Please note that the sine was initially located under the square, so in the first step it is also necessary to place it entirely under the square.

Let’s not forget about the theory: in the first two examples, finite numbers were obtained, which means numerators and denominators of the same order of smallness.

3) Let's find the limit. Let us replace the infinitesimal numerator function with the equivalent function , Where :

Here numerator of higher order of smallness than denominator. Lilliput (and the equivalent Lilliputian) reaches zero faster than .

4) Let's find the limit. Let us replace the infinitesimal numerator function with an equivalent function, where:

And here, on the contrary, the denominator higher order of smallness, than the numerator, the dwarf escapes to zero faster than the dwarf (and its equivalent dwarf).

Should remarkable equivalences be used in practice? It should, but not always. Thus, it is not advisable to solve not very complex limits (like those just considered) through remarkable equivalences. You may be accused of hackwork and forced to solve them in a standard way using trigonometric formulas and the first wonderful limit. However, using the tool in question, it is very beneficial to check the solution or even immediately find out the correct answer. Example No. 14 of the lesson is typical Methods for solving limits:

In the final version, it is advisable to draw up a rather large complete solution with a change of variable. But the ready answer lies on the surface - we mentally use equivalence: .

And once again the geometric meaning: why is it permissible to replace the function in the numerator with the function ? Infinitely close near zero their graphs can only be distinguished under a powerful microscope.

In addition to checking the solution, remarkable equivalences are used in two more cases:

– when the example is quite complex or generally unsolvable in the usual way;
– when remarkable equivalences need to be applied by condition.

Let's consider more meaningful tasks:

Example 4

Find the limit

The agenda is zero-to-zero uncertainty and the situation is borderline: the solution can be carried out in a standard way, but there will be a lot of transformations. From my point of view, it is quite appropriate to use the wonderful equivalences here:

Let us replace infinitesimal functions with equivalent functions. At :

That's all!

The only technical nuance: initially the tangent was squared, so after the replacement the argument must also be squared.

Example 5

Find the limit

This limit is solvable through trigonometric formulas and wonderful limits, but the solution again will not be very pleasant. This is an example for you to solve on your own, be especially careful when converting the numerator. If there is any confusion about degrees, represent it as a product:

Example 6

Find the limit

But this is a difficult case when it is very difficult to carry out a solution in a standard way. Let's use some wonderful equivalences:

Let us replace infinitesimals with equivalent ones. At :

The result is infinity, which means the denominator is of a higher order of smallness than the numerator.

Practice went briskly without outerwear =)

Example 7

Find the limit

This is an example for you to solve on your own. Think about how to deal with the logarithm ;-)

It is not uncommon for remarkable equivalences to be used in combination with other methods for solving limits:

Example 8

Find the limit of a function using equivalent infinitesimals and other transformations

Note that there are some remarkable equivalences required here.

We decide:

In the first step we use remarkable equivalences. At :

Everything is clear with sine: . What to do with the logarithm? Let's represent the logarithm in the form and apply the equivalence. As you understand, in this case and

In the second step, we will apply the technique discussed in the lesson.

As has been shown, the sum, difference and product of infinitesimal functions are infinitesimal, but the same cannot be said about the particular: dividing one infinitesimal by another can give different results.

For example, if a(x) = 2x, p(x) = 3x, then

If a(x) = x 2, P (l;) = x 3, then

It is advisable to introduce rules for comparing infinitesimal functions using appropriate terminology.

Let at X -» A the functions a(x) and p(.v) are infinitesimal. Then the following options for their comparison are distinguished, depending on the value With limit at a point A their relationship:

• 1. If With= I, then a(x) and P(x) are equivalent infinitesimals: a(x) - p(x).
• 2. If With= 0, then a(x) is an infinitesimal of a higher order than p(x) (or has a higher order of smallness).
• 3. If With = d* 0 (d- number), then Oh) and P(x) are infinitesimals of the same order.

Often it is not enough to know that one infinitesimal in relation to another is an infinitesimal of a higher order of smallness; one also needs to estimate the magnitude of this order. Therefore the following rule is used.

4. If Mm - - =d*0, then a(x) is an infinitesimal of the lth order with respect to - *->lp"(*)

literally P(x). In this case, use the symbol o "o" small"): a(x) = o(P(x)).

Note that similar rules for comparing infinitesimal functions for x -»oo are valid, X-" -oo, X-> +«>, as well as in the case of one-sided limits at x -» A left and right.

One important property follows from the comparison rules:

then there is a limit lim 1, and both of these limits are equal.

In a number of cases, the proven statement simplifies the calculation of limits and carrying out estimates.

Let's look at a few examples.

1. Sin functions X And X at X-» 0 are equivalent to infinitesimals due to the limit (8.11), i.e. at X -> 0 sin X ~ X.

Indeed, we have:

• 2. Sin functions kh and sin X are at q: -> 0 infinitesimals of the same order, since
• 3. Function a(x) = cos ah - cos bx (a * b) is at X-» 0 infinitesimal of the second order of smallness with respect to infinitesimal.v, since

Example 7. Find lim

*-+° x + x"

Solution. Since sin kh ~ kh And X + x 2 ~ X:

Comparison of infinitely large functions

For infinitely large functions, similar comparison rules also apply, with the only difference being that for them, instead of the term “order of smallness,” the term “order of growth” is used.

Let us explain what has been said with examples.

1. Functions f(x) = (2 + x)/x and g(x) = 2/x at X-» 0 are equivalent to infinitely large, since

Function data /(X) and #(*) have the same growth order.

2. Let’s compare the orders of growth of functions f(x) = 2x?+I and g(x)= x 3 + X at X-> why find the limit of their ratio:

It follows that the function g(x) has a higher order of growth than the function / (x).

3. Infinitely large functions for x -» °o /(x) = 3x 3 + X and #(x) = x 3 - 4x 2 have the same order of growth, since

4. The function /(x) = x 3 + 2x + 3 is infinitely large for x -»

third order with respect to an infinitely large function g(x) = x - I, since

Test

Discipline: Higher mathematics

Topic: Limits. Comparison of infinitesimals

1. Limit of number sequence

2. Function limit

3. The second wonderful limit

4. Comparison of infinitesimal quantities

Literature

1. Limit of number sequence

The solution of many mathematical and applied problems leads to a sequence of numbers specified in a certain way. Let's find out some of their properties.

Definition 1.1. If for every natural number

Based on Definition 1, it is clear that a number sequence always contains an infinite number of elements. The study of various number sequences shows that as the number increases, their members behave differently. They may increase or decrease indefinitely, may constantly approach a certain number, or may not show any pattern at all.

Definition 1.2. Number

A sequence that has a limit is called convergent. In this case they write

Obviously, to clarify the question of the convergence of a numerical sequence, it is necessary to have a criterion that would be based only on the properties of its elements.

Theorem 1.1.(Cauchy's theorem on the convergence of a number sequence). In order for a number sequence to be convergent, it is necessary and sufficient that for any number

Proof. Necessity. Given that the number sequence

It follows that

Adequacy. It is given that

Let's take an arbitrary

It follows that