The theory of limits is one of the branches of mathematical analysis. The question of solving limits is quite extensive, since there are dozens of methods for solving limits of various types. There are dozens of nuances and tricks that allow you to solve this or that limit. Nevertheless, we will still try to understand the main types of limits that are most often encountered in practice.

Let's start with the very concept of a limit. But first, a brief historical background. There lived a Frenchman, Augustin Louis Cauchy, in the 19th century, who gave strict definitions to many of the concepts of matan and laid its foundations. It must be said that this respected mathematician was, is, and will be in the nightmares of all students of physics and mathematics departments, since he proved a huge number of theorems of mathematical analysis, and one theorem is more lethal than the other. In this regard, we will not consider yet determination of the Cauchy limit, but let's try to do two things:

1. Understand what a limit is.
2. Learn to solve the main types of limits.

I apologize for some unscientific explanations, it is important that the material is understandable even to a teapot, which, in fact, is the task of the project.

So what is the limit?

And just an example of why to shaggy grandma....

Any limit consists of three parts:

1) The well-known limit icon.
2) Entries under the limit icon, in this case . The entry reads “X tends to one.” Most often - exactly, although instead of “X” in practice there are other variables. In practical tasks, the place of one can be absolutely any number, as well as infinity ().
3) Functions under the limit sign, in this case .

The recording itself reads like this: “the limit of a function as x tends to unity.”

Let's look at the next important question - what does the expression “x” mean? strives to one"? And what does “strive” even mean?
The concept of a limit is a concept, so to speak, dynamic. Let's build a sequence: first , then , , …, , ….
That is, the expression “x strives to one” should be understood as follows: “x” consistently takes on the values which approach unity infinitely close and practically coincide with it.

How to solve the above example? Based on the above, you just need to substitute one into the function under the limit sign:

So, the first rule: When given any limit, first we simply try to plug the number into the function.

We have considered the simplest limit, but these also occur in practice, and not so rarely!

Example with infinity:

Let's figure out what it is? This is the case when it increases without limit, that is: first, then, then, then, and so on ad infinitum.

What happens to the function at this time?
, , , …

So: if , then the function tends to minus infinity:

Roughly speaking, according to our first rule, instead of “X” we substitute infinity into the function and get the answer.

Another example with infinity:

Again we start increasing to infinity and look at the behavior of the function:

Conclusion: when the function increases without limit:

And another series of examples:

Please try to mentally analyze the following for yourself and remember the simplest types of limits:

, , , , , , , , ,
If you have doubts anywhere, you can pick up a calculator and practice a little.
In the event that , try to construct the sequence , , . If , then , , .

! Note: Strictly speaking, this approach to constructing sequences of several numbers is incorrect, but for understanding the simplest examples it is quite suitable.

Also pay attention to the following thing. Even if a limit is given with a large number at the top, or even with a million: , then it’s all the same , since sooner or later “X” will begin to take on such gigantic values ​​that a million in comparison will be a real microbe.

What do you need to remember and understand from the above?

1) When given any limit, first we simply try to substitute the number into the function.

2) You must understand and immediately solve the simplest limits, such as , , etc.

Moreover, the limit has a very good geometric meaning. For a better understanding of the topic, I recommend that you read the teaching material Graphs and properties of elementary functions. After reading this article, you will not only finally understand what a limit is, but also get acquainted with interesting cases when the limit of a function in general does not exist!

In practice, unfortunately, there are few gifts. And therefore we move on to consider more complex limits. By the way, on this topic there is intensive course in pdf format, which is especially useful if you have VERY little time to prepare. But the site materials, of course, are no worse:

Now we will consider the group of limits when , and the function is a fraction whose numerator and denominator contain polynomials

Example:

Calculate limit

According to our rule, we will try to substitute infinity into the function. What do we get at the top? Infinity. And what happens below? Also infinity. Thus, we have what is called species uncertainty. One might think that , and the answer is ready, but in the general case this is not at all the case, and it is necessary to apply some solution technique, which we will now consider.

How to solve limits of this type?

First we look at the numerator and find the highest power:

The leading power in the numerator is two.

