6.1. Limit and continuity of a function of several variables.
R n – metric space:
For M 0 (x, x,…, x) And M(X 1 , X 2 , …, X n) ( M 0 , M) = .
n= 2: for M 0
(x 0 ,
y 0),
M
(x,
y)
( M 0 ,
M)
=
.
Neighborhood of a point M 0 U (M 0) = – internal points of a circle of radius with center at M 0 .
6.1.1. Limit of a function of several variables. Repeat limits.
f: R n R is 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 point M 0 if >0 >0 M (0 < (M 0 , M ) < | f (M ) – A |< ).
F 
Recording forms: 
n
= 2: 
This double limit.
In the language of 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 on any path).
Repeat limits:
And 
.
(M approaching M 0 horizontally and vertically, respectively).
Theorem on the connection between double and repeated limits.
If double limit 
and limits 
,
,
then repeated limits 
,
and equal to double.
Note 1. The opposite statement 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, That f (x, y) = 0,5.
Note 2. Even if AR: f (x, y) A
when driving M To M 0 along 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 us set > 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. e.>0 >0 M
(0 < (M 0 ,
M)
<
|
f
(M)
– f
(M 0)|<
).
Comment. The function can vary continuously along some directions passing through the point M 0, and have discontinuities along other directions or paths of different shapes. If so, it is discontinuous at the point M 0 .
6.1.3. Properties of the limit of a function of several variables. Properties of functions continuous at a point.
Takes place uniqueness of the limit;
function having a finite limit at a point M 0 , bounded in some neighborhood of this point; are being carried out ordinal and algebraic properties limit
passage to the limit preserves equality signs and weak inequalities.
If the function is continuous at the point M 0 and f (M 0 ) 0 , That meaning signf (M ) is preserved in some U (M 0).
Sum, product, quotient(denominator 0) continuous functions also continuous functions, continuous complex function, composed of continuous ones.
6.1.4. Properties of functions continuous on a connected closed bounded set.n= 1, 2 and 3.
Definition 1. The set is called coherent, if, together with any two of its points, it also contains some continuous curve connecting them.
Definition 2. Set in R n called limited, if it is contained in some "ball" 
.
n
= 1
n
= 2 
n = 3 .
Examplesconnected closed bounded sets.
R 1 = R: segment [ a, b];
R 2: segment AB any continuous curve with ends at points A And IN;
closed continuous curve;
circle 
;
Definition 3. f: R n R is continuous on a connected closed set R n, if M 0
.
Theorem.Manyvalues continuous function
f: R n R on a closed bounded connected set is a segment [ m , M ] , Here m - least, A M - greatest its values at the points of the set.
Thus, on any closed bounded connected set inR n a continuous function is bounded, takes its smallest, largest, and all intermediate values.
| " |
The concepts of functions of two or three variables discussed above can be generalized to the case of variables.
Definition. Function 
variables 
called a function, domain of definition 
which belongs 
, and the range of values is the real axis.
Such a function for each set of variables 
from 
matches singular number 
.
In what follows, for definiteness, we will consider the functions 
variables, but all statements formulated for such functions remain true for functions of a larger number of variables.
Definition. Number 
called the limit of the function
at the point 
, if for each 
there is such a number 
that in front of everyone 
from the neighborhood 
, except for this point, the inequality holds
.
If the limit of the function 
at the point 
equals 
, then this is denoted in the form
.
Almost all the properties of limits that we considered earlier for functions of one variable remain valid for limits of functions of several variables, however, we will not deal with the practical determination of such limits.
Definition. Function 
called continuous at a point 
if three conditions are met:
1) exists 
2) there is a value of the function at the point 
3) these two numbers are equal to each other, i.e. .
In practice, we can study the continuity of a function using the following theorem.
Theorem. Any elementary function 
is continuous at all internal (i.e., non-boundary) points of its domain of definition.
Example. Let's find all the points at which the function
continuous.
As noted above, this function is defined in a closed circle
.
The internal points of this circle are the desired points of continuity of the function, i.e. function 
continuous in an open circle 
.
Definition of the concept of continuity at the boundary points of the domain of definition 
functions are possible, but we will not discuss this issue in the course.
1.3 Partial increments and partial derivatives
Unlike functions of one variable, functions of several variables have different types of increments. This is due to the fact that movements in the plane 
from point 
can be carried out in various directions.
