Limit of a function of two variables.
Concept and examples of solutions

Welcome to the third related lesson FNP, where all your fears finally began to come true =) As many suspected, the concept of a limit extends to a function of an arbitrary number of arguments, which is what we have to figure out today. However, there is optimistic news. It consists in the fact that at the limit is to a certain extent abstract and the corresponding tasks are extremely rare in practice. In this regard, our attention will be focused on the limits of the function of two variables or, as we often write it:.

Many of the ideas, principles and methods are similar to the theory and practice of "normal" limits, which means that at this point you should be able to find limits and most importantly, UNDERSTAND what is single variable function limit... And, as soon as fate brought you to this page, then, most likely, you already know a lot. And if not - it's okay, all the gaps can really be filled in a matter of hours and even minutes.

The events of this lesson unfold in our three-dimensional world, and therefore it would be simply a huge omission not to take an active part in them. First, let's build the well-known cartesian coordinate system in space... Let's get up and walk around the room a little ... ... the floor you walk on is a plane. Let's put an axis somewhere ... well, for example, in any corner so that it doesn't get in the way. Fine. Now, please look up and imagine that there is a spread blanket hanging there. it surface given by the function. Our movement on the floor, as it is easy to understand, simulates the change in independent variables, and we can move only under the blanket, i.e. v domains of a function of two variables... But the fun is just beginning. Just over the tip of your nose, a small cockroach crawls along the blanket, wherever you go, there he goes. Let's call him Freddie. Moving it simulates changing the corresponding function values. (except for those cases when the surface or its fragments are parallel to the plane and the height does not change)... Dear reader named Freddie, do not be offended, this is necessary for science.

Take an awl in our hands and pierce the blanket at an arbitrary point, the height of which we denote by, after which we stick the tool into the floor strictly under the hole - this will be a point. Now we start infinitely close approach a given point , and we have the right to approach on ANY trajectory (each point of which, of course, is included in the domain of definition)... If IN ALL cases Freddie will infinitely close crawl to the puncture to a height and EXACTLY TO THIS HEIGHT, then the function has a limit at the point at :

If, under these conditions, the punctured point is located on the edge of the blanket, then the limit will still exist - it is important that in an arbitrarily small neighborhood the tips of the awl were at least some points from the domain of the function definition. Moreover, as in the case with the limit of a function of one variable, does not matter whether the function is defined at a point or not. That is, our puncture can be closed up with chewing gum. (think that function of two variables is continuous) and this will not affect the situation - remember that the very essence of the limit implies infinitely close approximation, and not "exact approach" to the point.

However, the cloudless life is overshadowed by the fact that, unlike its younger brother, the limit is far more often non-existent. This is due to the fact that there are usually a lot of paths to one point or another on the plane, and each of them must lead Freddie strictly to a puncture (optionally "sealed with gum") and strictly to the height. And there are more than enough bizarre surfaces with equally bizarre breaks, which leads to a violation of this strict condition at some points.

Let's organize the simplest example - take a knife in our hands and cut the blanket so that the punctured point lies on the cut line. Note that the limit still exists, the only thing is that we have lost the right to step into points under the cut line, since this area "fell out" function domain... Now carefully lift the left side of the blanket along the axis, and the right side, on the contrary, move it down or even leave it in place. What changed? And the following has fundamentally changed: if we now approach the point from the left, then Freddie will be at a greater height than if we were approaching this point from the right. So there is no limit.

And of course wonderful limits, where without them. Consider an example that is instructive in every sense:

Example 11

We use a painfully familiar trigonometric formula, where we organize first wonderful limits :

Let's move on to polar coordinates:
If, then

It would seem that the decision goes to a logical outcome and nothing portends trouble, but at the very end there is a great risk of making a serious mistake, the nature of which I already hinted a little bit in Example 3 and described in detail after Example 6. First, the ending, then the comment:

Let's see why it would be bad to write down simply “infinity” or “plus infinity”. Let's look at the denominator: since, then the polar radius tends to infinitesimal positive value:. Besides, . Thus, the sign of the denominator and the entire limit depends only on the cosine:
if the polar angle (2nd and 3rd coordinate quarters:);
if the polar angle (1st and 4th coordinate quarters :).