Now we look at the denominator and also find it to the highest power:

The highest degree of the denominator is two.

Then we choose the highest power of the numerator and denominator: in this example, they are the same and equal to two.

So, the solution method is as follows: in order to reveal the uncertainty, it is necessary to divide the numerator and denominator by the highest power.

Here it is, the answer, and not infinity at all.

What is fundamentally important in the design of a decision?

First, we indicate uncertainty, if any.

Secondly, it is advisable to interrupt the solution for intermediate explanations. I usually use the sign, it does not have any mathematical meaning, but means that the solution is interrupted for an intermediate explanation.

Thirdly, in the limit it is advisable to mark what is going where. When the work is drawn up by hand, it is more convenient to do it this way:

It is better to use a simple pencil for notes.

Of course, you don’t have to do any of this, but then, perhaps, the teacher will point out shortcomings in the solution or start asking additional questions about the assignment. Do you need it?

Example 2

Find the limit
Again in the numerator and denominator we find in the highest degree:

Maximum degree in numerator: 3
Maximum degree in denominator: 4
Choose greatest value, in this case four.
According to our algorithm, to reveal uncertainty, we divide the numerator and denominator by .
The complete assignment might look like this:

Divide the numerator and denominator by

Example 3

Find the limit
Maximum degree of “X” in the numerator: 2
Maximum degree of “X” in the denominator: 1 (can be written as)
To reveal the uncertainty, it is necessary to divide the numerator and denominator by . The final solution might look like this:

Divide the numerator and denominator by

Notation does not mean division by zero (you cannot divide by zero), but division by an infinitesimal number.

Thus, by uncovering species uncertainty, we may be able to final number, zero or infinity.

Limits with uncertainty of type and method for solving them

The next group of limits is somewhat similar to the limits just considered: the numerator and denominator contain polynomials, but “x” no longer tends to infinity, but to finite number.

Example 4

Solve limit
First, let's try to substitute -1 into the fraction:

In this case, the so-called uncertainty is obtained.

General rule: if the numerator and denominator contain polynomials, and there is uncertainty of the form , then to disclose it you need to factor the numerator and denominator.

To do this, most often you need to solve a quadratic equation and/or use abbreviated multiplication formulas. If these things have been forgotten, then visit the page Mathematical formulas and tables and read the teaching material Hot formulas for school mathematics course. By the way, it is best to print it out; it is required very often, and information is absorbed better from paper.

So, let's solve our limit

Factor the numerator and denominator

In order to factor the numerator, you need to solve the quadratic equation:

First we find the discriminant:

And the square root of it: .

If the discriminant is large, for example 361, we use a calculator; the function of extracting the square root is on the simplest calculator.

! If the root is not extracted in its entirety (a fractional number with a comma is obtained), it is very likely that the discriminant was calculated incorrectly or there was a typo in the task.

Next we find the roots:

Thus:

All. The numerator is factorized.

Denominator. The denominator is already the simplest factor, and there is no way to simplify it.

Obviously, it can be shortened to:

Now we substitute -1 into the expression that remains under the limit sign:

Naturally, in a test, test, or exam, the solution is never described in such detail. In the final version, the design should look something like this:

Let's factorize the numerator.

Example 5

Calculate limit

First, the “finish” version of the solution

Let's factor the numerator and denominator.

Numerator:
Denominator:



,

What's important in this example?
Firstly, you must have a good understanding of how the numerator is revealed, first we took 2 out of brackets, and then used the formula for the difference of squares. This is the formula you need to know and see.

Recommendation: If in a limit (of almost any type) it is possible to take a number out of brackets, then we always do it.
Moreover, it is advisable to move such numbers beyond the limit icon. For what? Yes, just so that they don’t get in the way. The main thing is not to lose these numbers later during the solution.

Please note that at the final stage of the solution, I took the two out of the limit icon, and then the minus.

! Important
During the solution, the type fragment occurs very often. Reduce this fractionit is forbidden . First you need to change the sign of the numerator or denominator (put -1 out of brackets).
, that is, a minus sign appears, which is taken into account when calculating the limit and there is no need to lose it at all.

