How do I know if a feature is limited. Limits of monotone functions

Since the indicated areas are respectively equal

S Δ OAC=0,5∙OC∙OA∙ sin α= 0.5sinα, S sects. OAC \u003d0,5∙OC 2 ∙ α \u003d 0.5α, S Δ OBC=0,5∙OC∙BC \u003d0.5tgα.

Consequently,

sin α< α < tg α.

We divide all terms of the inequality by sin α\u003e 0:.

But. Therefore, based on Theorem 4 on the limits, we conclude that the derived formula is called the first remarkable limit.

Thus, the first remarkable limit serves to uncover uncertainty. Note that the resulting formula should not be confused with the limits Examples.

11.The Limit and associated limits.

SECOND WONDERFUL LIMIT

The second remarkable limit serves to reveal the uncertainty 1 ∞ and looks as follows

Let's pay attention to the fact that in the formula for the second remarkable limit in the exponent there should be an expression inverse to the one that is added to the unit at the base (since in this case you can introduce a change of variables and reduce the sought limit to the second remarkable limit)

Examples.

1. Function f (x)=(x-1) 2 is infinitesimal for x→ 1, since (see fig.).

2. Function f (x) \u003d tg x - infinitely small at x→0.

3. f (x) \u003d ln (1+ x) Is infinitely small for x→0.

4. f (x) = 1/x- infinitely small at x→∞.

Let us establish the following important relation:

Theorem. If the function y \u003d f (x) representable at x → aas the sum of a constant number b and infinitesimal value α (x): f (x) \u003d b + α (x) then.

Conversely, if, then f (x) \u003d b + α (x)where a (x) - infinitely small at x → a.

Evidence.

1. Let us prove the first part of the statement. From equality f (x) \u003d b + α (x) should | f (x) - b | \u003d | α |... But since a (x) Is infinitesimal, then for arbitrary ε there is δ - a neighborhood of the point a, with all x from which, the values a (x) satisfy the relation | α (x) |< ε. Then | f (x) - b |< ε. And that means that.

2. If, then for any ε >0 for all x from some δ is a neighborhood of the point a will be | f (x) - b |< ε. But if we denote f (x) - b \u003d αthen | α (x) |< ε, which means that a - infinitely small.

Consider the basic properties of infinitesimal functions.

Theorem 1. The algebraic sum of two, three and, in general, any finite number of infinitesimal is an infinitesimal function.

Evidence... Let us give a proof for two terms. Let be f (x) \u003d α (x) + β (x)where and. We need to prove that for an arbitrary arbitrarily small ε > 0 is found δ> 0 such that for xsatisfying the inequality | x - a |<δ , performed | f (x) |< ε.

So, we fix an arbitrary number ε > 0. Since by the hypothesis of the theorem α (x) Is an infinitesimal function, then there is δ 1 > 0, which for | x - a |< δ 1 we have | α (x) |< ε / 2. Similarly, since β (x) Is infinitely small, then there is δ 2 > 0, which for | x - a |< δ 2 we have | β (x) |< ε / 2.

Let's take δ \u003d min (δ 1 , δ 2 } . Then, in the vicinity of the point a radius δ each of the inequalities | α (x) |< ε / 2 and | β (x) |< ε / 2. Therefore, in this neighborhood there will be

| f (x) | \u003d | α (x) + β (x)| ≤ | α (x) | + | β (x) |< ε /2 + ε /2= ε,

those. | f (x) |< ε, as required.

Theorem 2. Product of an infinitesimal function a (x) for limited function f (x) at x → a (or at x → ∞) is an infinitesimal function.

Evidence... Since the function f (x) is limited, then there is a number M such that for all values x from some neighborhood of the point a | f (x) | ≤M. Moreover, since a (x) Is an infinitely small function at x → a, then for arbitrary ε > 0 there is a neighborhood of the point a, in which the inequality | α (x) |< ε / M... Then in the smaller of these neighborhoods we have | αf |< ε / M\u003d ε. And this means that af - infinitely small. For the occasion x → ∞ the proof is similar.

It follows from the proved theorem:

Corollary 1. If u, then

Corollary 2. If and c \u003dconst, then.

Theorem 3. The ratio of an infinitesimal function α (x) per function f (x), the limit of which is nonzero, is an infinitesimal function.