Geometrically, this means that if you approach the origin from the left, then the surface defined by the function , extends downward to infinity:

In order to give the concept of the limit of a function of several variables, we restrict ourselves to the case of two variables NS and at... By definition, the function f (x, y) has a limit at the point ( NS 0 , at 0) equal to the number A, denoted like this:

(write more f (x, y)>A at (x, y)> (NS 0 , at 0)) if it is defined in some neighborhood of the point ( NS 0 , at 0), with the possible exception of this point itself and if there is a limit

whatever the tending towards ( NS 0 , at 0) a sequence of points ( x k , y k).

Just as in the case of a function of one variable, you can introduce another equivalent definition of the limit of a function of two variables: the function f has at the point ( NS 0 , at 0) limit equal to A if it is defined in some neighborhood of the point ( NS 0 , at 0) with the possible exception of this point itself, and for any e> 0 there is q> 0 such that

| f (x, y) - A | < е (3)

for all (x, y)

0 < < д. (4)

This definition, in turn, is equivalent to the following: for any ε> 0 there is an q-neighborhood of the point ( NS 0 , at 0) such that for all ( x, y) from this neighborhood, other than ( NS 0 , at 0), inequality (3) holds.

Since the coordinates of an arbitrary point ( x, y) the neighborhood of the point ( NS 0 , at 0) can be written as x = x 0 + D NS, y = y 0 + D at, then equality (1) is equivalent to the following equality:

Consider some function defined in a neighborhood of the point ( NS 0 , at 0), except perhaps for this point itself.

Let u = (u NS, SCH at) is an arbitrary vector of length one (| u | 2 = u NS 2 + u at 2 = 1) and t> 0 is a scalar. Points of the form ( NS 0 + t SCH NS , y 0 + t SCH at) (0 < t)

form a ray emerging from ( NS 0 , at 0) in the direction of the vector u. For each u, we can consider the function

f (NS 0 + t SCH NS , y 0 + t SCH at) (0 < t < д)

from scalar variable t, where q is a sufficiently small number.

The limit of this function (one variable t)

f (NS 0 + t SCH NS , y 0 + t SCH at),

f at point ( NS 0 , at 0) in the direction of u.

Example 1. Functions

defined on the plane ( x, y) except for the point NS 0 = 0, at 0 = 0. We have (take into account that and):

(for e> 0 we put d = e / 2 and then | f (x, y)| < е, если < д).

from which it can be seen that the limit μ at the point (0, 0) in different directions is generally different (the unit vector of the ray y = kx, NS> 0, has the form

Example 2. Consider in R 2 function

(NS 4 + at 2 ? 0).

This function at the point (0, 0) on any straight line y = kx passing through the origin has a limit equal to zero:

at NS > 0.

However, this function has no limit at the points (0, 0), because for y = x 2

We will write if the function f is defined in some neighborhood of the point ( NS 0 , at 0), with the possible exception of the point itself ( NS 0 , at 0) and for any N> 0 there is q> 0 such that

| f (x, y)| > N,

since 0< < д.

You can also talk about the limit f, when NS, at > ?:

A equality (5) must be understood in the sense that for any ε> 0 there is such N> 0, which for all NS, at for which | x| > N, |y| > N, function f is defined and the inequality

| f (x, y) - A| < е.

Equalities are true

where could it be NS > ?, at>?. Moreover, as usual, limits (finite) in their left-hand sides exist if there are limits f and c.

Let us prove (7) for example.

Let be ( x k , y k) > (NS 0 , at 0) ((x k , y k) ? (NS 0 , at 0)); then

Thus, the limit on the left-hand side of (9) exists and is equal to the right-hand side of (9), and since the sequence ( x k , y k) tends to ( NS 0 , at 0) according to any law, then this limit is equal to the limit of the function f (x, y) c (x, y) at point ( NS 0 , at 0).

