Theorem 1. The limit of the algebraic sum of two, three and in general a certain number functions is equal algebraic sum the limits of these functions, i.e.
Proof. Let us carry out the proof for two terms, since it can be done in the same way for any number of terms. Let.Then f(x)=b+b(x) And g(x)=c+в(x), Where b And V- infinitesimal functions. Hence,
f(x) + g(x)=(b + c) + (b(x) + c(x)).
Because b+c There is constant, A b(x) + c(x)- the function is infinitesimal, then
Theorem 2. The limit of the product of two, three and in general finite number functions equal to the product the limits of these functions:
Proof. Let be. Hence, f(x)=b+b(x) And g(x)=c+в(x) And
fg = (b + b)(c + c) = bc + (bc + cb + bc).
Work bc there is a constant value. Function bв + c b + bv based on the properties of infinitesimal functions, there is an infinitesimal quantity. That's why.
Corollary 1. Constant multiplier can be taken beyond the limit sign:
Corollary 2. Degree limit equal to the power limit:
Example..
Theorem 3. The limit of the quotient of two functions is equal to the quotient of the limits of these functions if the limit of the denominator is different from zero, i.e.
Proof. Let be. Hence, f(x)=b+b(x) And g(x)=c+в(x), Where b, c- infinitesimal. Let's consider the quotient
A fraction is an infinitesimal function because the numerator is infinitely small function, and the denominator has a limit c 2 ?0.
3. Let's consider. At x>1 the numerator of the fraction tends to 1, and the denominator tends to 0. But since, i.e. is an infinitesimal function at x> 1, then.
Theorem 4. Let three functions be given f(x), u(x) And v(x), satisfying the inequalities u (x)?f(x)? v(x). If the functions u(x) And v(x) have the same limit at x>a(or x>?), then the function f(x) tends to the same limit, i.e. If
The meaning of this theorem is clear from the figure.
The proof of Theorem 4 can be found, for example, in the textbook: Piskunov N. S. Differential and integral calculus, vol. 1 - M.: Nauka, 1985.
Theorem 5. If at x>a(or x>?) function y=f(x) accepts non-negative values y?0 and at the same time tends to the limit b, then this limit cannot be negative: b?0.
Proof. We will carry out the proof by contradiction. Let's pretend that b<0 , Then |y - b|?|b| and, therefore, the difference modulus does not tend to zero when x>a. But then y does not reach the limit b at x>a, which contradicts the conditions of the theorem.
Theorem 6. If two functions f(x) And g(x) for all values of the argument x satisfy the inequality f(x)? g(x) and have limits, then there is inequality b?c.
Proof. According to the conditions of the theorem f(x)-g(x) ?0, therefore, by Theorem 5, or.
Basic theorems about limits.
1. The limit of the algebraic sum of two, three, and generally a certain number of variables is equal to the algebraic sum of the limits of these variables, i.e.
lim (u 1 + u 2 + … + u n) = lim u 1 + lim u 2 + … + lim u n
2. The limit of the product of a certain number of variables is equal to the product of the limits of these variables, i.e.
lim (u 1 × u 2 × … × u n) = lim u 1 × lim u 2 × … × lim u n
3. The limit of the quotient of two variables is equal to the quotient of the limits of these variables if the limit of the denominator is different from zero, i.e. If lim V ¹ 0 .
3. If for the corresponding function values u = u(x), z = z(x), v = v(x) the inequalities are satisfied u £ z £ v and wherein u(x) And v(x) at X ® a (or X ® ¥ ) tend to the same limit b, That z = z(x) at X ® a (or X ® ¥) tends to the same limit.
Theorem 4 allows us to prove the validity of an important relation called first remarkable limit
. 
(2.1)
From (2.1) follows the equivalence of infinitesimals X And sin x: sin x ~x.
| y |
| y = sin x |
| x |
| y = x |
| Rice. 2.3 |
Another important relation of the theory of limits, called the second remarkable limit is view: 
(2.2)
Number e– irrational (as well as number p) and can be written as an infinite decimal non-periodic fraction e = 2.71828…; plays important role V computational mathematics, serving, in particular, as the basis natural logarithm, denoted ln x = log e x. Function y = e x called exponential function (sometimes denoted as exp x). The following equalities can be useful in solving problems of the theory of limits: 
. You can also replace infinitesimal quantities with their equivalents:
Continuity of functions. Function y = f(x) A If:
1.This function is defined in a certain neighborhood of the point A and at the very point;
2.There is a limit to the function 
and it is equal to the value of the function at this point, i.e. 
. Another definition can be proposed. Let the argument x 0 will receive an increment Dx and will take the value x = x 0 + Dx. IN general case the function will also receive some increment Dу = f(х 0 + Dх) – f(х 0).
Function f(x) called continuous at a point x 0, if it is defined at this point and some neighborhood of it and if an infinitesimal increment of the argument corresponds to an infinitesimal increment of the function, i.e.
(2.3) or (2.3`)
Here is the formulation of the theorem: Every elementary function is continuous at every point at which it is defined and we obtain a corollary that is important for solving problems in the theory of limits. Let us write the continuity condition in the form 
or, what is the same, 
. But 
and therefore 
(2.4), i.e. for any continuous function at all points of its domain of definition, relation (2.4) is valid – limit of a function equal to function limit(the symbols (and corresponding operations) of the limit and function can be swapped): .