Definition. Partial increment by 
functions 
at the point 
corresponding increment 
called difference
This increment is essentially an increment of a function of one variable 
obtained from the function 
at a constant value 
.
Similarly, by partial increment
at the point 
functions 
corresponding increment 
called difference
This increment is calculated at a fixed value 
.
Example. Let 
,
,
. Let us find the partial increments of this function by 
and by 
In this example, with equal values of argument increments 
And 
, the partial increments of the function turned out to be different. This is due to the fact that the area of a rectangle with sides 
And 
when increasing the side 
on 
increases by the amount 
, and with increasing side 
on 
increases by 
(see Fig. 4).
From the fact that a function of two variables has two types of increments, it follows that two types of derivatives can be defined for it.
Definition. Partial derivative with respect to 
functions 
at the point 
is called the limit of the ratio of the partial increment by 
of this function at the specified point to the increment 
argument 
those.
.
(1)
Such partial derivatives are denoted by the symbols 
,
,
,
. In the latter cases, the round letter “ 
”
– “
” means the word “private”.
Similarly, the partial derivative with respect to 
at the point 
determined using the limit
.
(2)
Other notations for this partial derivative: 
,
,
.
Partial derivatives of functions are found according to the known rules for differentiating a function of one variable, while all variables except the one by which the function is differentiated are considered constant. So when you find 
variable 
is taken as a constant, and when found 
- constant 
.
Example. Let's find the partial derivatives of the function 
.
,
.
Example. Let's find the partial derivatives of a function of three variables
.
;
;
.
Partial derivative functions 
characterize the rate of change of this function in the case when one of the variables is fixed.
An example in economics.
The main concept of consumption theory is the utility function 
. This function expresses the utility of a set 
, where x is the quantity of product X, y is the quantity of product Y. Then the partial derivatives 
will be called the marginal utilities of x and y, respectively. Marginal rate of substitution 
one good to another is equal to the ratio of their marginal utilities:
.
(8)
Problem 1. Find the marginal rate of substitution h by y for the utility function at point A(3,12).
Solution: according to formula (8) we obtain
The economic meaning of the marginal rate of substitution lies in the substantiation of the formula 
, Where 
-price of product X, 
- price of goods U.
Definition. If the function 
there are partial derivatives, then its partial differentials are the expressions
And 
Here 
And 
.
Partial differentials are differentials of functions of one variable obtained from a function of two variables 
at fixed 
or 
.
Examples from economics. Let's take the Cobb-Douglas function as an example.
Magnitude 
- average labor productivity, since this is the amount of products (in value terms) produced by one worker.
Magnitude 
- average capital productivity - the number of products per machine.
Magnitude 
- average capital-labor ratio - the cost of funds per unit of labor resources.
Therefore the partial derivative 
is called the marginal productivity of labor because it is equal to the added value of output produced by one more additional worker.
Likewise, 
- marginal capital productivity.
In economics, questions are often asked: by what percentage will output change if the number of workers increases by 1% or if funds increase by 1%? The answers to such questions are given by the concepts of elasticity of a function with respect to argument or relative derivative. Find the elasticity of output with respect to labor 
. Substituting the partial derivative calculated above into the numerator 
, we get 
. So the parameter 
has a clear economic meaning - it is the elasticity of output with respect to labor.
The parameter has a similar meaning 
is the elasticity of output across funds.
Definition of a function of several variables. Basic concepts.
If each pair of numbers (x, y) independent of each other from a certain set, according to some rule, is associated with one value of the variable z, then it is called function of two variables. z=f(x,y,)
Domain of the function z- a set of pairs (x, y) for which the function z exists.
The set of values (range of values) of a function is all the values that the function takes in its domain of definition.
Graph of a function of two variables - a set of points P whose coordinates satisfy the equation z=f(x,y)
Neighborhood of a point M0 (x0;y0) of radius r– the set of all points (x,y) that satisfy the condition< r
The domain of definition and range of values of a function of several variables. Graph of a function of several variables.
Limit and continuity of a function of several variables.
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 X And at. By definition, function f(x,y) has a limit at the point ( X 0 , at 0), equal to the number A, denoted as follows:
(1)
(they also write f(x,y)→A at (x, y)→ (X 0 , at 0)), if it is defined in some neighborhood of the point ( X 0 , at 0), except perhaps at this point itself and if there is a limit
(2)
whatever the tending to ( X 0 , at 0) sequence of points ( x k ,y k).
