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 the limits in higher mathematics? Understanding comes with experience, so at the same time we will give a few detailed examples solutions of 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 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 in our numerator quadratic equation. 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.
Definition of sequence and function limits, properties of limits, first and second wonderful limits, examples.
Constant number A called limit sequences(x n), if for any arbitrarily small positive number ε > 0 there is a number N such that all values x n, for which n>N, satisfy the inequality
Write it down as follows: or x n → a.
Inequality (6.1) is equivalent to the double inequality
a - ε< x n < a + ε которое означает, что точки x n, starting from some number n>N, lie inside the interval (a-ε , a+ε), i.e. fall into any small ε-neighborhood of the point A.
A sequence having a limit is called convergent, otherwise - divergent.
The concept of a function limit is a generalization of the concept of a sequence limit, since the limit of a sequence can be considered as the limit of a function x n = f(n) of an integer argument n.
Let the function f(x) be given and let a - limit point domain of definition of this function D(f), i.e. such a point, any neighborhood of which contains points of the set D(f) other than a. Dot a may or may not belong to the set D(f).
Definition 1. The constant number A is called limit functions f(x) at x→ a, if for any sequence (x n ) of argument values tending to A, the corresponding sequences (f(x n)) have the same limit A.
This definition is called determining the limit of a function according to Heine, or " in sequence language”.
Definition 2. The constant number A is called limit functions f(x) at x→a, if, given an arbitrary, arbitrarily small positive number ε, one can find such δ >0 (depending on ε) that for all x, lying in the ε-neighborhood of the number A, i.e. For x, satisfying the inequality
0 < x-a < ε , значения функции f(x) будут лежать в
ε-окрестности числа А, т.е. |f(x)-A| < ε
This definition is called by defining the limit of a function according to Cauchy, or “in the language ε - δ"
Definitions 1 and 2 are equivalent. If the function f(x) as x → a has limit, equal to A, this is written in the form
In the event that the sequence (f(x n)) increases (or decreases) without limit for any method of approximation x to your limit A, then we will say that the function f(x) has infinite limit, and write it in the form:
A variable quantity (i.e. a sequence or function) whose limit equal to zero, called infinitely small.
A variable whose limit is equal to infinity is called infinitely large.
To find the limit in practice, the following theorems are used.
Theorem 1 . If every limit exists
(6.4)
(6.5)
(6.6)
Comment. Expressions of the form 0/0, ∞/∞, ∞-∞ 0*∞ are uncertain, for example, the ratio of two infinitesimal or infinitely large quantities, and finding a limit of this type is called “uncertainty disclosure.”
Theorem 2.
those. one can go to the limit based on the power with a constant exponent, in particular, 
Theorem 3.
(6.11)
Where e» 2.7 - base of natural logarithm. Formulas (6.10) and (6.11) are called the first remarkable limit and the second remarkable limit.
The consequences of formula (6.11) are also used in practice:
(6.12)
(6.13)
(6.14)
in particular the limit,
If x → a and at the same time x > a, then write x →a + 0. If, in particular, a = 0, then instead of the symbol 0+0 write +0. Similarly, if x→a and at the same time x 
. The function f(x) is called continuous at the point x 0 if limit
(6.15)
Condition (6.15) can be rewritten as:
that is, passage to the limit under the sign of a function is possible if it is continuous at a given point.
If equality (6.15) is violated, then we say that at x = x o function f(x) It has gap Consider the function y = 1/x. The domain of definition of this function is the set R, except for x = 0. The point x = 0 is a limit point of the set D(f), since in any neighborhood of it, i.e. in any open interval containing the point 0, there are points from D(f), but it itself does not belong to this set. The value f(x o)= f(0) is not defined, so at the point x o = 0 the function has a discontinuity.
The function f(x) is called continuous on the right at the point x o if the limit
And continuous on the left at the point x o, if the limit
Continuity of a function at a point x o is equivalent to its continuity at this point both to the right and to the left.
In order for the function to be continuous at a point x o, for example, on the right, it is necessary, firstly, that there be a finite limit, and secondly, that this limit be equal to f(x o). Therefore, if at least one of these two conditions is not met, then the function will have a discontinuity.
1. If the limit exists and is not equal to f(x o), then they say that function f(x) at the point x o has rupture of the first kind, or leap.
2. If the limit is +∞ or -∞ or does not exist, then they say that in point x o the function has a discontinuity second kind.
