In proving the properties of the limit of a function, we made sure that nothing really was required from the punctured neighborhoods in which our functions were defined and which arose in the course of proofs, except for the properties indicated in the introduction to the previous paragraph 2. This circumstance serves as a justification for singling out the following mathematical object.
A. Base; definition and main examples
Definition 11. A set B of subsets of a set X will be called a base in a set X if two conditions are met:
In other words, the elements of the collection B are non-empty sets, and the intersection of any two of them contains some element from the same collection.
Let us indicate some of the most commonly used bases in the analysis.
If then instead they write and say that x tends to a from the right or from the side large values(respectively, to the left or from the side of smaller values). When a short record is accepted instead of
The record will be used instead of It means that a; tends over the set E to a, remaining greater (less) than a.
then instead they write and say that x tends to plus infinity (respectively, to minus infinity).
The notation will be used instead
When instead of we (if this does not lead to misunderstanding) we will write, as is customary in the theory of the limit of a sequence,
Note that all the listed bases have the feature that the intersection of any two elements of the base is itself an element of this base, and not only contains some element of the base. We will meet with other bases when studying functions that are not given on the real axis.
We also note that the term “base” used here is a short designation of what is called “filter basis” in mathematics, and the base limit introduced below is the most essential part for analysis of the concept of filter limit created by the modern French mathematician A. Cartan
b. Base function limit
Definition 12. Let be a function on the set X; B is a base in X. A number is called the limit of a function with respect to the base B if for any neighborhood of the point A there is an element of the base whose image is contained in the neighborhood
If A is the limit of the function with respect to base B, then we write
Let's repeat the definition of the limit by the base in logical symbolism:
Since we are now considering functions with numeric values, it is useful to keep in mind the following form of this basic definition:
In this formulation, instead of an arbitrary neighborhood V(A), we take a neighborhood that is symmetric (with respect to the point A) (e-neighborhood). The equivalence of these definitions for real-valued functions follows from the fact that, as already mentioned, any neighborhood of a point contains some symmetric neighborhood of the same point (carry out the proof in full!).
We have given a general definition of the limit of a function with respect to the base. Above were considered examples of the most common bases in the analysis. In a specific problem where one or another of these bases appears, it is necessary to be able to decipher the general definition and write it down for a particular base.
Considering examples of bases, we, in particular, introduced the concept of a neighborhood of infinity. If this concept is used, then in accordance with common definition limit it is reasonable to adopt the following conventions:
or, which is the same,
Usually, by means a small value. In the above definitions, this is, of course, not the case. In accordance with the accepted conventions, for example, we can write
In order to be considered proven in the general case of a limit over an arbitrary base, all those theorems about limits that we proved in Section 2 for a special base , it is necessary to give the appropriate definitions: finally constant, finally bounded, and infinitely small for a given base of functions.
Definition 13. A function is called finally constant at base B if there exists a number and such an element of the base, at any point of which
Definition 14. A function is called bounded at base B or finally bounded at base B if there exists a number c and such an element of the base, at any point of which
Definition 15. A function is called infinitesimal with base B if
After these definitions and the basic observation that only base properties are needed to prove limit theorems, we can assume that all the properties of the limit established in Section 2 are valid for limits over any base.
In particular, we can now talk about the limit of a function at or at or at
In addition, we have secured the possibility of applying the theory of limits even in the case when the functions are not defined on numerical sets; this will prove to be especially valuable in the future. For example, the length of a curve is a numerical function defined on some class of curves. If we know this function on broken lines, then by passing to the limit we determine it for more complex curves, for example, for a circle.
At the moment, the main benefit of the observation made and the concept of base introduced in connection with it is that they save us from checks and formal proofs of limit theorems for each specific type of passage to the limit or, in our current terminology, for each specific type bases
In order to finally get used to the concept of a limit over an arbitrary base, we will prove the further properties of the limit of a function in a general form.
