Proving the properties of the limit of a function, we were convinced that from the punctured neighborhoods in which our functions were defined and which arose in the process of proof, in addition to the properties indicated in the introduction to previous point 2, really nothing was needed. This circumstance serves as a justification for identifying the following mathematical object.
A. Base; definition and basic examples
Definition 11. A collection B of subsets of a set X will be called a base in the set X if two conditions are met:
In other words, the elements of 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 analysis.
If then instead they write and say that x tends to a from the right or from the side large values(respectively, on the left or on the side of smaller values). When a short record is accepted instead
The entry will be used instead of She means that a; tends over the set E to a, remaining greater (smaller) than a.
then instead they write and say that x tends to plus infinity (respectively, to minus infinity).
The entry will be used instead
When instead of (if this does not lead to a misunderstanding) we will, as is customary in the theory of the limit of a sequence, write
Note that all of the listed bases have the peculiarity 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 encounter other bases when studying functions that are not specified on the number axis.
Note also that the term "base" used here is short designation what in mathematics is called the “filter basis”, 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. Function limit by base
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 base B if for any neighborhood of point A there is an element of the base whose image is contained in the neighborhood
If A is the limit of a function with respect to base B, then write
Let us repeat the definition of the limit by base in logical symbolism:
Since we are now looking at functions with numerical values, it is useful to keep in mind the following form this basic definition:
In this formulation, instead of an arbitrary neighborhood V (A), a symmetric (with respect to point A) neighborhood (e-neighborhood) is taken. 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 (perform the proof in full!).
We have given a general definition of the limit of a function over a base. Above we discussed examples of the most commonly used databases in analysis. IN specific task, where one or another of these bases appears, you must be able to decipher the general definition and write it down for a specific base.
Considering examples of bases, we, in particular, introduced the concept of a neighborhood of infinity. If we use this concept, then in accordance with general definition It is reasonable to accept the following agreements:
or, what is the same,
Usually we mean a small value. This is, of course, not the case in the above definitions. In accordance with accepted conventions, for example, we can write
In order to be considered proven in general case limit on an arbitrary base, all those theorems about limits that we proved in paragraph 2 for a special base, it is necessary to give the corresponding definitions: finally constant, finally limited and infinitesimal for a given base of functions.
Definition 13. A function is said to be finally constant with base B if there exists a number and an element of the base such that at any point
At the moment, the main benefit of the observation made and the concept of a base introduced in connection with it is that they save us from checks and formal proofs of limit theorems for each specific type of limit passages or, in our current terminology, for each specific type bases
In order to finally become familiar with the concept of a limit over an arbitrary base, we will carry out proofs of further properties of the limit of a function in a general form.
Consider the function %%f(x)%% defined at least in some punctured neighborhood %%\stackrel(\circ)(\text(U))(a)%% of the point %%a \in \overline( \mathbb(R))%% extended number line.
The concept of a Cauchy limit
The number %%A \in \mathbb(R)%% is called limit of the function%%f(x)%% at the point %%a \in \mathbb(R)%% (or at %%x%% tending to %%a \in \mathbb(R)%%), if, what no matter what positive number%%\varepsilon%%, there is a positive number %%\delta%% such that for all points of the punctured %%\delta%% neighborhood of the point %%a%% the function values belong to the %%\varepsilon%% neighborhood of the point %%A%%, or
$$ A = \lim\limits_(x \to a)(f(x)) \Leftrightarrow \forall\varepsilon > 0 ~\exists \delta > 0 \big(x \in \stackrel(\circ)(\text (U))_\delta(a) \Rightarrow f(x) \in \text(U)_\varepsilon (A) \big) $$
This definition is called the %%\varepsilon%% and %%\delta%% definition, proposed by the French mathematician Augustin Cauchy and used with early XIX century to the present, since it has the necessary mathematical rigor and accuracy.