For example, the function y = ctg x as x → +0 has a limit equal to +∞, which means that at the point x=0 it has a discontinuity of the second kind. Function y = E(x) (integer part of x) at points with whole abscissas has discontinuities of the first kind, or jumps.
A function that is continuous at every point in the interval is called continuous V . A continuous function is represented by a solid curve.
Many problems associated with the continuous growth of some quantity lead to the second remarkable limit. Such tasks, for example, include: growth of deposits according to the law of compound interest, growth of the country's population, decay of radioactive substances, proliferation of bacteria, etc.
Let's consider example of Ya. I. Perelman, giving an interpretation of the number e in the compound interest problem. Number e there is a limit 
. In savings banks, interest money is added to the fixed capital annually. If the accession is made more often, then the capital grows faster, since a larger amount is involved in the formation of interest. Let's take a purely theoretical, very simplified example. Let 100 deniers be deposited in the bank. units based on 100% per annum. If interest money is added to the fixed capital only after a year, then by this period 100 den. units will turn into 200 monetary units. Now let's see what 100 denize will turn into. units, if interest money is added to fixed capital every six months. After six months, 100 den. units will grow by 100 × 1.5 = 150, and after another six months - by 150 × 1.5 = 225 (den. units). If the accession is done every 1/3 of the year, then after a year 100 den. units will turn into 100 × (1 +1/3) 3 ≈ 237 (den. units). We will increase the terms for adding interest money to 0.1 year, to 0.01 year, to 0.001 year, etc. Then out of 100 den. units after a year it will be:
100×(1 +1/10) 10 ≈ 259 (den. units),
100×(1+1/100) 100 ≈ 270 (den. units),
100×(1+1/1000) 1000 ≈271 (den. units).
With an unlimited reduction in the terms for adding interest, the accumulated capital does not grow indefinitely, but approaches a certain limit equal to approximately 271. The capital deposited at 100% per annum cannot increase by more than 2.71 times, even if the accrued interest were added to the capital every second because the limit
Example 3.1. Using the definition of the limit of a number sequence, prove that the sequence x n =(n-1)/n has a limit equal to 1.
Solution. We need to prove that, no matter what ε > 0 we take, for it there is a natural number N such that for all n > N the inequality |x n -1|< ε
Take any ε > 0. Since x n -1 =(n+1)/n - 1= 1/n, then to find N it is sufficient to solve the inequality 1/n<ε. Отсюда n>1/ε and, therefore, N can be taken to be the integer part of 1/ε N = E(1/ε). We have thereby proven that the limit .
Example 3.2. Find the limit of a sequence given by a common term
.
Solution. Let's apply the limit of the sum theorem and find the limit of each term. As n → ∞, the numerator and denominator of each term tend to infinity, and we cannot directly apply the quotient limit theorem. Therefore, first we transform x n, dividing the numerator and denominator of the first term by n 2, and the second on n. Then, applying the limit of the quotient and the limit of the sum theorem, we find:
Example 3.3. 
. Find .
Here we used the limit of degree theorem: the limit of a degree is equal to the degree of the limit of the base.
Example 3.4. Find ( 
).
Solution. It is impossible to apply the limit of difference theorem, since we have an uncertainty of the form ∞-∞. Let's transform the general term formula:
Example 3.5. The function f(x)=2 1/x is given. Prove that there is no limit.
Solution. Let's use definition 1 of the limit of a function through a sequence. Let us take a sequence ( x n ) converging to 0, i.e. Let us show that the value f(x n)= behaves differently for different sequences. Let x n = 1/n. Obviously, then the limit 
Let us now choose as x n a sequence with a common term x n = -1/n, also tending to zero. 
Therefore there is no limit.
Example 3.6. Prove that there is no limit.
Solution. Let x 1 , x 2 ,..., x n ,... be a sequence for which
. How does the sequence (f(x n)) = (sin x n) behave for different x n → ∞
If x n = p n, then sin x n = sin (p n) = 0 for all n and the limit If
x n =2 p n+ p /2, then sin x n = sin(2 p n+ p /2) = sin p /2 = 1 for all n and therefore the limit. So it doesn't exist.
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.
ContentSecond 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 , If1) the function is defined on some punctured neighborhood 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 enter definitions infinite limits 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
Limit theorem 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.
Let's start with general things that are VERY important, but few people pay attention to them.
Limit of a function - basic concepts.
Infinity means symbol Essentially, infinity is either an infinitely large positive number or an infinitely large a negative number.