Theorem. if function f (x, y) has a nonzero limit at the point ( NS 0 , at 0), i.e.

then there exists q> 0 such that for all NS, at satisfying the inequalities

0 < < д, (10)

it satisfies the inequality

Therefore, for such (x, y)

those. inequality (11) holds. From inequality (12) for the indicated (x, y) it follows whence for A> 0 and at

A < 0 (сохранение знака).

By definition, the function f (x) = f (x 1 , …, x n ) = A has a limit at the point

x 0 = equal to number A, denoted like this:

(write more f (x) > A (x > x 0)) if it is defined on some neighborhood of the point x 0, except perhaps for herself, and if there is a limit

whatever the striving for x 0 sequence of points NS k from the specified neighborhood ( k= 1, 2, ...) other than x 0 .

Another equivalent definition is as follows: function f has at point x 0 limit equal to A if it is defined in some neighborhood of the point x 0, except, perhaps, for itself, and for any e> 0 there is q> 0 such that

for all NS satisfying the inequalities

0 < |x - x 0 | < д.

This definition, in turn, is equivalent to the following: for any ε> 0 there is a neighborhood U (x 0 ) points x 0 such that for all xU (x 0 ) , NS ? x 0, inequality (13) holds.

Obviously, if the number A there is a limit f (x) v x 0, then A there is a function limit f (x 0 + h) from h at zero point:

and vice versa.

Consider some function f given at all points of the neighborhood of the point x 0, except perhaps the point x 0; let u = (u 1, ..., u NS) is an arbitrary vector of length one (| u | = 1) and t> 0 is a scalar. View points x 0 + t u (0< t) form outgoing from x 0 ray in the direction of the vector u. For each u, we can consider the function

(0 < t < д щ)

from scalar variable t, where d u is a number depending on u. The limit of this function (from one variable t)

if it exists, it is natural to call it the limit f at the point x 0 in the direction of the vector u.

We will write if the function f defined in some neighborhood x 0, except maybe x 0, and for everyone N> 0 there is q> 0 such that | f (x)| > N, since 0< |x - x 0 | < д.

You can talk about the limit f, when NS > ?:

For example, in the case of a finite number A equality (14) must be understood in the sense that for any ε> 0 one can indicate such N> 0, which for points NS for which | x| > N, function f is defined and the inequality takes place.

So the limit of the function f (x) = f (x 1 , ..., NS NS ) from NS variables is defined by analogy in the same way as for a function of two variables.

Thus, we turn to the definition of the limit of a function of several variables.

Number A called the limit of the function f (M) at M > M 0 if for any number e> 0 there is always such a number g> 0 that for any points M other than M 0 and satisfying the condition | MM 0 | < д, будет иметь место неравенство | f (M) - A | < е.

The limit is denoted in the case of a function of two variables

Limit theorems. If functions f 1 (M) and f 2 (M) at M > M 0 each tend to a finite limit, then:

Example 1. Find the limit of a function:

Solution. We transform the limit as follows:

Let be y = kx, then

Example 2. Find the limit of a function:

Solution. We will use the first remarkable limit Then

Example 3. Find the limit of a function:

Solution. We will use the second remarkable limit Then

Consider the plane and the system Oxy Cartesian rectangular coordinates on it (you can also consider other coordinate systems).

We know from analytic geometry that each ordered pair of numbers (x, y) you can match a single point M plane and vice versa, to each point M the plane corresponds to a single pair of numbers.

Therefore, in what follows, speaking about a point, we will often mean the corresponding pair of numbers (x, y) and vice versa.

Definition 1.2 The set of pairs of numbers (x, y) satisfying the inequalities is called a rectangle (open).

On a plane, it will be depicted as a rectangle (Fig. 1.2) with sides parallel to the coordinate axes and centered at the point M 0 (x 0 y 0 ) .