Combining various neighborhoods of the point %%a%% of the form %%\stackrel(\circ)(\text(U))_\delta(a), \text(U)_\delta (\infty), \text(U) _\delta (-\infty), \text(U)_\delta (+\infty), \text(U)_\delta^+ (a), \text(U)_\delta^- (a) %% with surroundings %%\text(U)_\varepsilon (A), \text(U)_\varepsilon (\infty), \text(U)_\varepsilon (+\infty), \text(U) _\varepsilon (-\infty)%%, we get 24 definitions of the Cauchy limit.
Geometric meaning
Geometric meaning function limit
Let us find out what the geometric meaning of the limit of a function at a point is. Let's build a graph of the function %%y = f(x)%% and mark the points %%x = a%% and %%y = A%% on it.
The limit of the function %%y = f(x)%% at the point %%x \to a%% exists and is equal to A if for any %%\varepsilon%% neighborhood of the point %%A%% one can specify such a %%\ delta%%-neighborhood of the point %%a%%, such that for any %%x%% from this %%\delta%%-neighborhood the value %%f(x)%% will be in the %%\varepsilon%%-neighborhood points %%A%%.
Note that by the definition of the limit of a function according to Cauchy, for the existence of a limit at %%x \to a%%, it does not matter what value the function takes at the point %%a%%. Examples can be given where the function is not defined when %%x = a%% or takes a value other than %%A%%. However, the limit may be %%A%%.
Determination of the Heine limit
The element %%A \in \overline(\mathbb(R))%% is called the limit of the function %%f(x)%% at %% x \to a, a \in \overline(\mathbb(R))%% , if for any sequence %%\(x_n\) \to a%% from the domain of definition, the sequence of corresponding values %%\big\(f(x_n)\big\)%% tends to %%A%%.
The definition of a limit according to Heine is convenient to use when doubts arise about the existence of a limit of a function at a given point. If it is possible to construct at least one sequence %%\(x_n\)%% with a limit at the point %%a%% such that the sequence %%\big\(f(x_n)\big\)%% has no limit, then we can conclude that the function %%f(x)%% has no limit at this point. If for two various sequences %%\(x"_n\)%% and %%\(x""_n\)%% having same limit %%a%%, sequences %%\big\(f(x"_n)\big\)%% and %%\big\(f(x""_n)\big\)%% have various limits, then in this case there is also no limit of the function %%f(x)%%.
Example
Let %%f(x) = \sin(1/x)%%. Let's check whether the limit of this function exists at the point %%a = 0%%.
Let us first choose a sequence $$ \(x_n\) = \left\(\frac((-1)^n)(n\pi)\right\) converging to this point. $$
It is clear that %%x_n \ne 0~\forall~n \in \mathbb(N)%% and %%\lim (x_n) = 0%%. Then %%f(x_n) = \sin(\left((-1)^n n\pi\right)) \equiv 0%% and %%\lim\big\(f(x_n)\big\) = 0 %%.
Then take a sequence converging to the same point $$ x"_n = \left\( \frac(2)((4n + 1)\pi) \right\), $$
for which %%\lim(x"_n) = +0%%, %%f(x"_n) = \sin(\big((4n + 1)\pi/2\big)) \equiv 1%% and %%\lim\big\(f(x"_n)\big\) = 1%%. Similarly for the sequence $$ x""_n = \left\(-\frac(2)((4n + 1) \pi) \right\), $$
also converging to the point %%x = 0%%, %%\lim\big\(f(x""_n)\big\) = -1%%.
All three sequences gave different results, which contradicts the Heine definition condition, i.e. this function has no limit at the point %%x = 0%%.
Theorem
The Cauchy and Heine definitions of the limit are equivalent.
Let the function y = ƒ (x) be defined in some neighborhood of the point x o, except, perhaps, 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=ƒ(x) in the furnace x 0 (or at x® x o), if for any sequence acceptable values arguments x n, n є N (x n ¹ x 0), converging to x, the sequence of corresponding values of the function ƒ(x n), n є N, converges to the number A
In this case they write
or ƒ(x)->A at x→x o. The geometric meaning of the limit of a function: means that for all points x that are sufficiently close to the point xo, the corresponding values of the function differ as little as desired from the number A.
Definition 2 (in the “language of ε”, or according to Cauchy).
A number A is called the limit of a function at a 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.