What does this mean: when you see , it makes no difference whether it is or . But it’s better not to replace with , just as it’s better not to replace with .
Write the limit of a function f(x) taken as, the argument x is indicated below and, through an arrow, what value it is aiming for.
If it is a specific real number, then we speak of limit of the function at the point.
If or . then they talk about limit of a function at infinity.
The limit itself can be equal to a specific real number, in this case they say that the limit is finite.
If , 
or 
, then they say that the limit is infinite.
They also say that there is no limit, if it is impossible to determine a specific value of the limit or its infinite value (, or). For example, there is no limit on sine at infinity.
Limit of a function - basic definitions.
It's time to get busy finding the values of function limits at infinity and at a point. Several definitions will help us with this. These definitions are based on number sequences and their convergence or divergence.
Definition(finding the limit of a function at infinity).
The number A is called the limit of the function f(x) at , if for any infinitely large sequence of function arguments (infinitely large positive or negative), the sequence of values of this function converges to A. Denoted by .
Comment.
The limit of a function f(x) at is infinite if for any infinitely large sequence of function arguments (infinitely large positive or negative), the sequence of values of this function is infinitely positive or infinitely negative. Denoted by .
Example.
Using the definition of the limit at, prove the equality.
Solution.
Let's write down the sequence of function values for an infinitely large positive sequence of argument values. 
It is obvious that the terms of this sequence decrease monotonically towards zero.
Graphic illustration.
Now let's write down the sequence of function values for an infinitely large negative sequence of argument values. 
The terms of this sequence also decrease monotonically towards zero, which proves the original equality.
Graphic illustration.
Example.
Find the limit
Solution.
Let's write down the sequence of function values for an infinitely large positive sequence of argument values. For example, let's take .
The sequence of function values will be (blue dots on the graph) 
Obviously, this sequence is infinitely large positive, therefore, 
Now let’s write down the sequence of function values for an infinitely large negative sequence of argument values. For example, let's take .
The sequence of function values will be (green dots on the graph) 
Obviously, this sequence converges to zero, therefore, 
Graphic illustration
Answer:
Now let's talk about the existence and determination of the limit of a function at a point. Everything is based on defining one-sided limits. One cannot do without calculating one-sided limits when .
Definition(finding the limit of a function on the left).
The number B is called the limit of the function f(x) on the left at , if for any sequence of function arguments converging to a, the values of which remain less than a (), the sequence of values of this function converges to B.
Designated 
.
Definition(finding the limit of a function on the right).
The number B is called the limit of the function f(x) on the right at , if for any sequence of function arguments converging to a, the values of which remain greater than a (), the sequence of values of this function converges to B.
Designated 
.
Definition(existence of a limit of a function at a point).
The limit of the function f(x) at point a exists if there are limits to the left and right of a and they are equal to each other.
Comment.
The limit of the function f(x) at point a is infinite if the limits to the left and right of a are infinite.
Let us explain these definitions with an example.
Example.
Prove the existence of a finite limit of a function 
at point . Find its value.
Solution.
We will start from the definition of the existence of a limit of a function at a point.
First, we show the existence of a limit on the left. To do this, take a sequence of arguments converging to , and . An example of such a sequence would be
In the figure, the corresponding values are shown as green dots.
It's easy to see that this sequence converges to -2, so 
.
Secondly, we show the existence of a limit on the right. To do this, take a sequence of arguments converging to , and . An example of such a sequence would be
The corresponding sequence of function values will look like
In the figure, the corresponding values are shown as blue dots.
It's easy to see that this sequence also converges to -2, so 
.
By this we showed that the limits on the left and right are equal, therefore, by definition, there is a limit of the function at point , and 
Graphic illustration.
We recommend continuing your study of the basic definitions of the theory of limits with the topic.
Definition 1. Let E- an infinite number. If any neighborhood contains points of the set E, different from the point A, That A called ultimate point of the set E.
Definition 2. (Heinrich Heine (1821-1881)). Let the function 
defined on the set X And 
A called limit
functions 
at the point 
(or when 
, if for any sequence of argument values 
, converging to 
, the corresponding sequence of function values converges to the number A. They write: 
.
Examples. 1) Function 
has a limit equal to With, at any point on the number line.
Indeed, for any point 
and any sequence of argument values 
, converging to 
and consisting of numbers other than 
, the corresponding sequence of function values has the form 
, and we know that this sequence converges to With. That's why 
.