In general, I noticed that most often in finding limits of this type you have to solve two quadratic equations, that is, both the numerator and the denominator contain quadratic trinomials.

Method of multiplying the numerator and denominator by the conjugate expression

We continue to consider the uncertainty of the form

The next type of limits is similar to the previous type. The only thing, in addition to polynomials, we will add roots.

Example 6

Find the limit

Let's start deciding.

First we try to substitute 3 into the expression under the limit sign
I repeat once again - this is the first thing you need to do for ANY limit. This action is usually carried out mentally or in draft form.

An uncertainty of the form has been obtained that needs to be eliminated.

As you probably noticed, our numerator contains the difference of the roots. And in mathematics it is customary to get rid of roots, if possible. For what? And life is easier without them.

Limit of a function at a point and at

The limit of a function is the main apparatus of mathematical analysis. With its help, the continuity of a function, derivative, integral, and sum of a series are subsequently determined.

Let the function y=f(x)defined in some neighborhood of the point, except perhaps the point 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). Number b called limit of the function y=f(x) at the point (or for ), if for any sequence of admissible argument values ​​converging to (i.e. ), the sequence of corresponding function values ​​converges to the number b(ie).

In this case they write or at. The geometric meaning of the limit of a function: means that for all points X, sufficiently close to the point , the corresponding values ​​of the function differ as little as desired from the number b.

Definition 2 (in "language e-d ", or according to Cauchy). Number b called limit of the function y=f(x) at the point (or for ), if for any positive number e there is a positive number d such that for all satisfying the inequality , the inequality .

Recorded.

This definition can be briefly written as follows:

Note that you can write it like this.

The geometric meaning of the limit of the function: if for any e-neighborhood of the point b there is such a d-neighborhood of the point , that for all from this d-neighborhood the corresponding values ​​of the function f(x) lie in the e-neighborhood of the point b. In other words, the points on the graph of the function y=f(x) lie inside a strip of width 2e bounded by straight lines at = b+e, at = b- e (Figure 17). Obviously, the value of d depends on the choice of e, so we write d = d(e).

In determining the limit of a function it is assumed that X strives for in any way: remaining less than (on the left of ), greater than (to the right of ), or fluctuating around a point .

There are cases when the method of approximating an argument X To significantly affects the value of the function limit. Therefore, the concepts of one-sided limits are introduced.

Definition. The number is called limit of the function y=f(x) left at the point , if for any number e > 0 there is a number d = d(e) > 0 such that for , the inequality .

The limit on the left is written this way or briefly (Dirichlet notation) (Figure 18).

Defined similarly limit of the function on the right , let's write it using symbols:

Briefly, the limit on the right is denoted by .

The left and right limits of a function are called one-way limits . Obviously, if , then both one-sided limits exist, and .

The converse is also true: if both limits exist and and they are equal, then there exists a limit and .

If, then it does not exist.

Definition. Let the function y=f(x) is defined in the interval . Number b called limit of the function y=f(x) at X® ¥, if for any number e > 0 there is such a number M = M(e) > 0, which for all X, satisfying the inequality the inequality is satisfied. Briefly this definition can be written as follows:

If X® +¥, then write if X® -¥, then they write , if = , then their common meaning is usually denoted .

The geometric meaning of this definition is as follows: for , that for and the corresponding values ​​of the function y=f(x) fall into the e-neighborhood of the point b, i.e. the points of the graph lie in a strip of width 2e, bounded by straight lines and (Figure 19).

Infinitely large functions (b.b.f)

Infinitesimal functions (infinitesimal functions)

Definition. Function y=f(x) is called infinitely large at , if for any number M> 0 there is a number d = d( M) > 0, which is for everyone X, satisfying the inequality, the inequality is satisfied. Write or at .

For example, the function is b.b.f. at .

If f(x) tends to infinity at and takes only positive values, then write ; if only negative values, then .

Definition. Function y=f(x), defined on the entire numerical axis, is called infinitely large at , if for any number M> 0 there is such a number N = N(M) > 0, which is for everyone X satisfying the inequality, the inequality is written. Short:

For example, there is b.b.f. at .