Geometric meaning of the limit of a function:
if for any ε-neighborhood of point A there is a δ-neighborhood of the point x o such that for all x1 xo from this δ-neighborhood the corresponding values of the function ƒ(x) lie in the ε-neighborhood 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 |x-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 defining the limit of a 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 oscillating around the point x 0.
There are cases when the method of approximating the argument x to x o significantly affects the value of the function limit. Therefore, the concepts of one-sided limits are introduced.
The number A 1 is called the limit of the function y=ƒ(x) on the left at the point x o if for any number ε>0 there is a number δ=δ(ε)> 0 such that at 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) = A 1 (Dirichlet notation) (see Fig. 111).
The limit of the function on the right is determined similarly; we write it using symbols:
Briefly, the limit on the right is denoted by ƒ(x o +0)=A.
The left and right limits of a function are called one-sided limits. Obviously, if exists, then both one-sided limits exist, and A = A 1 = A 2.
The converse 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 = ƒ(x 0 -0).
If A 1 ¹ A 2, then this chapel 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 limit of the functionƒ(x) at x→ ∞ , if for any positive number ε there is a number M=M()>0 such that for all x satisfying the inequality |x|>M the inequality |ƒ(x)-A|<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 point A , that is, the points of the graph lie in a strip of width 2ε, limited by the straight lines y=A+ε and y=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 b.b.f. for x->2.
If ƒ(x) tends to infinity as x→x o and takes only positive values, then they write
if only negative values, then
The function y=ƒ(x), defined on the entire number line, called infinitely large as x→∞, if for any number M>0 there is a number N=N(M)>0 such that for all x satisfying the inequality |x|>N, the inequality |ƒ(x)|>M holds. Short:
For example, y=2x has b.b.f. as x→∞.
Note that if the argument x, tending to infinity, takes only natural values, i.e. xєN, 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 a point x o is unbounded in this neighborhood. The converse is not true: an unbounded function may not be 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 limited in the vicinity of the point x o.
Indeed, from the definition of the limit of a function it follows that as 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.
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A little theory.
Limit of the function at x->x 0
Let the function f(x) be defined on some set X and let the point \(x_0 \in X\) or \(x_0 \notin X\)
Let us take from X a sequence of points different from x 0:
x 1 , x 2 , x 3 , ..., x n , ... (1)
converging to x*. The function values at the points of this sequence also form a numerical sequence
f(x 1), f(x 2), f(x 3), ..., f(x n), ... (2)
and one can raise the question of the existence of its limit.
Definition. The number A is called the limit of the function f(x) at the point x = x 0 (or at x -> x 0), if for any sequence (1) of values of the argument x different from x 0 converging to x 0, the corresponding sequence (2) of values function converges to number A.
$$ \lim_(x\to x_0)( f(x)) = A $$
The function f(x) can have only one limit at the point x 0. This follows from the fact that the sequence
(f(x n)) has only one limit.
There is another definition of the limit of a function.
Definition The number A is called the limit of the function f(x) at the point x = x 0 if for any number \(\varepsilon > 0\) there is a number \(\delta > 0\) such that for all \(x \in X, \; x \neq x_0 \), satisfying the inequality \(|x-x_0| Using logical symbols, this definition can be written as
\((\forall \varepsilon > 0) (\exists \delta > 0) (\forall x \in X, \; x \neq x_0, \; |x-x_0| Note that the inequalities \(x \neq x_0 , \; |x-x_0| The first definition is based on the concept of the limit of a number sequence, so it is often called the “in the language of sequences” definition. The second definition is called the “in the language of \(\varepsilon - \delta \)”.
These two definitions of the limit of a function are equivalent and you can use either of them depending on which is more convenient for solving a particular problem.
Note that the definition of the limit of a function “in the language of sequences” is also called the definition of the limit of a function according to Heine, and the definition of the limit of a function “in the language \(\varepsilon - \delta \)” is also called the definition of the limit of a function according to Cauchy.
Limit of the function at x->x 0 - and at x->x 0 +
In what follows, we will use the concepts of one-sided limits of a function, which are defined as follows.