2) For function 
.
This is obvious, because if 
, then 
.
3) Dirichlet function 
has no limit at any point.
Indeed, let 
And 
, and all 
– rational numbers. Then 
for all n, That's why 
. If 
and that's all 
are irrational numbers, then 
for all n, That's why 
. We see that the conditions of Definition 2 are not satisfied, therefore 
does not exist.
4)
.
Indeed, let us take an arbitrary sequence 
, converging to
number 2. Then . Q.E.D.
Definition 3. (Cauchy (1789-1857)). Let the function 
defined on the set X And 
is the limit point of this set. Number A called limit
functions 
at the point 
(or when 
, if for any 
there will be 
, such that for all values of the argument X, satisfying the inequality
,
inequality is true
.
They write: 
.
Cauchy's definition can also be given using neighborhoods, if we note that , a:
let function 
defined on the set X And 
is the limit point of this set. Number A called limit
functions 
at the point 
, if for any 
-neighborhood of a point A 
there is a pierced one 
- neighborhood of a point 
,such that 
.
It is useful to illustrate this definition with a drawing.
Example 5.
.
Indeed, let's take 
randomly and find 
, such that for everyone X, satisfying the inequality 
inequality holds 
. The last inequality is equivalent to the inequality 
, so we see that it is enough to take 
. The statement has been proven.
Fair
Theorem 1. The definitions of the limit of a function according to Heine and according to Cauchy are equivalent.
Proof. 1) Let 
according to Cauchy. Let us prove that the same number is also a limit according to Heine.
Let's take 
arbitrarily. According to Definition 3 there is 
, such that for everyone 
inequality holds 
. Let 
– an arbitrary sequence such that 
at 
. Then there is a number N such that for everyone 
inequality holds 
, That's why 
for all 
, i.e. 
according to Heine.
2) Let now 
according to Heine. Let's prove that 
and according to Cauchy.
Let's assume the opposite, i.e. What 
according to Cauchy. Then there is 
such that for anyone 
there will be 
,
And 
. Consider the sequence 
. For the specified 
and any n exists 
And 
. It means that 
, Although 
, i.e. number A is not the limit 
at the point 
according to Heine. We have obtained a contradiction, which proves the statement. The theorem has been proven.
Theorem 2 (on the uniqueness of the limit). If there is a limit of a function at a point 
, then he is the only one.
Proof. If a limit is defined according to Heine, then its uniqueness follows from the uniqueness of the limit of the sequence. If a limit is defined according to Cauchy, then its uniqueness follows from the equivalence of the definitions of a limit according to Cauchy and according to Heine. The theorem has been proven.
Similar to the Cauchy criterion for sequences, the Cauchy criterion for the existence of a limit of a function holds. Before formulating it, let us give
Definition 4. They say that the function 
satisfies the Cauchy condition at the point 
, if for any 
exists 
, such that 
And 
, the inequality holds 
.
Theorem 3 (Cauchy criterion for the existence of a limit). In order for the function 
had at the point 
finite limit, it is necessary and sufficient that at this point the function satisfies the Cauchy condition.
Proof.Necessity. Let 
. We must prove that 
satisfies at the point 
Cauchy condition.
Let's take 
arbitrarily and put 
. By definition of the limit for 
exists 
, such that for any values 
, satisfying the inequalities 
And 
, the inequalities are satisfied 
And 
. Then
The need has been proven.
Adequacy. Let the function 
satisfies at the point 
Cauchy condition. We must prove that it has at the point 
final limit.
Let's take 
arbitrarily. By definition there is 4 
, such that from the inequalities 
,
follows that 
- this is given.
Let us first show that for any sequence 
, converging to 
, subsequence 
function values converges. Indeed, if 
, then, by virtue of the definition of the limit of the sequence, for a given 
there is a number N, such that for any 
And 
. Because the 
at the point 
satisfies the Cauchy condition, we have 
. Then, by the Cauchy criterion for sequences, the sequence 
converges. Let us show that all such sequences 
converge to the same limit. Let's assume the opposite, i.e. what are sequences 
And 
,
,
, such that. Let's consider the sequence. It is clear that it converges to 
, therefore, by what was proven above, the sequence converges, which is impossible, since the subsequences 
And 
have different limits 
And 
. The resulting contradiction shows that 
=
. Therefore, by Heine’s definition, the function has at the point 
final limit. The sufficiency, and hence the theorem, has been proven.