In this article, we will explain 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.
The concept of a limit
In mathematics, the concept of infinity, denoted by the symbol ∞, is fundamentally important. It should be understood as an infinitely large + ∞ or an infinitely small - ∞ number. When we talk about infinity, we often mean both of these meanings at once, but the notation of the form + ∞ or - ∞ should not be replaced simply with ∞.
The function limit is written as lim x → x 0 f (x) . At the bottom, we write the main argument x, and use the arrow to indicate which value x 0 it will tend to. If the value x 0 is a specific real number, then we are dealing with the limit of the function at a point. If the value x 0 tends to infinity (it does not matter, ∞, + ∞ or - ∞), then we should talk about the limit of the function at infinity.
The limit is finite and infinite. If it is equal to a specific real number, i.e. lim x → x 0 f (x) = A , then it is called the 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 define either a finite or an infinite value, this 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 is number sequences, 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 will converge to A for any infinitely large sequence of arguments (negative or positive).
The function limit is written as follows: lim x → ∞ f (x) = A .
Definition 2
As x → ∞, the limit of the 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 notation looks like lim x → ∞ f (x) = ∞ .
Example 1
Prove the equality lim x → ∞ 1 x 2 = 0 using the basic definition of a 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 picture:
x = - 1 , - 2 , - 3 , . . . , - n , . . .
1 1 > 1 4 > 1 9 > 1 16 > . . . > 1 - n 2 > . . .
Here, too, one can see a monotonic decrease to zero, which confirms the correctness of the given in the equality condition:
Answer: The correctness of the given in the condition of equality is confirmed.
Example 2
Calculate limit lim x → ∞ e 1 10 x .
Solution
Let's start, as before, by writing 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, so f (x) = lim x → + ∞ e 1 10 x = + ∞
We proceed to write 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 of the problem is clearly shown in the illustration. The blue dots mark the sequence of positive values, the green dots mark the sequence of negative ones.
Answer: lim x → ∞ e 1 10 x = + ∞ , pr and x → + ∞ 0 , pr and x → - ∞ .
Let us pass to the method of calculating the limit of a function at a point. To do this, we need to know how to properly define the one-sided limit. This will also be useful to us in order to find the vertical asymptotes of the function graph.
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 written in writing as lim x → a - 0 f (x) = B .
Now we formulate what is the limit of the function on the right.
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 the function f (x) at some point when it has equal limits on the left and right side, i.e. lim x → a f (x) = lim x → a - 0 f (x) = lim x → a + 0 f (x) = B . In the case of infinity of both limits, the limit of the function at the starting point will also be infinite.
Now we will explain these definitions by writing down the solution of 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 the 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 , so lim x → 2 + 0 1 6 (x - 8) 2 - 8 = - 2 .
We have obtained 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 exists at the point x 0 = 2, 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 in more depth, we advise you to read the article about the continuity of a function at a point and the main types of discontinuity points.
If you notice a mistake in the text, please highlight it and press Ctrl+Enter
Definition of sequence and function limits, properties of limits, first and second remarkable limits, examples.
constant number A called limit sequences(x n) if for any arbitrarily small positive number ε > 0 there exists a number N such that all values x n, for which n>N, satisfy the inequality
Write it 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 that has a limit is called converging, otherwise - divergent.
The concept of the limit of a function is a generalization of the concept of the limit of a sequence, since the limit of a sequence can be considered as the limit of the function x n = f(n) of an integer argument n.
Let a function f(x) be given and let a - limit point the domain of definition of this function D(f), i.e. such a point, any neighborhood of which contains points of the set D(f) different from 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 defining the limit of a function according to Heine, or " in the language of sequences”.
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 δ >0 (depending on ε) such 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 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 as
In the event that the sequence (f(x n)) increases (or decreases) indefinitely 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 as:
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, use the following theorems.
