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 accurate upper limit
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:
.
Determining the limit of a function
Determination of the limit of a function according to Cauchy
Finite limits of function at end points
Let the function be defined in some neighborhood of the end point, with the possible exception of the point itself. at a point, if for any there is such a thing, depending on , that for all x for which , the inequality holds
.
The limit of a function is denoted as follows:
.
Or at .
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.
.
.
.
They are often referred to as:
;
;
.
Using the concept of neighborhood of a point
If we introduce the concept of a punctured neighborhood of a point, then we can give a unified definition of the finite limit of a function at finite and infinitely distant points:
.
Here for endpoints
;
;
.
Any neighborhood of points at infinity is punctured:
;
;
.
Infinite Function Limits
Definition
Let the function be defined in some punctured neighborhood of a point (finite or at infinity). Limit of function f (x) as x → x 0
equals infinity, if for any arbitrarily large number M > 0
, there is a number δ M > 0
, depending on M, that for all x belonging to the punctured δ M - neighborhood of the point: , the following inequality holds:
.
The infinite limit is denoted as follows:
.
Or at .
Using the logical symbols of existence and universality, the definition of the infinite limit of a function can be written as follows:
.
You can also introduce definitions of infinite limits of certain signs equal to and :
.
.
Universal definition of the limit of a function
Using the concept of a neighborhood of a point, we can give a universal definition of the finite and infinite limit of a function, applicable both for finite (two-sided and one-sided) and infinitely distant points:
.
Determination of the limit of a function according to Heine
Let the function be defined on some set X:.
The number a is called the limit of the function at point:
,
if for any sequence converging to x 0
:
,
whose elements belong to the set X: ,
.
Let us write this definition using the logical symbols of existence and universality:
.
If we take the left-sided neighborhood of the point x as a set X 0 , then we obtain the definition of the left limit. If it is right-handed, then we get the definition of the right limit. If we take the neighborhood of a point at infinity as a set X, we obtain the definition of the limit of a function at infinity.
Theorem
The Cauchy and Heine definitions of the limit of a function are equivalent.
Proof
Properties and theorems of the limit of a function
Further, we assume that the functions under consideration are defined in the corresponding 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 limits of a function."
Arithmetic 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 limits 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 the point , then the limit sign can be applied to the argument 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 (t) as t → t 0
, and it is equal to x 0
:
.
Here is point t 0
can be finite or infinitely distant: .
And let the function f (x) is continuous at point x 0
.
Then there is a limit of the complex function f (g(t)), and it is equal to f (x0):
.
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 - infinitely small 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 infinite great 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".
References:
L.D. Kudryavtsev. Well mathematical analysis. Volume 1. Moscow, 2003.
CM. Nikolsky. Course of mathematical analysis. Volume 1. Moscow, 1983.
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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 when 
), if for any sequence of valid argument values 
converging to
(those. 
), sequence of corresponding function values 
converges to a number b(those. 
).
In this case they write 
or 
at 
. 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", or according to Cauchy). Number b called limit of the function
y=f(x)
at the point
(or when 
), if for any positive number there is a positive number such that for all 
satisfying the inequality 
, the inequality holds 
.
Write down 
.
This definition can be briefly written as follows:
notice, that 
can be written like this 
.
G 
geometric meaning of the limit of a function: 
, if for any neighborhood of the point b there is such a neighborhood of the point
that's for everyone 
from this neighborhood the corresponding values of the function f
(x) lie in the 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 2 bounded by straight lines at
= b + ,
at = b (Figure 17). Obviously, the value of depends on the choice of , so they write = ().
Example Prove that 
Solution
. Let’s take an arbitrary 0 and find = () 0 such that for all X 
, the inequality holds 
. Since from 
those. 
, then taking 
, we see that for everyone X, satisfying the inequality 
, the inequality holds 
. Hence, 
Example Prove that if f
(x) =
With, That 
.
Solution
. For 
you can take it 
. Then at 
we have . Hence, 
.