Just as in the case of a function of one variable, another equivalent definition of the limit of a function of two variables can be introduced: function f has at point ( X 0 , at 0) limit equal to A, if it is defined in some neighborhood of the point ( X 0 , at 0) except, perhaps, for this point itself, and for any ε > 0 there is a δ > 0 such that
| f(x,y) – A| < ε (3)
for everyone (x, y), satisfying the inequalities
0 < 
< δ. (4)
This definition, in turn, is equivalent to the following: for any ε > 0 there is a δ-neighborhood of the point ( X 0 , at 0) such that for all ( x, y) from this neighborhood, different from ( X 0 , at 0), inequality (3) is satisfied.
Since the coordinates of an arbitrary point ( x, y) neighborhood of point ( X 0 , at 0) can be written as x = x 0 + Δ X, y = y 0 + Δ at, then equality (1) is equivalent to the following equality:
Let us consider some function defined in a neighborhood of the point ( X 0 , at 0), except, perhaps, this point itself.
Let ω = (ω X, ω at) – an arbitrary vector of length one (|ω| 2 = ω X 2 + ω at 2 = 1) and t> 0 – scalar. View points
(X 0 + tω X, y 0 + tω at) (0 < t)
form a ray emerging from ( X 0 , at 0) in the direction of the vector ω. For each ω we can consider the function
f(X 0 + tω X, y 0 + tω at) (0 < t< δ)
from a scalar variable t, where δ is a fairly small number.
The limit of this function (one variable) t)
f(X 0 + tω X, y 0 + tω at),
if it exists, it is natural to call it a limit f at point ( X 0 , at 0) in the direction ω.
Example 1. Functions
defined on the plane ( x, y) except for the point X 0 = 0, at 0 = 0. We have (take into account that 
And 
):
(for ε > 0 we set δ = ε/2 and then | f(x,y)| < ε, если < δ).
from which it is clear that the limit φ at the point (0, 0) in different directions is generally different (the unit ray vector y = kx, X> 0, has the form
).
Number A called the limit of the function f(M) at M → M 0 if for any number ε > 0 there is always a number δ > 0 such that for any points M, different from M 0 and satisfying the condition | MM 0 | < δ, будет иметь место неравенство |f(M) – A | < ε.
Limit denote 
In the case of a function of two variables 
Limit theorems. If the functions f 1 (M) And f 2 (M) at M → M 0 each tend to a finite limit, then:
V) 
Continuity of a function of several variables
By definition, function f(x,y) is continuous at the point ( X 0 , at 0), if it is defined in some of its neighborhood, including at the point itself ( X 0 , at 0) and if the limit f(x,y) at this point is equal to its value at it:
(1)
Continuity condition f at point ( X 0 , at 0) can be written in equivalent form:
(1")
those. function f is continuous at the point ( X 0 , at 0), if the function is continuous f(x 0 + Δ X, at 0 + Δ y) on variables Δ X, Δ at at Δ X = Δ y = 0.
You can enter an increment Δ And functions And = f(x,y) at the point (x, y), corresponding to increments Δ X, Δ at arguments
Δ And = f(x + Δ X, at + Δ y) – f(x,y)
and in this language define continuity f V (x, y): function f continuous at a point (x, y), If
(1"")
Theorem. Sum, difference, product and quotient of continuous at a point ( X 0 ,at 0) functions f and φ is a continuous function at this point, unless, of course, in the case of a quotient φ ( X 0 , at 0) ≠ 0.
Constant With can be considered as a function f(x,y) = With from variables x,y. It is continuous in these variables because
|f(x,y) – f (X 0 , at 0) | = |s – s| = 0 0.
The next most difficult functions are f(x,y) = X And f(x,y) = at. They can also be considered as functions of (x, y), and at the same time they are continuous. For example, the function f(x,y) = X matches each point (x, y) a number equal to X. Continuity of this function at an arbitrary point (x, y) can be proven like this:
| f(x + Δ X, at + Δ y) – f(x,y) | = |f(x + Δ x) – x| = | Δ X | ≤ 0.
If you produce over functions x, y and constant actions of addition, subtraction and multiplication in a finite number, then we will obtain functions called polynomials in x, y. Based on the properties formulated above, polynomials in variables x, y– continuous functions of these variables for all points (x, y) R 2 .
Attitude P/Q two polynomials from (x, y) is a rational function of (x,y), obviously continuous everywhere on R 2, excluding points (x, y), Where Q(x, y) = 0.