Note that if the argument X, tending to infinity, takes only natural values, i.e. , then the corresponding b.b.f. becomes an infinitely large sequence. For example, the sequence is an infinitely large sequence. Obviously, any b.b.f. in the surrounding area points is unlimited in this vicinity. The converse is not true: an unbounded function may not be b.b.f. (For example, )

However, if where b - final number, then the function f(x limited in the vicinity of the point.

Indeed, from the definition of the limit of a function it follows that when the condition is satisfied. Therefore, for , and this means that the function f(x) is limited.

Definition. Function y=f(x) is called infinitesimal at , If

By definition of the limit of a function, this equality means: for any number there is a number such that for all X satisfying the inequality, the inequality is satisfied.

The b.m.f. is determined similarly. at

: In all these cases.

Infinitesimal functions are often called infinitesimal quantities or infinitesimal ; usually denoted by the Greek letters a, b, etc.

Examples of b.m.f. serve functions when

Another example: - an infinitesimal sequence.

Example Prove that .

Solution . Feature 5+ X can be represented as the sum of the number 7 and b.m.f. X- 2 (at ), i.e. equality is satisfied. Therefore, by Theorem 3.4.6 we obtain .

Basic theorems about limits

Let's consider theorems (without proof) that make it easier to find the limits of a function. The formulation of the theorems for the cases when and is similar. In the theorems presented, we will assume that the limits exist.

Theorem 5.8 The limit of the sum (difference) of two functions is equal to the sum (difference) of their limits: .

Theorem 5.9 The limit of the product of two functions is equal to the product of their limits:

Note that the theorem is valid for the product of any finite number of functions.

Corollary 3 The constant factor can be taken beyond the limit sign: .

Corollary 4 The limit of a degree with a natural exponent is equal to the same degree of the limit: . In particular,

Theorem 5.10 The limit of a fraction is equal to the limit of the numerator divided by the limit of the denominator, unless the limit of the denominator is zero:

Example Calculate

Solution .

Example Calculate

Solution . Here the theorem on the limit of a fraction cannot be applied, because the limit of the denominator, at is equal to 0. In addition, the limit of the numerator is equal to 0. In such cases we say that we have type uncertainty. To expand it, we factorize the numerator and denominator of the fraction, then reduce it by:

Example Calculate

Solution . Here we are dealing with type uncertainty. To find the limit of a given fraction, divide the numerator and denominator by:

The function is the sum of the number 2 and b.m.f., therefore

Signs of Limits

Not every function, even a limited one, has a limit. For example, the function at has no limit. In many questions of analysis, it is enough just to verify the existence of a limit of a function. In such cases, signs of the existence of a limit are used.

The first and second remarkable limits

Definition. When calculating the limits of expressions containing trigonometric functions, the limit is often used

called the first remarkable limit .

It reads: the limit of the ratio of a sine to its argument is equal to one when the argument tends to zero.

Example Find

Solution . We have uncertainty of the form . The fraction limit theorem does not apply. Let us then denote at and

Example 3 Find

Solution.

Definition. Equalities are called second remarkable limit .

Comment. It is known that the limit of a number sequence

Has a limit equal to e: . The number e is called the Neper number. The number e is irrational, its approximate value is 2.72 (e = 2, 718281828459045...). Some properties of the number e make it especially convenient to choose this number as the base of logarithms. Logarithms to base e are called natural logarithms and are denoted by Note that

Let us accept without proof the statement that the function also tends to the number e

If you put it then follows. These equalities are widely used in calculating limits. In applications of analysis, an important role is played by the exponential function with base e. The function is called exponential, and the notation is also used

Example Find

Solution . We denote obviously, with We have

Calculation of limits

To reveal uncertainties of the form, it is often useful to apply the principle of replacing infinitesimals with equivalent ones and other properties of equivalent infinitesimal functions. As is known ~ x when ~ x at , because

In this article we will tell you what the limit of a function is. First, let us explain the general points that are very important for understanding the essence of this phenomenon.