The rectangle is usually denoted by the following symbol:

Let us introduce an important concept for the further presentation: a neighborhood of a point.

Definition 1.3 Rectangular δ -neighborhood ( delta neighborhood ) points M 0 (x 0 y 0 ) called a rectangle

centered at point M 0 and with sides of the same length 2δ .

Definition 1.4 Circular δ - the neighborhood of the point M 0 (x 0 y 0 ) called a circle of radius δ centered at point M 0 , i.e., the set of points M (xy) whose coordinates satisfy the inequality:

You can introduce the concept of neighborhoods and other types, but for the purposes of mathematical analysis of technical problems, basically, only rectangular and circular neighborhoods are used.

Let us introduce the following concept of the limit of a function of two variables.

Let the function z = f (x, y) defined in some area ζ and M 0 (x 0 y 0 ) - a point lying inside or on the border of this area.

Definition 1.5 Finite number A called the limit of the function f (x, y) at

if for any positive number ε you can find such a positive number δ that inequality

is performed for all points M (x, y) from the area ζ other than M 0 (x 0 y 0 ) whose coordinates satisfy the inequalities:

The meaning of this definition is that the values ​​of the function f (x, y) differ arbitrarily little from the number A at points in a sufficiently small neighborhood of the point M 0 .

Here, the definition is based on rectangular neighborhoods M 0 ... One could consider the circular neighborhoods of the point M 0 and then it would be necessary to require the fulfillment of the inequality

at all points M (x, y) areas ζ other than M 0 and satisfying the condition:

Distance between points M and M 0 .

The following limit notations are used:

Taking into account the definition of the limit of a function of two variables, one can transfer the basic theorems on limits for functions of one variable to functions of two variables.

For example, theorems about the limit of the sum, product and quotient of two functions.

§3 Continuity of a function of two variables

Let the function z = f (x, y) defined at point M 0 (x 0 y 0 ) and its surroundings.

Definition 1.6 A function is called continuous at a point M 0 (x 0 y 0 ) , if

If the function f (x, y) continuous at the point M 0 (x 0 y 0 ) , then

Insofar as

That is, if the function f (x, y) continuous at the point M 0 (x 0 y 0 ) , then infinitesimal increments of arguments in this region correspond to infinitesimal increments Δz functions z .

The converse is also true: if infinitesimal increments of arguments correspond to infinitesimal increments of a function, then the function is continuous

A function that is continuous at each point of the region is called continuous in the region. For continuous functions of two variables, as well as for a function of one variable, continuous on an interval, the fundamental theorems of Weierstrass and Bolzano - Cauchy are valid.

Reference: Karl Theodor Wilhelm Weierstrass (1815 - 1897) - German mathematician. Bernard Bolzano (1781 - 1848) - Czech mathematician and philosopher. Augustin Louis Cauchy (1789 - 1857) - French mathematician, president of the French Academy of Sciences (1844 - 1857).

Example 1.4. Investigate the continuity of a function

This function is defined for all values ​​of the variables x and y except for the origin, where the denominator vanishes.

Polynomial x 2 + y 2 is continuous everywhere, and hence the square root of a continuous function is continuous.

The fraction will be continuous everywhere, except for the points where the denominator is zero. That is, the function under consideration is continuous on the entire coordinate plane Ooh excluding the origin.

Example 1.5. Investigate the continuity of a function z = tg (x, y) ... The tangent is defined and continuous for all finite values ​​of the argument, except for values ​​equal to an odd number of the value π / 2 , i.e. excluding the points where

For every fixed "k" equation (1.11) defines a hyperbola. Therefore, the function under consideration is a continuous function x and y , excluding points lying on curves (1.11).