P(x,y) = X 3 – at 2 + X 2 at – 4
could be an example of a polynomial from (x, y) third degree, and the function
P(x,y) = X 4 – 2X 2 at 2 +at 4
there is an example of a polynomial from (x, y) fourth degree.
Let us give an example of a theorem stating the continuity of a function of continuous functions.
Theorem. Let the function f(x, y, z) continuous at a point (x 0 , y 0 , z 0 ) space R 3 (points (x, y, z)), and the functions
x = φ (u, v), y= ψ (u, v), z= χ (u, v)
continuous at a point (u 0 , v 0 ) space R 2 (points (u, v)). Let, in addition,
x 0 = φ (u 0 , v 0 ), y 0 = ψ (u 0 , v 0 ), z 0 = χ (u 0 , v 0 ) .
Then the function F(u, v) = f[ φ (u, v),ψ (u, v),χ (u, v)] is continuous (by
(u, v)) at point (u 0 , v 0 ) .
Proof. Since the sign of the limit can be placed under the sign of the characteristic of a continuous function, then
Theorem. Function f(x,y), continuous at the point ( X 0 , at 0) and not equal to zero at this point, preserves the sign of the number f(X 0 , at 0) in some neighborhood of the point ( X 0 , at 0).
By definition, function f(x) = f(x 1 , ..., x p) continuous at a point X 0 =(X 0 1 , ..., X 0 p), if it is defined in some of its neighborhood, including at the point itself X 0, and if its limit is at the point X 0 is equal to its value in it:
(2)
Continuity condition f at the point X 0 can be written in equivalent form:
(2")
those. function f(x) continuous at a point X 0 if the function is continuous f(x 0 +h) from h at the point h = 0.
You can enter an increment f at the point X 0 corresponding to increment h = (h 1 , ..., h p),
Δ h f (x 0 ) = f (x 0 + h) – f(x 0 )
and in his language define continuity f V X 0: function f continuous in X 0 if
Theorem. Sum, difference, product and quotient of continuous at a point X 0 functions f(x) and φ (x) is a continuous function at this point, if, of course, in the case of a particular φ (X 0 ) ≠ 0.
Comment. Increment Δ h f (x 0 ) also called the complete increment of the function f at the point X 0 .
In space Rn points X = (x 1 , ..., x p) let's set a set of points G.
By definition X 0 = (X 0 1 , ..., X 0 p) is the interior point of the set G, if there is an open ball with center in it, completely belonging to G.
Many G Rn is called open if all its points are interior.
They say that the functions
X 1 = φ 1 (t), ..., x n =φ p(t) (a ≤ t ≤ b)
continuous on the segment [ a, b], define a continuous curve in Rn, connecting the points X 1 = (X 1 1 , ..., X 1 p) And X 2 = (X 2 1 , ..., X 2 p), Where X 1 1 = φ 1 (A), ..., X 1 n =φ p(a), X 2 1 = φ 1 (b), ..., X 2 n =φ p(b). Letter t called the curve parameter.
Consider the plane and the system Oxy Cartesian rectangular coordinates on it (other coordinate systems can be considered).
From analytical geometry we know that for each ordered pair of numbers (x, y) you can compare a single point M plane and vice versa, to each point M The plane corresponds to a single pair of numbers.
Therefore, in the future, when speaking about a point, we will often mean the corresponding pair of numbers (x, y) and vice versa.
Definition 1.2 Set of pairs of numbers (x, y) , satisfying the inequalities, is called a rectangle (open).
On the 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 ) .
A rectangle is usually denoted by the following symbol:
Let us introduce an important concept for further discussion: the neighborhood of a point.
Definition 1.3 Rectangular δ -surroundings ( delta neighborhood ) points M 0 (x 0 y 0 ) called a rectangle
centered at a point M 0 and with sides of equal length 2δ .
Definition 1.4 Circular δ - neighborhood of a point M 0 (x 0 y 0 ) called a circle of radius δ centered at a point M 0 , i.e. a set of points M(xy) , whose coordinates satisfy the inequality:
It is possible to introduce the concepts of neighborhoods and other types, but for the purposes of mathematical analysis of technical problems, mainly 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 1.5Finite number A called limit of the function f (x, y) at
if for any positive number ε can you find such a positive number δ that inequality
performed for all points M(x,y) from the region ζ , different from 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 as little as desired 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 circular neighborhoods of the point M 0 and then it would be necessary to require the inequality
at all points M(x,y) region ζ , different from M 0 and satisfying the condition:
Distance between points M And M 0 .