Limit concept

In mathematics, the concept of infinity, denoted by the symbol ∞, is fundamentally important. It should be understood as an infinitely large + ∞ or an infinitesimal - ∞ number. When we talk about infinity, we often mean both of these meanings at once, but notation of the form + ∞ or - ∞ should not be replaced simply by ∞.

The limit of a function is written as lim x → x 0 f (x) . At the bottom we write the main argument x, and with the help of an arrow we indicate which value x0 it will tend to. If the value x 0 is a concrete real number, then we are dealing with the limit of the function at a point. If the value x 0 tends to infinity (it doesn’t matter whether ∞, + ∞ or - ∞), then we should talk about the limit of the function at infinity.

The limit can be finite or infinite. If it is equal to a specific real number, i.e. lim x → x 0 f (x) = A, then it is called a finite limit, but if lim x → x 0 f (x) = ∞, lim x → x 0 f (x) = + ∞ or lim x → x 0 f (x) = - ∞ , then infinite.

If we cannot determine either a finite or an infinite value, it means that such a limit does not exist. An example of this case would be the limit of sine at infinity.

In this paragraph we will explain how to find the value of the limit of a function at a point and at infinity. To do this, we need to introduce basic definitions and remember what number sequences are, as well as their convergence and divergence.

Definition 1

The number A is the limit of the function f (x) as x → ∞ if the sequence of its values ​​converges to A for any infinitely large sequence of arguments (negative or positive).

Writing the limit of a function looks like this: lim x → ∞ f (x) = A.

Definition 2

As x → ∞, the limit of a function f(x) is infinite if the sequence of values ​​for any infinitely large sequence of arguments is also infinitely large (positive or negative).

The entry looks like lim x → ∞ f (x) = ∞ .

Example 1

Prove the equality lim x → ∞ 1 x 2 = 0 using the basic definition of the limit for x → ∞.

Solution

Let's start by writing a sequence of values ​​of the function 1 x 2 for an infinitely large positive sequence of values ​​of the argument x = 1, 2, 3, . . . , n , . . . .

1 1 > 1 4 > 1 9 > 1 16 > . . . > 1 n 2 > . . .

We see that the values ​​will gradually decrease, tending to 0. See in the picture:

x = - 1 , - 2 , - 3 , . . . , - n , . . .

1 1 > 1 4 > 1 9 > 1 16 > . . . > 1 - n 2 > . . .

Here we can also see a monotonic decrease towards zero, which confirms the validity of this in the equality condition:

Answer: The correctness of this in the equality condition is confirmed.

Example 2

Calculate the limit lim x → ∞ e 1 10 x .

Solution

Let's start, as before, by writing down sequences of values ​​f (x) = e 1 10 x for an infinitely large positive sequence of arguments. For example, x = 1, 4, 9, 16, 25, . . . , 10 2 , . . . → + ∞ .

e 1 10 ; e 4 10 ; e 9 10 ; e 16 10 ; e 25 10 ; . . . ; e 100 10 ; . . . = = 1, 10; 1, 49; 2, 45; 4, 95; 12, 18; . . . ; 22026, 46; . . .

We see that this sequence is infinitely positive, which means f (x) = lim x → + ∞ e 1 10 x = + ∞

Let's move on to writing the values ​​of an infinitely large negative sequence, for example, x = - 1, - 4, - 9, - 16, - 25, . . . , - 10 2 , . . . → - ∞ .

e - 1 10 ; e - 4 10 ; e - 9 10 ; e - 16 10 ; e - 25 10 ; . . . ; e - 100 10 ; . . . = = 0, 90; 0, 67; 0, 40; 0, 20; 0, 08; . . . ; 0.000045; . . . x = 1, 4, 9, 16, 25, . . . , 10 2 , . . . → ∞

Since it also tends to zero, then f (x) = lim x → ∞ 1 e 10 x = 0 .

The solution to the problem is clearly shown in the illustration. Blue dots indicate a sequence of positive values, green dots indicate a sequence of negative values.

Answer: lim x → ∞ e 1 10 x = + ∞ , pr and x → + ∞ 0 , pr and x → - ∞ .