5.1. Vector function and coordinate functions.

5.2. Continuity of a vector function. Limit of a vector function.

5. Derivative and differential of a vector function, geometric interpretation. Equations of a tangent to a curve in space. (5.3)

5.3. Derivative and differential of a vector function.

5.3.1. Definition and geometric interpretation of the derivative of a vector function.

5.3.2. Differential of a vector function.

5.3.3. Differentiation rules.

5.3.4. Equations of the tangent line to a curve in three-dimensional space.

6. F: Rnr - real functions of several (many) real variables.

6.1. Limit and continuity of a function of several variables.

6.1.1. Limit of a function of several variables. Repeated limits.

6.1.2. Continuity of a function of several variables.

6.1.3. Limit properties of a function of several variables. Properties of functions continuous at a point.

8. Limit of a function of two variables. Linking double limit with repetitions. (6.1.1)

6.1.1. Limit of a function of several variables. Repeated limits.

9.Definition of a partial derivative. Partial derivatives of higher orders. The mixed derivative theorem. (6.2.3, 6.3.1)

6.2.3. Partial derivatives.

10. Definition of a differentiable function of two variables. Relationship between differentiability and continuity and existence of partial derivatives. (6.2.4)

6.2.4. The connection between differentiability and the existence of partial derivatives. Differential uniqueness.

11. Differential of a function of two variables. Approximate calculations using the differential. Tangent plane. (6.2.1, 6.2.5, 6.2.6)

6.2.1. Differentiated function. Differential.

6.2.6. Geometric interpretation of the differentiability of a function of two variables. The tangent plane to the graph of the function.

12. Invariance of the form of the differential. Partial Differential Formulas for Complex Functions (6.2.9)

13. Invariance of the form of the differential. Formulas for partial derivatives of implicit functions. (6.2.10)

6.2.10. An implicit function existence theorem. Derivative (partial derivatives) of an implicit function.

14. Directional derivative. The formula for calculating it. (6.2.7)

15. The gradient of the function at a point. The geometric meaning of the direction and length of the gradient. The orientation of the gradient in relation to the line or surface of the level. (6.2.8)

17. Differentials of higher orders. Taylor's formula for f (X, y). (6.4)

18. Necessary and sufficient conditions for the extremum of the function f (X, y). (6.5.1-6.5.3)

6.5.2. A necessary condition for a local extremum of a function of several variables.

6.5.3. A sufficient condition for a local extremum of a function of several variables.

20. The largest and the smallest values ​​of a differentiable function of two variables in a closed bounded area. Algorithm for finding them. (6.7)

21. Least squares method. (6.8)

6.1. Limit and continuity of a function of several variables.

R n - metric space:

for M 0 (x, x,…, x) and M(NS 1 , NS 2 , …, NS n) ( M 0 , M) = .

n= 2: for M 0 (x 0 , y 0), M (x, y) ( M 0 , M) =
.

Point neighborhood M 0 U  (M 0) = are the interior points of a circle of radius centered at M 0 .

6.1.1. Limit of a function of several variables. Repeated limits.

f: R n R given in some neighborhood of the point M 0, except maybe the point itself M 0 .

Definition. Number A called limit functions

f(x 1 , x 2 , …, x n) at the point M 0 if  >0  >0  M (0 <  (M 0 , M ) <   | f (M ) – A |<  ).

F Record Forms:

n = 2:

it double limit.

In the language of the neighborhoods of points:

>0  >0  M (x , y ) (M  U  (M 0 )\ M 0  f (x , y )  U  (A )).

(M may be approaching M 0 by any path).

Repeated limits:
and
.

(M approaching M 0 horizontally and vertically, respectively).

A theorem on the relationship between double and repeated limits.

If  double limit
and limits
,
,

then  repeated limits
,
and are equal to double.

Remark 1. The converse is not true.

Example. f (x, y) =

,

.

However, the double limit

=

does not exist, since in any neighborhood of the point (0, 0) the function also takes values ​​"far" from zero, for example, if x = y, then f (x, y) = 0,5.

Remark 2. Even if  AR: f (x, y) A

when driving M To M 0 in any straight line, the double limit may not exist.

Example.f (x, y) =
,M 0 (0, 0). M (x, y)  M 0 (0, 0)

Conclusion: the (double) limit does not exist.