Theorem 1 . If every limit exists
(6.4)
(6.5)
(6.6)
Comment. Expressions of the form 0/0, ∞/∞, ∞-∞ 0*∞ are indefinite, for example, the ratio of two infinitesimal or infinitely large quantities, and finding a limit of this kind is called “uncertainty disclosure”.
Theorem 2.
those. it is possible to pass to the limit at the base of the degree at a constant exponent, in particular, 
Theorem 3.
(6.11)
Where e» 2.7 is the base of the natural logarithm. Formulas (6.10) and (6.11) are called the first remarkable limit and the second remarkable limit.
The corollaries 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 write +0 instead of the symbol 0+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 = xo function f(x) It has gap. Consider the function y = 1/x. The domain 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 of its neighborhoods, i.e., any open interval containing the point 0 contains points from D(f), but it does not itself belong to this set. The value f(x o)= f(0) is not defined, so the function has a discontinuity at the point x o = 0.
The function f(x) is called continuous on the right at a point x o if limit
And continuous on the left at a point x o if limit
Continuity of a function at a point x o is equivalent to its continuity at this point both on the right and on the left.
For a function to be continuous at a point x o, for example, on the right, it is necessary, firstly, that there is 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 gap.
1. If the limit exists and is not equal to f(x o), then they say that function f(x) at the point xo has break of the first kind, or jump.
2. If the limit is +∞ or -∞ or does not exist, then they say that in point x o the function has a break 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 integer abscissas has discontinuities of the first kind, or jumps.
A function that is continuous at every point of 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: the growth of the contribution according to the law of compound interest, the growth of the country's population, the decay of a radioactive substance, the multiplication of bacteria, etc.
Consider example of Ya. I. Perelman, which gives the 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 connection is made more often, then the capital grows faster, since a large amount is involved in the formation of interest. Let's take a purely theoretical, highly simplified example. Let the bank put 100 den. units at the rate of 100% per annum. If interest-bearing money is added to the fixed capital only after a year, then by this time 100 den. units will turn into 200 den. Now let's see what 100 den will turn into. units, if interest money is added to the fixed capital every six months. After half a year 100 den. units will grow by 100 × 1.5 = 150, and in another six months - by 150 × 1.5 = 225 (money 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 timeframe for adding interest money to 0.1 year, 0.01 year, 0.001 year, and so on. Then out of 100 den. units a year later:
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 of joining interest, the accrued capital does not grow indefinitely, but approaches a certain limit equal to approximately 271. The capital placed at 100% per annum cannot increase 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 whatever ε > 0 we take, there is a natural number N for it, 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 enough to solve the inequality 1/n<ε. Отсюда n>1/ε and, therefore, N can be taken as the integer part of 1/ε N = E(1/ε). We thus proved that the limit .
Example 3.2. Find the limit of a sequence given by a common term
.
Solution. Apply the limit sum theorem and find the limit of each term. As n → ∞, the numerator and denominator of each term tends to infinity, and we cannot apply the quotient limit theorem directly. Therefore, we first transform x n, dividing the numerator and denominator of the first term by n 2, and the second n. Then, applying the quotient limit theorem and the sum limit theorem, we find:
Example 3.3. 
. Find .
Here we have used the degree limit 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 difference limit theorem, since we have an uncertainty of the form ∞-∞. Let's transform the formula of the general term:
Example 3.5. Given a function f(x)=2 1/x . Prove that the limit does not exist.
Solution. We use the definition 1 of the limit of a function in terms of a sequence. 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's choose now 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 the limit does not exist.
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 \u003d p n, then sin x n \u003d sin (p n) = 0 for all n and limit If
xn=2 p n+ p /2, then sin x n = sin(2 p n+ p /2) = sin p /2 = 1 for all n and hence the limit. Thus does not exist.
The definition of the finite limit of a sequence is given. Related properties and an equivalent definition are considered. A definition is given that a point a is not a limit of a sequence. Examples are considered in which the existence of a limit is proved using the definition.