Definition The number A is called the right (left) limit of the function f(x) at the point x 0 if for any sequence (1) converging to x 0, the elements x n of which are greater (less than) x 0, the corresponding sequence (2) converges to A.
Symbolically it is written like this:
$$ \lim_(x \to x_0+) f(x) = A \; \left(\lim_(x \to x_0-) f(x) = A \right) $$
We can give an equivalent definition of one-sided limits of a function “in the language \(\varepsilon - \delta \)”:
Definition a number A is called the right (left) limit of the function f(x) at the point x 0 if for any \(\varepsilon > 0\) there exists \(\delta > 0\) such that for all x satisfying the inequalities \(x_0 Symbolic entries:
In this article we will tell you what the limit of a function is. First, let us explain the general points that are very important for understanding the essence of this phenomenon.
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Limit concept
In mathematics, the concept of infinity, denoted by the symbol ∞, is fundamentally important. It should be understood as an infinitely large + ∞ or an infinitesimal - ∞ number. When we talk about infinity, we often mean both of these meanings at once, but notation of the form + ∞ or - ∞ should not be replaced simply by ∞.
The limit of a function is written as lim x → x 0 f (x) . At the bottom we write the main argument x, and with the help of an arrow we indicate which value x0 it will tend to. If the value x 0 is a concrete real number, then we are dealing with the limit of the function at a point. If the value x 0 tends to infinity (it doesn’t matter whether ∞, + ∞ or - ∞), then we should talk about the limit of the function at infinity.
The limit can be finite or infinite. If it is equal to a specific real number, i.e. lim x → x 0 f (x) = A, then it is called a finite limit, but if lim x → x 0 f (x) = ∞, lim x → x 0 f (x) = + ∞ or lim x → x 0 f (x) = - ∞ , then infinite.
If we cannot determine either a finite or an infinite value, it means that such a limit does not exist. An example of this case would be the limit of sine at infinity.
In this paragraph we will explain how to find the value of the limit of a function at a point and at infinity. To do this, we need to introduce basic definitions and remember what number sequences are, as well as their convergence and divergence.
Definition 1
The number A is the limit of the function f (x) as x → ∞ if the sequence of its values converges to A for any infinitely large sequence of arguments (negative or positive).
Writing the limit of a function looks like this: lim x → ∞ f (x) = A.
Definition 2
As x → ∞, the limit of a function f(x) is infinite if the sequence of values for any infinitely large sequence of arguments is also infinitely large (positive or negative).
The entry looks like lim x → ∞ f (x) = ∞ .
Example 1
Prove the equality lim x → ∞ 1 x 2 = 0 using the basic definition of the limit for x → ∞.
Solution
Let's start by writing a sequence of values of the function 1 x 2 for an infinitely large positive sequence of values of the argument x = 1, 2, 3, . . . , n , . . . .
1 1 > 1 4 > 1 9 > 1 16 > . . . > 1 n 2 > . . .
We see that the values will gradually decrease, tending to 0. See in the picture:
x = - 1 , - 2 , - 3 , . . . , - n , . . .
1 1 > 1 4 > 1 9 > 1 16 > . . . > 1 - n 2 > . . .
Here we can also see a monotonic decrease towards zero, which confirms the validity of this in the equality condition:
Answer: The correctness of this in the equality condition is confirmed.
Example 2
Calculate the limit lim x → ∞ e 1 10 x .
Solution
Let's start, as before, by writing down sequences of values f (x) = e 1 10 x for an infinitely large positive sequence of arguments. For example, x = 1, 4, 9, 16, 25, . . . , 10 2 , . . . → + ∞ .
e 1 10 ; e 4 10 ; e 9 10 ; e 16 10 ; e 25 10 ; . . . ; e 100 10 ; . . . = = 1, 10; 1, 49; 2, 45; 4, 95; 12, 18; . . . ; 22026, 46; . . .