An example of finding the limit.

f (x, y) =
, M 0 (0, 0).

Let us show that the number 0 is the limit of the function at the point M 0 .

=
,

 - distance between points M and M 0. (Used the inequality
,

which follows from the inequalities
)

Let > 0 and let  = 2. <  

6.1.2. Continuity of a function of several variables.

Definition. f (x, y) is continuous at the point M 0 (x 0 , y 0) if it is defined in some U  (M 0) and
,T. i.e.> 0 > 0  M (0 < (M 0 , M) <   | f (M) – f (M 0)|< ).

Comment. The function can change continuously along some directions passing through the point M 0, and have discontinuities along other directions or paths of a different shape. If so, it is discontinuous at a point M 0 .

6.1.3. Limit properties of a function of several variables. Properties of functions continuous at a point.

Occurs uniqueness limit;

function having a finite limit at a point M 0 , bounded in some neighborhood of this point; are carried out ordinal and algebraic properties limit,

passage to the limit preserves the equal and nonstrict inequality signs.

If the function is continuous at the point M 0 and f (M 0 )  0 , then value signf (M ) is preserved in some U  (M 0).

Sum, product, quotient(denominator  0) continuous functions are also continuous functions, continuous complex function composed of continuous.

6.1.4. Properties of functions that are continuous on a connected closed bounded set.n= 1, 2 and 3.

Definition 1. The set  is called connected if, together with any two of its points, it also contains some continuous curve connecting them.

Definition 2. The set  in R n called limited if it is contained in some "ball"
.

n = 1 

n = 2 

n = 3  .

Examples ofconnected closed bounded sets.

R 1 = R: section [ a, b];

R 2: segment AB any continuous curve with endpoints A and V;

closed continuous curve;

circle
;

Definition 3. f: R n  R is continuous on a connected closed set   R n if  M 0 

.

Theorem.Lots ofvalues continuous function

f: R n  R on a closed bounded connected set is a segment [ m , M ] , here m - least, a M - the greatest its values ​​at points of the set.

Thus, on any closed bounded connected set inR n continuous function is limited, takes its smallest, largest, as well as all intermediate values.

"

Department: Higher Mathematics

abstract

in the discipline "Higher Mathematics"

Topic: "Limit and continuity of functions of several variables"

Togliatti, 2008

Introduction

The concept of a function of one variable does not cover all the dependencies that exist in nature. Even in the simplest problems, there are quantities whose values ​​are determined by the combination of values ​​of several quantities.

To study such dependencies, the concept of a function of several variables is introduced.

The concept of a function of several variables

Definition. The magnitude u is called a function of several independent variables ( x, y, z, …, t), if each set of values ​​of these variables is associated with a certain value of the quantity u.

If a variable is a function of two variables NS and at, then the functional dependence is denoted by

z = f (x, y).

Symbol f defines here a set of actions or a rule for calculating a value z for a given pair of values NS and at.

So, for the function z = x 2 + 3xy

at NS= 1 and at= 1 we have z = 4,

at NS= 2 and at= 3 we have z = 22,

at NS= 4 and at= 0 we have z= 16, etc.

The quantity u function of three variables x, y, z, if a rule is given, as for a given triple of values x, y and z calculate the corresponding value u:

u = F (x, y, z).

Here the symbol F defines a set of actions or a rule for calculating a value u corresponding to these values x, y and z.

So, for the function u = xy + 2xz – 3yz

at NS = 1, at= 1 and z= 1 we have u = 0,

at NS = 1, at= -2 and z= 3 we have u = 22,

at NS = 2, at= -1 and z= -2 we have u = -16, etc.

Thus, if by virtue of some law of each set NS numbers ( x, y, z, …, t) from some set E assigns a specific value to a variable u, then u called a function of NS variables x, y, z, …, t defined on the set E, and denoted

u = f(x, y, z, …, t).

Variables x, y, z, …, t are called function arguments, the set E- the scope of the function.