In defining the limit of a function 
It is believed 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. Number 
called limit of the function
y=f(x)
left
at the point
, if for any number 0 there is a number = () 0 such that for 
, the inequality holds 
.
The limit on the left is written as follows 
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 
.
P 
The limits of a function on the left and right are called one-way limits
. Obviously, if there is 
, then both one-sided limits exist, and 
.
The converse is also true: if both limits exist 
And 
and they are equal, then there is a limit 
And .
If 
, That 
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 0 there is such a number M
= M() 0, which for all X, satisfying the inequality 
inequality holds 
. Briefly this definition can be written as follows:
E 
if X +, then they write 
, If X , then they write 
, If 
=
, then their general meaning is usually denoted 
.
The geometric meaning of this definition is as follows: for 
, that at 
And 
corresponding function values y=f(x) fall into the neighborhood of the point b, i.e. the graph points lie in a strip 2 wide, bounded by straight lines 
And 
(Figure 19).
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. Substitute here in pure form 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 roots 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.
Constant number A called limit sequences(x n ), if for any arbitrarily small positive numberε > 0 there is a number N that has all the values x n, for which n>N, satisfy the inequality
|x n - a|< ε. (6.1)
Write it down as follows: or x n → a.
Inequality (6.1) is equivalent to the double inequality
a- ε< x n < a + ε, (6.2)
which means that the points x n, starting from some number n>N, lie inside the interval (a-ε, a+ ε ), i.e. fall into any smallε -neighborhood of a 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 by defining 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, by specifying an arbitrary arbitrarily small positive number ε, one can find such δ>0 (depending on ε), which is for everyone x, lying inε-neighborhoods of the number A, i.e. For x, satisfying the inequality
0 <
x-a< ε
, the values of the function f(x) will lie inε-neighborhood of the number A, i.e.|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
. (6.3)
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 (i.e. a sequence or function) whose limit is zero is 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 like 0/0, ∞/∞, ∞-∞ , 0*∞ , - are uncertain, for example, the ratio of two infinitely small or infinitely large quantities, and finding a limit of this type is called “uncovering uncertainties.”
Theorem 2. (6.7)
those. one can go to the limit based on the power with a constant exponent, in particular, 
;
(6.8)
(6.9)
Theorem 3.
(6.10)
(6.11)
Where e » 2.7 - base of natural logarithm. Formulas (6.10) and (6.11) are called the first wonderful 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 
and are called accordingly right limit And left limit functions f(x) at the point A. For there to be a limit of the function f(x) as x→a is necessary and sufficient so that 
. 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 = xo 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 xo 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 the point xo, 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 xo the function has a discontinuity second kind.
For example, function y = cot x at 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 to 100×
1.5 = 150, and after another six months - at 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 whatε > 0 no matter what we take, there is something for him natural number N, such that for all n N the inequality holds|x n -1|< ε.
Let's take any e > 0. Since ; x n -1 =(n+1)/n - 1= 1/n, then to find N it is enough to solve the inequality 1/n< e. Hence n>1/ e and, therefore, N can be taken as an integer part of 1/ e , N = E(1/ e ). 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. When 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 .
Solution. 
.
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.
Widget for calculating limits on-line
In the upper window, instead of sin(x)/x, enter the function whose limit you want to find. In the lower window, enter the number to which x tends and click the Calcular button, get the desired limit. And if in the result window you click on Show steps in the upper right corner, you will get a detailed solution.
Rules for entering functions: sqrt(x)- Square root, cbrt(x) - cube root, exp(x) - exponent, ln(x) - natural logarithm, sin(x) - sine, cos(x) - cosine, tan(x) - tangent, cot(x) - cotangent, arcsin(x) - arcsine, arccos(x) - arccosine, arctan(x) - arctangent. Signs: * multiplication, / division, ^ exponentiation, instead infinity Infinity. Example: the function is entered as sqrt(tan(x/2)).