ContentSee also: Sequence limit - basic theorems and properties
Main types of inequalities and their properties
Here we consider the definition of the finite limit of a sequence. The case of a sequence converging to infinity is discussed on the page "Definition of an infinitely large sequence".
The limit of a sequence is a number a if for any positive number ε > 0 there exists a natural number N ε depending on ε such that for all natural numbers n > N ε the inequality| x n - a|< ε .
Here x n is the element of the sequence with number n . Sequence limit denoted like this:
.
Or at .
Let's transform the inequality:
;
;
.
It follows from the definition that if the sequence has a limit a, then no matter what ε - neighborhood of the point a we choose, only a finite number of elements of the sequence, or none at all (empty set), can be outside of it. And any ε - neighborhood contains an infinite number of elements. Indeed, by setting a certain number ε , we thereby have a number . So all elements of the sequence with numbers , by definition, are in the ε - neighborhood of the point a . The first elements can be anywhere. That is, outside the ε - neighborhood there can be no more than elements - that is, a finite number.
We also note that the difference does not have to monotonously tend to zero, that is, to decrease all the time. It can tend to zero not monotonically: it can either increase or decrease, having local maxima. However, these maxima, with increasing n, should tend to zero (perhaps also not monotonously).
Using the logical symbols of existence and universality, the definition of the limit can be written as follows:
(1)
.
Determining that a is not a limit
Now consider the converse assertion that the number a is not the limit of the sequence.
Number a is not the limit of the sequence, if there exists such that for any natural n there exists such a natural m >n, What
.
Let's write this statement using logical symbols.
(2)
.
The assertion that the number a is not the limit of the sequence, means that
you can choose such an ε - neighborhood of the point a, outside of which there will be an infinite number of elements of the sequence.
Consider an example. Let a sequence with a common element be given
(3)
Any neighborhood of a point contains an infinite number of elements. However, this point is not the limit of the sequence, since any neighborhood of the point also contains an infinite number of elements. Take ε - a neighborhood of a point with ε = 1
. This will be the interval (-1, +1)
. All elements except the first one with even n belong to this interval. But all elements with odd n are outside this interval because they satisfy the inequality x n > 2
. Since the number of odd elements is infinite, there will be an infinite number of elements outside the selected neighborhood. Therefore, the point is not the limit of the sequence.
Let us now show this by strictly adhering to assertion (2). The point is not the limit of the sequence (3), since there exists such , so that, for any natural n , there is an odd n for which the inequality
.
It can also be shown that any point a cannot be the limit of this sequence. We can always choose an ε - neighborhood of the point a that does not contain either the point 0 or the point 2. And then there will be an infinite number of elements of the sequence outside the chosen neighborhood.
Equivalent definition of sequence limit
We can give an equivalent definition of the limit of a sequence if we expand the concept of ε - neighborhood. We will get an equivalent definition if instead of ε-neighbourhood, any neighborhood of the point a will appear in it. The neighborhood of a point is any open interval containing that point. Mathematically point neighborhood is defined as follows: , where ε 1 and ε 2 are arbitrary positive numbers.
Then the equivalent definition of the limit is as follows.
The limit of a sequence is such a number a if for any of its neighborhoods there exists such a natural number N so that all elements of the sequence with numbers belong to this neighborhood.
This definition can also be presented in expanded form.
The limit of a sequence is a number a if for any positive numbers and there exists a natural number N depending on and such that the inequalities hold for all natural numbers
.
Proof of the equivalence of definitions
Let us prove that the above two definitions of the limit of a sequence are equivalent.
Let the number a be the limit of the sequence according to the first definition. This means that there is a function , so that for any positive number ε the following inequalities hold:
(4)
at .
Let us show that the number a is the limit of the sequence by the second definition as well. That is, we need to show that there is such a function , so that for any positive numbers ε 1
and ε 2
the following inequalities hold:
(5)
at .