The particular value of a function is the value of a function at some point M 0 (x 0 , y 0 , z 0 , …, t 0) and is denoted f (M 0) = f (x 0 , y 0 , z 0 , …, t 0).

The domain of a function is the set of all argument values ​​that correspond to any real values ​​of the function.

Function of two variables z = f (x, y) in space is represented by a certain surface. That is, when the point with coordinates NS, at runs through the entire domain of the function located in the plane hoy, the corresponding spatial point, generally speaking, describes the surface.

Function of three variables u = F (x, y, z) considered as a function of a point of some set of points in three-dimensional space. Similarly, the function NS variables u = f(x, y, z, …, t) is considered as a function of a point of some NS-dimensional space.

Limit of a function of several variables

In order to give the concept of the limit of a function of several variables, we restrict ourselves to the case of two variables NS and at... By definition, the function f (x, y) has a limit at the point ( NS 0 , at 0) equal to the number A, denoted like this:

(1)

(write more f (x, y) →A at (x, y) → (NS 0 , at 0)) if it is defined in some neighborhood of the point ( NS 0 , at 0), with the possible exception of this point itself and if there is a limit

(2)

whatever the tending towards ( NS 0 , at 0) a sequence of points ( x k, y k).

Just as in the case of a function of one variable, you can introduce another equivalent definition of the limit of a function of two variables: the function f has at the point ( NS 0 , at 0) limit equal to A if it is defined in some neighborhood of the point ( NS 0 , at 0) except, perhaps, this point itself, and for any ε> 0 there is δ> 0 such that

| f (x, y) – A| < ε(3)

for all (x, y) satisfying the inequalities

< δ. (4)

This definition, in turn, is equivalent to the following: for any ε> 0 there is a δ-neighborhood of the point ( NS 0 , at 0) such that for all ( x, y) from this neighborhood, other than ( NS 0 , at 0), inequality (3) holds.

Since the coordinates of an arbitrary point ( x, y) the neighborhood of the point ( NS 0 , at 0) can be written as x = x 0 + Δ NS, y = y 0 + Δ at, then equality (1) is equivalent to the following equality:

Consider some function defined in a neighborhood of the point ( NS 0 , at 0), except perhaps for this point itself.

Let ω = (ω NS, ω at) Is an arbitrary vector of length one (| ω | 2 = ω NS 2 + ω at 2 = 1) and t> 0 is a scalar. View points

(NS 0 + tω NS, y 0 + tω at) (0 < t)

form a ray emerging from ( NS 0 , at 0) in the direction of the vector ω. For each ω, we can consider the function

f(NS 0 + tω NS, y 0 + tω at) (0 < t< δ)

from scalar variable t, where δ is a sufficiently small number.

The limit of this function (one variable t)

f(NS 0 + tω NS, y 0 + tω at),

if it exists, it is natural to call it the limit f at point ( NS 0 , at 0) in the direction ω.

Example 1. Functions

defined on the plane ( x, y) except for the point NS 0 = 0, at 0 = 0. We have (take into account that

and ):

(for ε> 0 we put δ = ε / 2 and then | f (x, y) | < ε, если

< δ).

from which it can be seen that the limit φ at the point (0, 0) in different directions is generally different (the unit vector of the ray y = kx, NS> 0, has the form

).

Example 2. Consider in R 2 function

(NS 4 + at 2 ≠ 0).

This function at the point (0, 0) on any straight line y = kx passing through the origin has a limit equal to zero:

at NS → 0.

However, this function has no limit at the points (0, 0), because for y = x 2

and

Will write

if the function f is defined in some neighborhood of the point ( NS 0 , at 0), with the possible exception of the point itself ( NS 0 , at 0) and for any N> 0 there is δ> 0 such that

|f (x, y) | > N,

since 0<

< δ.

You can also talk about the limit f, when NS, at → ∞:

(5)

For example, in the case of a finite number A equality (5) must be understood in the sense that for any ε> 0 there is such N> 0, which for all NS, at for which | x| > N, |y| > N, function f is defined and the inequality