Let's move on to the method of calculating the limit of a function at a point. To do this, we need to know how to correctly define a one-sided limit. This will also be useful to us in order to find the vertical asymptotes of the graph of a function.

Definition 3

The number B is the limit of the function f (x) on the left as x → a in the case when the sequence of its values ​​converges to a given number for any sequence of arguments of the function x n converging to a, if its values ​​remain less than a (x n< a).

Such a limit is denoted in writing as lim x → a - 0 f (x) = B.

Now let’s formulate what the limit of a function on the right is.

Definition 4

The number B is the limit of the function f (x) on the right as x → a in the case when the sequence of its values ​​converges to a given number for any sequence of arguments of the function x n converging to a, if its values ​​remain greater than a (x n > a) .

We write this limit as lim x → a + 0 f (x) = B .

We can find the limit of a function f (x) at a certain point when it has equal limits on the left and right sides, i.e. lim x → a f (x) = lim x → a - 0 f (x) = lim x → a + 0 f (x) = B . If both limits are infinite, the limit of the function at the starting point will also be infinite.

Now we will clarify these definitions by writing down the solution to a specific problem.

Example 3

Prove that there is a finite limit of the function f (x) = 1 6 (x - 8) 2 - 8 at the point x 0 = 2 and calculate its value.

Solution

In order to solve the problem, we need to recall the definition of the limit of a function at a point. First, let's prove that the original function has a limit on the left. Let's write down a sequence of function values ​​that will converge to x 0 = 2 if x n< 2:

f(-2); f (0) ; f (1) ; f 1 1 2 ; f 1 3 4 ; f 1 7 8 ; f 1 15 16 ; . . . ; f 1 1023 1024 ; . . . = = 8, 667; 2, 667; 0, 167; - 0, 958; - 1, 489; - 1, 747; - 1, 874; . . . ; - 1,998; . . . → - 2

Since the above sequence reduces to - 2, we can write that lim x → 2 - 0 1 6 x - 8 2 - 8 = - 2.

6 , 4 , 3 , 2 1 2 , 2 1 4 , 2 1 8 , 2 1 16 , . . . , 2 1 1024 , . . . → 2

The function values ​​in this sequence will look like this:

f (6) ; f (4) ; f (3) ; f 2 1 2 ; f 2 3 4 ; f 2 7 8 ; f 2 15 16 ; . . . ; f 2 1023 1024 ; . . . = = - 7, 333; - 5, 333; - 3, 833; - 2, 958; - 2, 489; - 2, 247; - 2, 124; . . . , - 2,001, . . . → - 2

This sequence also converges to - 2, which means lim x → 2 + 0 1 6 (x - 8) 2 - 8 = - 2.

We found that the limits on the right and left sides of this function will be equal, which means that the limit of the function f (x) = 1 6 (x - 8) 2 - 8 at the point x 0 = 2 exists, and lim x → 2 1 6 (x - 8) 2 - 8 = - 2 .

You can see the progress of the solution in the illustration (green dots are a sequence of values ​​converging to x n< 2 , синие – к x n > 2).

Answer: The limits on the right and left sides of this function will be equal, which means that the limit of the function exists, and lim x → 2 1 6 (x - 8) 2 - 8 = - 2.

To study the theory of limits more deeply, we advise you to read the article on the continuity of a function at a point and the main types of discontinuity points.

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Consider the function %%f(x)%% defined at least in some punctured neighborhood %%\stackrel(\circ)(\text(U))(a)%% of the point %%a \in \overline( \mathbb(R))%% extended number line.

The concept of a Cauchy limit

The number %%A \in \mathbb(R)%% is called limit of the function%%f(x)%% at the point %%a \in \mathbb(R)%% (or at %%x%% tending to %%a \in \mathbb(R)%%), if, what Whatever the positive number %%\varepsilon%%, there is a positive number %%\delta%% such that for all points in the punctured %%\delta%% neighborhood of the point %%a%% the function values ​​belong to %%\varepsilon %%-neighborhood of point %%A%%, or

$$ A = \lim\limits_(x \to a)(f(x)) \Leftrightarrow \forall\varepsilon > 0 ~\exists \delta > 0 \big(x \in \stackrel(\circ)(\text (U))_\delta(a) \Rightarrow f(x) \in \text(U)_\varepsilon (A) \big) $$

This definition is called the %%\varepsilon%% and %%\delta%% definition, proposed by the French mathematician Augustin Cauchy and used from the beginning of the 19th century to the present day because it has the necessary mathematical rigor and accuracy.