Let we have two positive numbers: ε 1
and ε 2
. And let ε be the smallest of them: . Then ; ; . We use this in (5):
.
But the inequalities hold for . Then inequalities (5) also hold for .
That is, we have found a function such that inequalities (5) hold for any positive numbers ε 1
and ε 2
.
The first part is proven.
Now let the number a be the limit of the sequence according to the second definition. This means that there is a function , so that for any positive numbers ε 1
and ε 2
the following inequalities hold:
(5)
at .
Let us show that the number a is the limit of the sequence and by the first definition. For this you need to put . Then, for , the following inequalities hold:
.
This corresponds to the first definition with .
The equivalence of the definitions is proved.
Examples
Example 1
Prove that .
(1)
.
In our case ;
.
.
Let's use the properties of inequalities. Then if and , then
.
.
Then
at .
This means that the number is the limit of the given sequence:
.
Example 2
Using the definition of the limit of a sequence, prove that
.
We write down the definition of the limit of a sequence:
(1)
.
In our case , ;
.
We enter positive numbers and:
.
Let's use the properties of inequalities. Then if and , then
.
That is, for any positive , we can take any natural number greater than or equal to :
.
Then
at .
.
Example 3
.
We introduce the notation , .
Let's transform the difference:
.
For natural n = 1, 2, 3, ...
we have:
.
We write down the definition of the limit of a sequence:
(1)
.
We enter positive numbers and:
.
Then if and , then
.
That is, for any positive , we can take any natural number greater than or equal to :
.
Wherein
at .
This means that the number is the limit of the sequence:
.
Example 4
Using the definition of the limit of a sequence, prove that
.
We write down the definition of the limit of a sequence:
(1)
.
In our case , ;
.
We enter positive numbers and:
.
Then if and , then
.
That is, for any positive , we can take any natural number greater than or equal to :
.
Then
at .
This means that the number is the limit of the sequence:
.
References:
L.D. Kudryavtsev. Well mathematical analysis. Volume 1. Moscow, 2003.
CM. Nikolsky. Course of mathematical analysis. Volume 1. Moscow, 1983.
Let the function y=ƒ(x) be defined in some neighborhood of the point x o, except, perhaps, for the point x o itself.
Let us formulate two equivalent definitions of the limit of a function at a point.
Definition 1 (in the "language of sequences", or according to Heine).
The number A is called the limit of the function y \u003d ƒ (x) in the furnace x 0 (or at x® x o), if for any sequence of admissible values of the argument x n, n є N (x n ¹ x 0) converging to x o the sequence of corresponding values of the function ƒ(х n), n є N, converges to the number A
In this case, write
or ƒ(x)->A at x→x o. The geometric meaning of the limit of a function: means that for all points x sufficiently close to the point x o, the corresponding values of the function differ arbitrarily little from the number A.
Definition 2 (in the "language of ε", or after Cauchy).
The number A is called the limit of the function at the point x o (or at x → x o) if for any positive ε there is a positive number δ such that for all x¹ x o satisfying the inequality |x-x o |<δ, выполняется неравенство |ƒ(х)-А|<ε.
The geometric meaning of the function limit:
if for any ε-neighbourhood of the point A there is such a δ-neighborhood of the point x o such that for all x¹ ho from this δ-neighborhood the corresponding values of the function ƒ(x) lie in the ε-neighbourhood of the point A. In other words, the points of the graph of the function y = ƒ(x) lie inside a strip of width 2ε bounded by straight lines y=A+ ε , y=A-ε (see Fig. 110). Obviously, the value of δ depends on the choice of ε, so we write δ=δ(ε).
<< Пример 16.1
Prove that
Solution: Take an arbitrary ε>0, find δ=δ(ε)>0 such that for all x satisfying the inequality |х-3|< δ, выполняется неравенство |(2х-1)-5|<ε, т. е. |х-3|<ε.
Taking δ=ε/2, we see that for all x satisfying the inequality |x-3|< δ, выполняется неравенство |(2х-1)-5|<ε. Следовательно, lim(2x-1)=5 при х –>3.
<< Пример 16.2
16.2. One-sided limits
In the definition of the limit of the function, it is considered that x tends to x 0 in any way: remaining less than x 0 (to the left of x 0), greater than x o (to the right of x o), or fluctuating around the point x 0 .
There are cases when the method of approaching the argument x to xo significantly affects the value of the limit of the function. Therefore, the concept of one-sided limits is introduced.
The number A 1 is called the limit of the function y \u003d ƒ (x) on the left at the point x o, if for any number ε> 0 there is a number δ \u003d δ (ε)> 0 such that for x є (x 0 -δ; x o), the inequality |ƒ(x)-A|<ε. Предел слева записывают так: limƒ(х)=А при х–>x 0 -0 or briefly: ƒ (x o- 0) \u003d A 1 (Dirichlet notation) (see Fig. 111).
The limit of the function on the right is defined similarly, we write it using symbols:
Briefly, the limit on the right is denoted by ƒ(x o +0)=A.
The limits of a function on the left and right are called one-sided limits. Obviously, if exists, then both one-sided limits exist, and A=A 1 =A 2 .
The converse statement is also true: if both limits ƒ(x 0 -0) and ƒ(x 0 +0) exist and they are equal, then there is a limit and A \u003d ƒ(x 0 -0).
If A 1 ¹ A 2, then this aisle does not exist.
16.3. Limit of the function at x ® ∞
Let the function y=ƒ(x) be defined in the interval (-∞;∞). The number A is called function limitƒ(x) at x→ ∞ , if for any positive number ε there is such a number М=М()>0 that for all х satisfying the inequality |х|>М the inequality |ƒ(х)-А|<ε. Коротко это определение можно записать так:
The geometric meaning of this definition is as follows: for "ε>0 $ M>0, that for x є(-∞; -M) or x є(M; +∞) the corresponding values of the function ƒ(x) fall into the ε-neighborhood of the point A , i.e., the points of the graph lie in a strip of width 2ε, bounded by straight lines y \u003d A + ε and y \u003d A-ε (see Fig. 112).
16.4. Infinitely large function (b.b.f.)
The function y=ƒ(x) is called infinitely large for x→x 0 if for any number M>0 there is a number δ=δ(M)>0, which for all x satisfying the inequality 0<|х-хо|<δ, выполняется неравенство |ƒ(х)|>M.
For example, the function y=1/(x-2) is a b.b.f. at x->2.
If ƒ(x) tends to infinity as x→x o and takes only positive values, then we write
if only negative values, then
The function y \u003d ƒ (x), given on the entire number line, called infinite for x→∞, if for any number M>0 there is such a number N=N(M)>0 that for all x satisfying the inequality |x|>N, the inequality |ƒ(x)|>M is satisfied. Short:
For example, y=2x has a b.b.f. at x→∞.
Note that if the argument х, tending to infinity, takes only natural values, i.e., хєN, then the corresponding b.b.f. becomes an infinitely large sequence. For example, the sequence v n =n 2 +1, n є N, is an infinitely large sequence. Obviously, every b.b.f. in a neighborhood of the point x o is unbounded in this neighborhood. The converse is not true: an unbounded function may not be a b.b.f. (For example, y=xsinx.)
However, if limƒ(x)=A for x→x 0 , where A is a finite number, then the function ƒ(x) is bounded in the vicinity of the point x o.
Indeed, from the definition of the limit of the function it follows that for x → x 0 the condition |ƒ(x)-A|<ε. Следовательно, А-ε<ƒ(х)<А+ε при х є (х о -ε; х о +ε), а это и означает, что функция ƒ (х) ограничена.
Loading...