We see that this sequence is infinitely positive, which means f (x) = lim x → + ∞ e 1 10 x = + ∞
Let's move on to writing the values of an infinitely large negative sequence, for example, x = - 1, - 4, - 9, - 16, - 25, . . . , - 10 2 , . . . → - ∞ .
e - 1 10 ; e - 4 10 ; e - 9 10 ; e - 16 10 ; e - 25 10 ; . . . ; e - 100 10 ; . . . = = 0, 90; 0, 67; 0, 40; 0, 20; 0, 08; . . . ; 0.000045; . . . x = 1, 4, 9, 16, 25, . . . , 10 2 , . . . → ∞
Since it also tends to zero, then f (x) = lim x → ∞ 1 e 10 x = 0 .
The solution to the problem is clearly shown in the illustration. Blue dots indicate a sequence of positive values, green dots indicate a sequence of negative values.
Answer: lim x → ∞ e 1 10 x = + ∞ , pr and x → + ∞ 0 , pr and x → - ∞ .
Let's move on to the method of calculating the limit of a function at a point. To do this, we need to know how to correctly define a one-sided limit. This will also be useful to us in order to find the vertical asymptotes of the graph of a function.
Definition 3
The number B is the limit of the function f (x) on the left as x → a in the case when the sequence of its values converges to a given number for any sequence of arguments of the function x n converging to a, if its values remain less than a (x n< a).
Such a limit is denoted in writing as lim x → a - 0 f (x) = B.
Now let’s formulate what the limit of a function on the right is.
Definition 4
The number B is the limit of the function f (x) on the right as x → a in the case when the sequence of its values converges to a given number for any sequence of arguments of the function x n converging to a, if its values remain greater than a (x n > a) .
We write this limit as lim x → a + 0 f (x) = B .
We can find the limit of a function f (x) at a certain point when it has equal limits on the left and right sides, i.e. lim x → a f (x) = lim x → a - 0 f (x) = lim x → a + 0 f (x) = B . If both limits are infinite, the limit of the function at the starting point will also be infinite.
Now we will clarify these definitions by writing down the solution to a specific problem.
Example 3
Prove that there is a finite limit of the function f (x) = 1 6 (x - 8) 2 - 8 at the point x 0 = 2 and calculate its value.
Solution
In order to solve the problem, we need to recall the definition of the limit of a function at a point. First, let's prove that the original function has a limit on the left. Let's write down a sequence of function values that will converge to x 0 = 2 if x n< 2:
f(-2); f (0) ; f (1) ; f 1 1 2 ; f 1 3 4 ; f 1 7 8 ; f 1 15 16 ; . . . ; f 1 1023 1024 ; . . . = = 8, 667; 2, 667; 0, 167; - 0, 958; - 1, 489; - 1, 747; - 1, 874; . . . ; - 1,998; . . . → - 2
Since the above sequence reduces to - 2, we can write that lim x → 2 - 0 1 6 x - 8 2 - 8 = - 2.
6 , 4 , 3 , 2 1 2 , 2 1 4 , 2 1 8 , 2 1 16 , . . . , 2 1 1024 , . . . → 2
The function values in this sequence will look like this:
f (6) ; f (4) ; f (3) ; f 2 1 2 ; f 2 3 4 ; f 2 7 8 ; f 2 15 16 ; . . . ; f 2 1023 1024 ; . . . = = - 7, 333; - 5, 333; - 3, 833; - 2, 958; - 2, 489; - 2, 247; - 2, 124; . . . , - 2,001, . . . → - 2
This sequence also converges to - 2, which means lim x → 2 + 0 1 6 (x - 8) 2 - 8 = - 2.
We found that the limits on the right and left sides of this function will be equal, which means that the limit of the function f (x) = 1 6 (x - 8) 2 - 8 at the point x 0 = 2 exists, and lim x → 2 1 6 (x - 8) 2 - 8 = - 2 .
You can see the progress of the solution in the illustration (green dots are a sequence of values converging to x n< 2 , синие – к x n > 2).
Answer: The limits on the right and left sides of this function will be equal, which means that the limit of the function exists, and lim x → 2 1 6 (x - 8) 2 - 8 = - 2.
To study the theory of limits more deeply, we advise you to read the article on the continuity of a function at a point and the main types of discontinuity points.
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