Example:
In some cases it is convenient to use the following relationship:
They say that if a function f(x) continuous at every point of some interval (a, b), Where a< b
, then the function is continuous on this interval. The point inside or on the boundary of the domain of definition at which the continuity condition is violated is called breaking point. If there are finite limits 
And 
, and not all three numbers b 1, b 2 And f(a) equal to each other, period A called discontinuity point of the first kind. These points are divided into points jump, When b 1 ¹ b 2(the jump is b 2 - b 1) and points repairable gap, When b 1 = b 2. Discontinuity points that are not discontinuity points of the first kind are called points rupture of the second kind. At these points at least one of the one-sided limits does not exist (Example - “infinite” gap: 
).
Let's consider some properties of continuous functions (proofs of the theorems can be found in the recommended literature).
1. If the function f(x) continuous on some segment , then there is at least one point on this segment x = x 1 such that the value of the function at this point will satisfy the relation f(x 1) ³f(x) , Where X– any other point on the segment, and there is at least one point x 2 such that the value of the function at this point will satisfy the relationf(x 2) ≤ f(x).
| y 1 |
| y 2 |
| y 3 |
| x |
| a |
| m |
| M |
| V |
| Rice. 2.4 |
(Note that on the interval (a, b) the theorem may not be true. Example: y = x– the function does not have on the interval (a, b) the greatest and lowest values, because does not reach values A And b!)
| at |
| at 2 |
| A |
| V |
| X |
| at 1 |
| Rice. 2.5 |
| X |
3. If the function f(x) defined and continuous on the segment and at the ends of this segment takes unequal values f(a) = A And f(b) = B whatever the number ism , enclosed between numbers A And IN, there is such a point x = c, concluded between a And b, What f(c) = m (it is easy to see that Theorem 2 is a special case of Theorem 3).
FUNCTIONS AND LIMITS IX
§ 212. Basic theorems on the limits of functions
First of all, note that not for every function at = f (X ) there is a limit f (X ). So, for example, when x -> π / 2 function values at = tg X (Fig. 303) or grow unlimitedly (with X < π / 2), or decrease without limit (with X > π / 2).
Therefore, no number can be specified b , to which the values of this function would tend.
Another example. Let
The graph of this function is presented in Figure 304.
When the argument values X approach 0, remaining negative, the corresponding function values tend to 1. When the argument values X approach 0, remaining positive, the corresponding function values tend to -2. At the very same point X = 0 the function turns to 0. Obviously, indicate one number to which all values would tend at when approaching X to 0, no. That's why this function has no limit at X -> 0.
When talking about the limit of a function in the future, we will always assume that this limit exists.
Assumption of the existence of a limit f (X ) does not mean that this limit coincides with the value of the function f (X ) at point x = a . For example, consider the function whose graph is presented in Figure 305.
Obviously the limit f (X ) exists and is equal to 1. But at the point itself X = 0 the function takes a value equal to 2. Therefore, in in this case
f (X ) =/= f (0).
If the function y = f (X ) satisfies the condition
f (X ) = f (a ),
then it's called continuous at the point x = a . If the specified condition is not met, then the function f (X ) is called explosive at the point x = a ."
All elementary functions(For example, y = x n , at = sin X , at = tg X , at = tan 2 X + tg X etc.) are continuous at every point at which they are defined.
Function at = f (X ) is called continuous in the interval [a, b ] if it is continuous at every point of this interval. For example, the function at = tg x is continuous in the interval[- π / 4 , π / 4 ], functions at = sin x And y =cos x continuous in any interval, etc.
We present without proof the main theorems on the limits of functions. These theorems are quite similar to those that we considered (also without proof) earlier when studying the limits of number sequences.
1. The limit of a constant is equal to this constant itself:
c = c .
2. The constant factor can be taken beyond the limit sign:
[ k f (X )] = k f (X ).
3. Limit on the sum (difference) of functions equal to the sum(differences) between the limits of these functions:
[ f (X ) ± g (X )] = f (X ) ± g (x ).
4. The limit of a product of functions is equal to the product of the limits of these functions:
[ f (X ) g (X )] = f (X ) g (x ).
5. Limit of the ratio of two functions equal to the ratio limits of these functions, unless the limit of the divisor equal to zero:
Let's look at several typical examples of finding the limits of functions.
Example 1. Find
At X -> 3 The numerator and denominator of this fraction tend to zero. Therefore, direct application of the theorem on the limit of a quotient is impossible here. However given fraction can be shortened:
(Please note the following important feature, characteristic of the considered example. When we talk about the limit f (X ), then we usually assume that the function f (X ) is defined at all points sufficiently close to the point x = a . However, the function is only defined for positive values X . Therefore, when considering the limit of this function, we are actually assuming that X -> 0, remaining positive all the time. In such cases, they talk not just about the limit, but about unilaterally limit. We will encounter similar examples later in the exercises for this section.)
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. Arithmetic properties are considered; theorems related to inequalities; Cauchy convergence criterion; limit complex function; properties of infinitely small, infinitely large and monotonic functions. The definition of a function is given.
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:
.
Numeric function called limited, if there is a number M such that for all:
.
Top edge or exact upper bound real function is the smallest number that limits the range of its 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′: .
Top edge functions can be denoted as follows:
.
Respectively bottom edge or accurate 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 end point except perhaps for 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). f (x) as x → x 0
equals infinity, if for anyone, 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:
.
Demon final limit 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 neighborhood of a point, we can give universal definition finite and infinite limit of a function, applicable for both 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 in any way 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 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
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 (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 is an infinitesimal function at .
"Properties of infinitesimal functions".
Infinitely large functions
Definition
A function is said to be infinitely large if
.
Sum or difference limited function, on some punctured neighborhood of the point , and an infinitely large function at is infinitely 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 infinitesimals and infinitely great features can be supplemented with the following relations:
,
,
,
.
Additional formulas, linking infinity symbols can be found on the page
"Points at infinity and their properties."
Limits of monotonic functions
Definition
Function defined on some set 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.