The following limit designations are used:
Given the definition of the limit of a function of two variables, it is possible to transfer the basic theorems about limits for functions of one variable to functions of two variables.
For example, theorems on 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 said to be continuous at a point M 0 (x 0 y 0 ) , If
If the function f(x,y) continuous at a point M 0 (x 0 y 0 ) , That
Because
That is, if the function f(x,y) continuous at a 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 the function, then the function is continuous
A function that is continuous at every point in a domain is called continuous in the domain. 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. Examine the continuity of a function
This function is defined for all values of the variables x And y , except at the origin, where the denominator goes to zero.
Polynomial x 2 +y 2 is continuous everywhere, and therefore the square root of a continuous function is continuous.
The fraction will be continuous everywhere except at points where the denominator is zero. That is, the function under consideration is continuous on the entire coordinate plane Ohoo , excluding the origin.
Example 1.5. Examine 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 quantity π/2 , i.e. excluding 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).
Limit of a function of two variables.
Concept and examples of solutions
Welcome to the third lesson on the topic FNP, where all your fears finally began to come true =) As many suspected, the concept of a limit also extends to a function of an arbitrary number of arguments, which is what we have to figure out today. However, there is some optimistic news. It consists in the fact that 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 a function of two variables or, as we more often write it: .
Many ideas, principles and methods are similar to the theory and practice of “ordinary” limits, which means that at the moment you should be able to find limits and most importantly UNDERSTAND what it is limit of a function of one variable. And, since fate brought you to this page, then, most likely, you already understand and 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 take place 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 a 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 the axis somewhere... well, for example, in any corner, so that it doesn't get in the way. Great. Now please look up and imagine that the blanket is hanging there, spread out. This surface, given by the function. Our movement on the floor, as is easy to understand, imitates a change in independent variables, and we can move exclusively under the blanket, i.e. V domain of definition of a function of two variables. But the fun is just beginning. A small cockroach is crawling on the blanket right above the tip of your nose, and wherever you go, so does it. Let's call him Freddy. Its movement simulates a change in 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, don’t be offended, this is necessary for science.
Let's take an awl in our hands and pierce the blanket at an arbitrary point, the height of which we will denote by , after which we will stick the tool into the floor strictly under the hole - this will be the point. Now let's begin infinitely close approach a given point 
, and we have the right to approach along ANY trajectory (each point of which, of course, is included in the domain of definition). If in ALL cases Freddy will be infinitely close crawl to the puncture to a height and EXACTLY THIS HEIGHT, then the function has a limit at the point at 
:
If, under the specified conditions, the pierced point is located on the edge of the blanket, then the limit will still exist - it is important that in arbitrarily small neighborhood the tips of the awl were at least some points from the domain of definition of the function. Moreover, as is the case with limit of a function of one variable, doesn't matter, whether the function is defined at a point or not. That is, our puncture can be sealed with chewing gum (assume that function of two variables is continuous) and this will not affect the situation - we remember that the very essence of the limit implies infinitely close approximation, and not a “precise approach” to a point.
However, a cloudless life is overshadowed by the fact that, unlike its younger brother, the limit much more often does not exist. This is due to the fact that there are usually many paths to a particular point on the plane, and each of them must lead Freddy strictly to the puncture (optional “sealed with chewing gum”) and strictly to the height. And there are more than enough bizarre surfaces with equally bizarre discontinuities, which leads to the violation of this strict condition at some points.
Let's organize simplest example– take a knife in your hands and cut the blanket so that the pierced 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” of function domain. Now let’s carefully lift the left part of the blanket along the axis, and, on the contrary, move the right part down or even leave it in place. What has changed? And the following has fundamentally changed: if we now approach a point on the left, then Freddy will be at a higher altitude than if we were approaching a given point on the right. So there is no limit.
And of course wonderful limits Where would we be without them? Let's look at an example that is instructive in every sense:
Example 11
We use the painfully familiar trigonometric formula, where we organize using a standard artificial technique first remarkable limits :
Let's move on to polar coordinates:
If , then
It would seem that the solution is heading towards a natural outcome and nothing foretells trouble, but at the very end there is a great risk of making a serious flaw, the nature of which I already hinted at a little in Example 3 and described in detail after Example 6. First the ending, then the comment:
Let's figure out why it would be bad to write simply “infinity” or “plus infinity.” Let's look at the denominator: since , 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 down to infinity: 