Combining various neighborhoods of the point %%a%% of the form %%\stackrel(\circ)(\text(U))_\delta(a), \text(U)_\delta (\infty), \text(U) _\delta (-\infty), \text(U)_\delta (+\infty), \text(U)_\delta^+ (a), \text(U)_\delta^- (a) %% with surroundings %%\text(U)_\varepsilon (A), \text(U)_\varepsilon (\infty), \text(U)_\varepsilon (+\infty), \text(U) _\varepsilon (-\infty)%%, we get 24 definitions of the Cauchy limit.

Geometric meaning

Geometric meaning of the limit of a function

Let us find out what the geometric meaning of the limit of a function at a point is. Let's build a graph of the function %%y = f(x)%% and mark the points %%x = a%% and %%y = A%% on it.

The limit of the function %%y = f(x)%% at the point %%x \to a%% exists and is equal to A if for any %%\varepsilon%% neighborhood of the point %%A%% one can specify such a %%\ delta%%-neighborhood of the point %%a%%, such that for any %%x%% from this %%\delta%%-neighborhood the value %%f(x)%% will be in the %%\varepsilon%%-neighborhood points %%A%%.

Note that by the definition of the limit of a function according to Cauchy, for the existence of a limit at %%x \to a%%, it does not matter what value the function takes at the point %%a%%. Examples can be given where the function is not defined when %%x = a%% or takes a value other than %%A%%. However, the limit may be %%A%%.

Determination of the Heine limit

The element %%A \in \overline(\mathbb(R))%% is called the limit of the function %%f(x)%% at %% x \to a, a \in \overline(\mathbb(R))%% , if for any sequence %%\(x_n\) \to a%% from the domain of definition, the sequence of corresponding values ​​%%\big\(f(x_n)\big\)%% tends to %%A%%.

The definition of a limit according to Heine is convenient to use when doubts arise about the existence of a limit of a function at a given point. If it is possible to construct at least one sequence %%\(x_n\)%% with a limit at the point %%a%% such that the sequence %%\big\(f(x_n)\big\)%% has no limit, then we can conclude that the function %%f(x)%% has no limit at this point. If for two various sequences %%\(x"_n\)%% and %%\(x""_n\)%% having same limit %%a%%, sequences %%\big\(f(x"_n)\big\)%% and %%\big\(f(x""_n)\big\)%% have various limits, then in this case there is also no limit of the function %%f(x)%%.

Example

Let %%f(x) = \sin(1/x)%%. Let's check whether the limit of this function exists at the point %%a = 0%%.

Let us first choose a sequence $$ \(x_n\) = \left\(\frac((-1)^n)(n\pi)\right\) converging to this point. $$

It is clear that %%x_n \ne 0~\forall~n \in \mathbb(N)%% and %%\lim (x_n) = 0%%. Then %%f(x_n) = \sin(\left((-1)^n n\pi\right)) \equiv 0%% and %%\lim\big\(f(x_n)\big\) = 0 %%.

Then take a sequence converging to the same point $$ x"_n = \left\( \frac(2)((4n + 1)\pi) \right\), $$

for which %%\lim(x"_n) = +0%%, %%f(x"_n) = \sin(\big((4n + 1)\pi/2\big)) \equiv 1%% and %%\lim\big\(f(x"_n)\big\) = 1%%. Similarly for the sequence $$ x""_n = \left\(-\frac(2)((4n + 1) \pi) \right\), $$

also converging to the point %%x = 0%%, %%\lim\big\(f(x""_n)\big\) = -1%%.

All three sequences gave different results, which contradicts the Heine definition condition, i.e. this function has no limit at the point %%x = 0%%.

Theorem

The Cauchy and Heine definitions of the limit are equivalent.

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 , depending 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.

See also: