Explanation of the definition of limit. Universal definition of the limit of a function according to Hein and Cauchy. Limit of a function

Are you here: Eye and face protection

Function limit- number a will be the limit of some variable quantity if, in the process of its change, this variable quantity indefinitely approaches a.

Or in other words, the number A is the limit of the function y = f(x) at the point x 0, if for any sequence of points from the domain of definition of the function , not equal x 0, and which converges to the point x 0 (lim x n = x0), the sequence of corresponding function values converges to the number A.

The graph of a function whose limit, given an argument that tends to infinity, is equal to L:

Meaning A is limit (limit value) of the function f(x) at the point x 0 in case for any sequence of points , which converges to x 0, but which does not contain x 0 as one of its elements (i.e. in the punctured vicinity x 0), sequence of function values converges to A.

Limit of a Cauchy function.

Meaning A will be limit of the function f(x) at the point x 0 if for any non-negative number taken in advance ε the corresponding non-negative number will be found δ = δ(ε) such that for each argument x, satisfying the condition 0 < | x - x0 | < δ , the inequality will be satisfied | f(x)A |< ε .

It will be very simple if you understand the essence of the limit and the basic rules for finding it. What is the limit of the function f (x) at x striving for a equals A, is written like this:

Moreover, the value to which the variable tends x, can be not only a number, but also infinity (∞), sometimes +∞ or -∞, or there may be no limit at all.

To understand how find the limits of a function, it is best to look at examples of solutions.

It is necessary to find the limits of the function f (x) = 1/x at:

x→ 2, x→ 0, x→ ∞.

Let's find a solution to the first limit. To do this, you can simply substitute x the number it tends to, i.e. 2, we get:

Let's find the second limit of the function. Here substitute pure 0 instead x it is impossible, because You cannot divide by 0. But we can take values close to zero, for example, 0.01; 0.001; 0.0001; 0.00001 and so on, and the value of the function f (x) will increase: 100; 1000; 10000; 100,000 and so on. Thus, it can be understood that when x→ 0 the value of the function that is under the limit sign will increase without limit, i.e. strive towards infinity. Which means:

Regarding the third limit. The same situation as in the previous case, it is impossible to substitute ∞ in its purest form. We need to consider the case of unlimited increase x. We substitute 1000 one by one; 10000; 100000 and so on, we have that the value of the function f (x) = 1/x will decrease: 0.001; 0.0001; 0.00001; and so on, tending to zero. That's why:

It is necessary to calculate the limit of the function

Starting to solve the second example, we see uncertainty. From here we find the highest degree of the numerator and denominator - this is x 3, we take it out of brackets in the numerator and denominator and then reduce it by:

Answer

The first step in finding this limit, substitute the value 1 instead x, resulting in uncertainty. To solve it, let’s factorize the numerator and do this using the method of finding the roots of a quadratic equation x 2 + 2x - 3:

D = 2 2 - 4*1*(-3) = 4 +12 = 16→ √ D=√16 = 4

x 1.2 = (-2±4)/2→ x 1 = -3;x 2= 1.

So the numerator will be:

Answer

This is the definition of its specific value or a certain area where the function falls, which is limited by the limit.

To solve limits, follow the rules:

Having understood the essence and main rules for solving the limit, you will get a basic understanding of how to solve them.

Limits give all mathematics students a lot of trouble. To solve a limit, sometimes you have to use a lot of tricks and choose from a variety of solution methods exactly the one that is suitable for a particular example.

In this article we will not help you understand the limits of your capabilities or comprehend the limits of control, but we will try to answer the question: how to understand limits in higher mathematics? Understanding comes with experience, so at the same time we will give several detailed examples of solving limits with explanations.

The concept of limit in mathematics

The first question is: what is this limit and the limit of what? We can talk about the limits of numerical sequences and functions. We are interested in the concept of the limit of a function, since this is what students most often encounter. But first, the most general definition of a limit:

Let's say there is some variable value. If this value in the process of change unlimitedly approaches a certain number a , That a – the limit of this value.

For a function defined in a certain interval f(x)=y such a number is called a limit A , which the function tends to when X , tending to a certain point A . Dot A belongs to the interval on which the function is defined.

It sounds cumbersome, but it is written very simply:

Lim- from English limit- limit.

There is also a geometric explanation for determining the limit, but here we will not delve into the theory, since we are more interested in the practical rather than the theoretical side of the issue. When we say that X tends to some value, this means that the variable does not take on the value of a number, but approaches it infinitely close.

Let's give a specific example. The task is to find the limit.

To solve this example, we substitute the value x=3 into a function. We get:

By the way, if you are interested in basic operations on matrices, read a separate article on this topic.

In examples X can tend to any value. It can be any number or infinity. Here's an example when X tends to infinity:

Intuitively, the larger the number in the denominator, the smaller the value the function will take. So, with unlimited growth X meaning 1/x will decrease and approach zero.

As you can see, to solve the limit, you just need to substitute the value to strive for into the function X . However, this is the simplest case. Often finding the limit is not so obvious. Within the limits there are uncertainties of the type 0/0 or infinity/infinity . What to do in such cases? Resort to tricks!

Uncertainties within

Uncertainty of the form infinity/infinity

Let there be a limit:

If we try to substitute infinity into the function, we will get infinity in both the numerator and the denominator. In general, it is worth saying that there is a certain element of art in resolving such uncertainties: you need to notice how you can transform the function in such a way that the uncertainty goes away. In our case, we divide the numerator and denominator by X in the senior degree. What will happen?

From the example already discussed above, we know that terms containing x in the denominator will tend to zero. Then the solution to the limit is:

To resolve type uncertainties infinity/infinity divide the numerator and denominator by X to the highest degree.

By the way! For our readers there is now a 10% discount on any type of work

Another type of uncertainty: 0/0

As always, substituting values into the function x=-1 gives 0 in the numerator and denominator. Look a little more closely and you will notice that we have a quadratic equation in the numerator. Let's find the roots and write:

Let's reduce and get:

So, if you are faced with type uncertainty 0/0 – factor the numerator and denominator.

To make it easier for you to solve examples, we present a table with the limits of some functions:

L'Hopital's rule within

Another powerful way to eliminate both types of uncertainty. What is the essence of the method?

If there is uncertainty in the limit, take the derivative of the numerator and denominator until the uncertainty disappears.

L'Hopital's rule looks like this:

Important point : the limit in which the derivatives of the numerator and denominator stand instead of the numerator and denominator must exist.

And now - a real example:

There is typical uncertainty 0/0 . Let's take the derivatives of the numerator and denominator:

Voila, uncertainty is resolved quickly and elegantly.

We hope that you will be able to usefully apply this information in practice and find the answer to the question “how to solve limits in higher mathematics.” If you need to calculate the limit of a sequence or the limit of a function at a point, and there is absolutely no time for this work, contact a professional student service for a quick and detailed solution.

Let the function y = ƒ (x) be defined in some neighborhood of the point x o, except, perhaps, the point x o itself.

Let us formulate two equivalent definitions of the limit of a function at a point.

Definition 1 (in the “language of sequences”, or according to Heine).

The number A is called the limit of the function y=ƒ(x) in the furnace x 0 (or at x® x o), if for any sequence of permissible values of the argument x n, n є N (x n ¹ x 0), converging to x, the sequence of corresponding values of the function ƒ(x n), n є N, converges to the number A

In this case they write
or ƒ(x)->A at x→x o. The geometric meaning of the limit of a function: means that for all points x that are sufficiently close to the point xo, the corresponding values of the function differ as little as desired from the number A.

Definition 2 (in the “language of ε”, or according to Cauchy).

A number A is called the limit of a function at a point x o (or at x→x o) if for any positive ε there is a positive number δ such that for all x¹ x o satisfying the inequality |x-x o |<δ, выполняется неравенство |ƒ(х)-А|<ε.

Geometric meaning of the limit of a function:

if for any ε-neighborhood of point A there is a δ-neighborhood of the point x o such that for all x1 xo from this δ-neighborhood the corresponding values of the function ƒ(x) lie in the ε-neighborhood of the point A. In other words, the points of the graph of the function y = ƒ(x) lie inside a strip of width 2ε, bounded by straight lines y=A+ ε, y=A-ε (see Fig. 110). Obviously, the value of δ depends on the choice of ε, so we write δ=δ(ε).

<< Пример 16.1

Prove that

Solution: Take an arbitrary ε>0, find δ=δ(ε)>0 such that for all x satisfying the inequality |x-3|< δ, выполняется неравенство |(2х-1)-5|<ε, т. е. |х-3|<ε.

Taking δ=ε/2, we see that for all x satisfying the inequality |x-3|< δ, выполняется неравенство |(2х-1)-5|<ε. Следовательно, lim(2x-1)=5 при х –>3.

<< Пример 16.2

16.2. One-sided limits

In defining the limit of a function, it is considered that x tends to x 0 in any way: remaining less than x 0 (to the left of x 0), greater than x o (to the right of x o), or oscillating around the point x 0.

There are cases when the method of approximating the argument x to x o significantly affects the value of the function limit. Therefore, the concepts of one-sided limits are introduced.

The number A 1 is called the limit of the function y=ƒ(x) on the left at the point x o if for any number ε>0 there is a number δ=δ(ε)> 0 such that at x є (x 0 -δ;x o), the inequality |ƒ(x)-A|<ε. Предел слева записывают так: limƒ(х)=А при х–>x 0 -0 or briefly: ƒ(x o- 0) = A 1 (Dirichlet notation) (see Fig. 111).

The limit of the function on the right is determined similarly; we write it using symbols:

Briefly, the limit on the right is denoted by ƒ(x o +0)=A.

The left and right limits of a function are called one-sided limits. Obviously, if exists, then both one-sided limits exist, and A = A 1 = A 2.

The converse is also true: if both limits ƒ(x 0 -0) and ƒ(x 0 +0) exist and they are equal, then there is a limit and A = ƒ(x 0 -0).

If A 1 ¹ A 2, then this chapel does not exist.

16.3. Limit of the function at x ® ∞

Let the function y=ƒ(x) be defined in the interval (-∞;∞). The number A is called limit of the functionƒ(x) at x→ ∞ , if for any positive number ε there is a number M=M()>0 such that for all x satisfying the inequality |x|>M the inequality |ƒ(x)-A|<ε. Коротко это определение можно записать так:

The geometric meaning of this definition is as follows: for " ε>0 $ M>0, that for x є(-∞; -M) or x є(M; +∞) the corresponding values of the function ƒ(x) fall into the ε-neighborhood of point A , that is, the points of the graph lie in a strip of width 2ε, limited by the straight lines y=A+ε and y=A-ε (see Fig. 112).

16.4. Infinitely large function (b.b.f.)

The function y=ƒ(x) is called infinitely large for x→x 0 if for any number M>0 there is a number δ=δ(M)>0, which for all x satisfying the inequality 0<|х-хо|<δ, выполняется неравенство |ƒ(х)|>M.

For example, the function y=1/(x-2) is b.b.f. for x->2.

If ƒ(x) tends to infinity as x→x o and takes only positive values, then they write

if only negative values, then

The function y=ƒ(x), defined on the entire number line, called infinitely large as x→∞, if for any number M>0 there is a number N=N(M)>0 such that for all x satisfying the inequality |x|>N, the inequality |ƒ(x)|>M holds. Short:

For example, y=2x has b.b.f. as x→∞.

Note that if the argument x, tending to infinity, takes only natural values, i.e. xєN, then the corresponding b.b.f. becomes an infinitely large sequence. For example, the sequence v n =n 2 +1, n є N, is an infinitely large sequence. Obviously, every b.b.f. in a neighborhood of a point x o is unbounded in this neighborhood. The converse is not true: an unbounded function may not be b.b.f. (For example, y=xsinx.)

However, if limƒ(x)=A for x→x 0, where A is a finite number, then the function ƒ(x) is limited in the vicinity of the point x o.

Indeed, from the definition of the limit of a function it follows that as x→ x 0 the condition |ƒ(x)-A|<ε. Следовательно, А-ε<ƒ(х)<А+ε при х є (х о -ε; х о +ε), а это и означает, что функция ƒ (х) ограничена.

By proving the properties of the limit of a function, we were convinced that nothing was really required from the punctured neighborhoods in which our functions were defined and which arose in the process of proof, except for the properties indicated in the introduction to the previous paragraph 2. This circumstance serves as a justification for identifying the following mathematical object.

A. Base; definition and basic examples

Definition 11. A collection B of subsets of a set X will be called a base in the set X if two conditions are met:

In other words, the elements of collection B are non-empty sets, and the intersection of any two of them contains some element from the same collection.

Let us indicate some of the most commonly used bases in analysis.

If then instead they write and say that x tends to a from the right or from the side of larger values (respectively, from the left or from the side of smaller values). When a short record is accepted instead

The entry will be used instead of She means that a; tends over the set E to a, remaining greater (smaller) than a.

then instead they write and say that x tends to plus infinity (respectively, to minus infinity).

The entry will be used instead

When instead of (if this does not lead to a misunderstanding) we will, as is customary in the theory of the limit of a sequence, write

Note that all of the listed bases have the peculiarity that the intersection of any two elements of the base is itself an element of this base, and not only contains some element of the base. We will encounter other bases when studying functions that are not specified on the number axis.

Note also that the term “base” used here is a short designation of what is called in mathematics “filter basis”, and the base limit introduced below is the most essential part for analysis of the concept of filter limit created by the modern French mathematician A. Cartan

b. Function limit by base

Definition 12. Let be a function on the set X; B is a base in X. A number is called the limit of a function with respect to base B if for any neighborhood of point A there is an element of the base whose image is contained in the neighborhood

If A is the limit of a function with respect to base B, then write

Let us repeat the definition of the limit by base in logical symbolism:

Since we are now looking at functions with numeric values, it is useful to keep in mind the following form of this basic definition:

In this formulation, instead of an arbitrary neighborhood V (A), a symmetric (with respect to point A) neighborhood (e-neighborhood) is taken. The equivalence of these definitions for real-valued functions follows from the fact that, as already mentioned, any neighborhood of a point contains some symmetric neighborhood of the same point (perform the proof in full!).

We have given a general definition of the limit of a function over a base. Above we discussed examples of the most commonly used databases in analysis. In a specific problem where one or another of these bases appears, it is necessary to be able to decipher the general definition and write it down for a specific base.

Considering examples of bases, we, in particular, introduced the concept of a neighborhood of infinity. If we use this concept, then in accordance with the general definition of the limit it is reasonable to accept the following conventions:

or, what is the same,

Usually we mean a small value. This is, of course, not the case in the above definitions. In accordance with accepted conventions, for example, we can write

In order for all the theorems on limits that we proved in paragraph 2 for a special base to be considered proven in the general case of a limit over an arbitrary base, it is necessary to give the appropriate definitions: finally constant, finally bounded and infinitesimal for a given base of functions.

Definition 13. A function is said to be finally constant with base B if there exists a number and an element of the base such that at any point

Definition 14. A function is called bounded with base B or finally bounded with base B if there exists a number c and an element of the base at any point of which

Definition 15. A function is said to be infinitesimal with base B if

After these definitions and the basic observation that to prove limit theorems, only the properties of the base are needed, we can assume that all the properties of the limit established in paragraph 2 are valid for limits on any base.

In particular, we can now talk about the limit of a function at or at or at

In addition, we have ensured that we can apply the theory of limits in the case where functions are not defined on numerical sets; this will prove especially valuable in the future. For example, the length of a curve is a numerical function defined on a certain class of curves. If we know this function on broken lines, then by passing to the limit we determine it for more complex curves, for example, for a circle.

At the moment, the main benefit of the observation made and the concept of a base introduced in connection with it is that they save us from checks and formal proofs of limit theorems for each specific type of limit passages or, in our current terminology, for each specific type bases

In order to finally become familiar with the concept of a limit over an arbitrary base, we will carry out proofs of further properties of the limit of a function in a general form.

The formulation of the main theorems and properties of the limit of a function is given. Definitions of finite and infinite limits at finite points and at infinity (two-sided and one-sided) according to Cauchy and Heine are given. Arithmetic properties are considered; theorems related to inequalities; Cauchy convergence criterion; limit of a complex function; properties of infinitesimal, infinitely large and monotonic functions. The definition of a function is given.

Content

Second definition according to Cauchy

The limit of a function (according to Cauchy) as its argument x tends to x 0 is a finite number or point at infinity a for which the following conditions are met:
1) there is such a punctured neighborhood of the point x 0 , on which the function f (x) determined;
2) for any neighborhood of the point a belonging to , there is such a punctured neighborhood of the point x 0 , on which the function values belong to the selected neighborhood of point a:
at .

Here a and x 0 can also be either finite numbers or points at infinity. Using the logical symbols of existence and universality, this definition can be written as follows:
.

If we take the left or right neighborhood of an end point as a set, we obtain the definition of a Cauchy limit on the left or right.

Theorem
The Cauchy and Heine definitions of the limit of a function are equivalent.
Proof

Applicable neighborhoods of points

Then, in fact, the Cauchy definition means the following.
For any positive numbers , there are numbers , so that for all x belonging to the punctured neighborhood of the point : , the values of the function belong to the neighborhood of the point a: ,
Where , .

This definition is not very convenient to work with, since neighborhoods are defined using four numbers. But it can be simplified by introducing neighborhoods with equidistant ends. That is, you can put , . Then we will get a definition that is easier to use when proving theorems. Moreover, it is equivalent to the definition in which arbitrary neighborhoods are used. The proof of this fact is given in the section “Equivalence of Cauchy definitions of the limit of a function”.

Then we can give a unified definition of the limit of a function at finite and infinitely distant points:
.
Here for endpoints
; ;
.
Any neighborhood of points at infinity is punctured:
; ; .

Finite limits of function at end points

The number a is called the limit of the function f (x) at point x 0 , If
1) the function is defined on some punctured neighborhood of the end point;
2) for any there exists such that depends on , such that for all x for which , the inequality holds
.

Using the logical symbols of existence and universality, the definition of the limit of a function can be written as follows:
.

One-sided limits.
Left limit at a point (left-sided limit):
.
Right limit at a point (right-hand limit):
.
The left and right limits are often denoted as follows:
; .

Finite limits of a function at points at infinity

Limits at points at infinity are determined in a similar way.
.
.
.

Infinite Function Limits

You can also introduce definitions of infinite limits of certain signs equal to and :
.
.

Properties and theorems of the limit of a function

We further assume that the functions under consideration are defined in the corresponding punctured neighborhood of the point , which is a finite number or one of the symbols: . It can also be a one-sided limit point, that is, have the form or . The neighborhood is two-sided for a two-sided limit and one-sided for a one-sided limit.

Basic properties

If the values of the function f (x) change (or make undefined) a finite number of points x 1, x 2, x 3, ... x n, then this change will not affect the existence and value of the limit of the function at an arbitrary point x 0 .

If there is a finite limit, then there is a punctured neighborhood of the point x 0 , on which the function f (x) limited:
.

Let the function have at point x 0 finite non-zero limit:
.
Then, for any number c from the interval , there is such a punctured neighborhood of the point x 0 , what for ,
, If ;
, If .

If, on some punctured neighborhood of the point, , is a constant, then .

If there are finite limits and and on some punctured neighborhood of the point x 0
,
That .

If , and on some neighborhood of the point
,
That .
In particular, if in some neighborhood of a point
,
then if , then and ;
if , then and .

If on some punctured neighborhood of a point x 0 :
,
and there are finite (or infinite of a certain sign) equal limits:
, That
.

Proofs of the main properties are given on the page
"Basic properties of the limit of a function."

Let the functions and be defined in some punctured neighborhood of the point . And let there be finite limits:
And .
And let C be a constant, that is, a given number. Then
;
;
;
, If .

If, then.

Proofs of arithmetic properties are given on the page
"Arithmetic properties of the limit of a function".

Cauchy criterion for the existence of a limit of a function

Theorem
In order for a function defined on some punctured neighborhood of a finite or at infinity point x 0 , had a finite limit at this point, it is necessary and sufficient that for any ε > 0 there was such a punctured neighborhood of the point x 0 , that for any points and from this neighborhood, the following inequality holds:
.

Limit of a complex function

Theorem on the limit of a complex function
Let the function have a limit and map a punctured neighborhood of a point onto a punctured neighborhood of a point. Let the function be defined on this neighborhood and have a limit on it.
Here are the final or infinitely distant points: . Neighborhoods and their corresponding limits can be either two-sided or one-sided.
Then there is a limit of a complex function and it is equal to:
.

The limit theorem of a complex function is applied when the function is not defined at a point or has a value different from the limit. To apply this theorem, there must be a punctured neighborhood of the point where the set of values of the function does not contain the point:
.

If the function is continuous at point , then the limit sign can be applied to the argument of the continuous function:
.
The following is a theorem corresponding to this case.

Theorem on the limit of a continuous function of a function
Let there be a limit of the function g (x) as x → x 0 , and it is equal to t 0 :
.
Here is point x 0 can be finite or infinitely distant: .
And let the function f (t) continuous at point t 0 .
Then there is a limit of the complex function f (g(x)), and it is equal to f (t 0):
.

Proofs of the theorems are given on the page
"Limit and continuity of a complex function".

Infinitesimal and infinitely large functions

Infinitesimal functions

Definition
A function is said to be infinitesimal if
.

Sum, difference and product of a finite number of infinitesimal functions at is an infinitesimal function at .

Product of a function bounded on some punctured neighborhood of the point , to an infinitesimal at is an infinitesimal function at .

In order for a function to have a finite limit, it is necessary and sufficient that
,
where is an infinitesimal function at .

"Properties of infinitesimal functions".

Infinitely large functions

Definition
A function is said to be infinitely large if
.

The sum or difference of a bounded function, on some punctured neighborhood of the point , and an infinitely large function at is an infinitely large function at .

If the function is infinitely large for , and the function is bounded on some punctured neighborhood of the point , then
.

If the function , on some punctured neighborhood of the point , satisfies the inequality:
,
and the function is infinitesimal at:
, and (on some punctured neighborhood of the point), then
.

Proofs of the properties are presented in section
"Properties of infinitely large functions".

Relationship between infinitely large and infinitesimal functions

From the two previous properties follows the connection between infinitely large and infinitesimal functions.

If a function is infinitely large at , then the function is infinitesimal at .

If a function is infinitesimal for , and , then the function is infinitely large for .

The relationship between an infinitesimal and an infinitely large function can be expressed symbolically:
, .

If an infinitesimal function has a certain sign at , that is, it is positive (or negative) on some punctured neighborhood of the point , then this fact can be expressed as follows:
.
In the same way, if an infinitely large function has a certain sign at , then they write:
.

Then the symbolic connection between infinitely small and infinitely large functions can be supplemented with the following relations:
, ,
, .

Additional formulas relating infinity symbols can be found on the page
"Points at infinity and their properties."

Limits of monotonic functions

Definition
A function defined on some set of real numbers X is called strictly increasing, if for all such that the following inequality holds:
.
Accordingly, for strictly decreasing function the following inequality holds:
.
For non-decreasing:
.
For non-increasing:
.

It follows that a strictly increasing function is also non-decreasing. A strictly decreasing function is also non-increasing.

The function is called monotonous, if it is non-decreasing or non-increasing.

Theorem
Let the function not decrease on the interval where .
If it is bounded above by the number M: then there is a finite limit. If not limited from above, then .
If it is limited from below by the number m: then there is a finite limit. If not limited from below, then .

If points a and b are at infinity, then in the expressions the limit signs mean that .
This theorem can be formulated more compactly.

Let the function not decrease on the interval where . Then there are one-sided limits at points a and b:
;
.

A similar theorem for a non-increasing function.

Let the function not increase on the interval where . Then there are one-sided limits:
;
.

The proof of the theorem is presented on the page
"Limits of monotonic functions".

Function Definition

Function y = f (x) is a law (rule) according to which each element x of the set X is associated with one and only one element y of the set Y.

Element x ∈ X called function argument or independent variable.
Element y ∈ Y called function value or dependent variable.

The set X is called domain of the function.
Set of elements y ∈ Y, which have preimages in the set X, is called area or set of function values.

The actual function is called limited from above (from below), if there is a number M such that the inequality holds for all:
.
The number function is called limited, if there is a number M such that for all:
.

Top edge or exact upper bound A real function is called the smallest number that limits its range of values from above. That is, this is a number s for which, for everyone and for any, there is an argument whose function value exceeds s′: .
The upper bound of a function can be denoted as follows:
.

Respectively bottom edge or exact lower limit A real function is called the largest number that limits its range of values from below. That is, this is a number i for which, for everyone and for any, there is an argument whose function value is less than i′: .
The infimum of a function can be denoted as follows:
.

References:
L.D. Kudryavtsev. Course of mathematical analysis. Volume 1. Moscow, 2003.
CM. Nikolsky. Course of mathematical analysis. Volume 1. Moscow, 1983.