Evidence... Let be . Then 1 / f (x) there is a limited function. Therefore, the fraction is the product of an infinitesimal function and a bounded function, i.e. the function is infinitesimal.

Examples.

1. It is clear that for x → + ∞function y \u003d x 2 +1 is infinitely large. But then, according to the theorem formulated above, the function is infinitely small for x → + ∞, i.e. ...

The converse theorem can also be proved.

Theorem 2. If the function f (x) - infinitely small at x → a (or x → ∞) and does not vanish, then y \u003d1/ f (x) is an infinitely large function.

Prove the theorem yourself.

Examples.

3., since the functions and are infinitesimal for x → + ∞, then, how the sum of infinitesimal functions is an infinitesimal function. The function is the sum of a constant number and an infinitesimal function. Hence, by Theorem 1, for infinitesimal functions, we obtain the required equality.

Thus, the simplest properties of infinitely small and infinitely large functions can be written using the following conditional relations: A≠ 0

13. Infinitesimal functions of the same order, equivalent to infinitesimal ones.

Infinitesimal functions and are called infinitesimal of the same order of smallness, if, denote. And, finally, if it does not exist, then infinitesimal functions are incomparable.

EXAMPLE 2. Comparison of infinitesimal functions

Equivalent infinitesimal functions.

If, then infinitesimal functions are called equivalent, denote ~.

Locally equivalent functions:

If if

Some equivalences(at):

One-sided limits.

So far, we have considered the definition of the limit of a function when x → a in an arbitrary way, i.e. the limit of the function did not depend on how the x towards a, left or right of a... However, quite often you can find functions that do not have a limit under this condition, but they do have a limit if x → astaying on one side of and, left or right (see fig.). Therefore, the concept of one-sided limits is introduced.

If f (x)tends to the limit b at x tending to a certain number a so that xtakes only values \u200b\u200bless than athen write and call b by the limit of the function f (x) at the point a on the left.

So the number b called the limit of the function y \u003d f (x) at x → aon the left, if whatever the positive number ε, there is such a number δ (less a

Similarly, if x → a and takes on large values athen write and call b the limit of the function at the point and on right. Those. number b called limit of the function y \u003d f (x) as x → a on the right, if whatever the positive number ε, there is such a number δ (greater and) such that the inequality holds for all.

Note that if the limits on the left and right at the point a for function f (x) do not coincide, then the function has no (two-sided) limit at the point and.

Examples.

1. Consider the function y \u003d f (x)defined on the segment as follows

Find the limits of the function f (x) at x →3. Obviously, but

In other words, for any arbitrarily small number of epsilons, there is such a number of deltas, depending on epsilon, that from the fact that for any x satisfying the inequality it follows that the difference between the values \u200b\u200bof the function at these points will be arbitrarily small.

Continuity criterion for a function at a point:

Function will be continuous at point A if and only if it is continuous at point A both to the right and to the left, that is, so that at point A there are two one-sided limits, they are equal to each other and equal to the value of the function at point A.

Definition 2: The function is continuous on a set if it is continuous at all points of this set.

Derivative of a function at a point

Let the given be defined in the neighborhood. Consider

If this limit exists, then it is called derivative of the function f at the point.

Function derivative - the limit of the ratios of the function increment to the argument increment, with the argument increment.

The operation of calculating or finding a derivative at a point is called differentiation .

Differentiation rules.

Derivative function f (x) at the point x \u003d x 0 is the ratio of the increment of the function at this point to the increment of the argument, as the latter tends to zero. Finding the derivative is called differentiation... The derivative of the function is calculated according to the general rule of differentiation: f (x) \u003d u, g (x) \u003d v- functions differentiable at the point x. Basic rules for differentiation 1) (the derivative of the sum is equal to the sum of the derivatives) 2) (this, in particular, implies that the derivative of the product of a function and a constant is equal to the product of the derivative of this function by a constant) 3) Derivative of the quotient: if g  0 4) Derivative of a complex function: 5) If the function is given parametrically:, then

Examples.

1. y = x a is a power function with an arbitrary exponent.

Implicitly specified function

If the function is given by the equation y \u003d ƒ (x), resolved with respect to y, then the function is given explicitly (explicit function).

Under implicit assignment functions understand the definition of a function in the form of an equation F (x; y) \u003d 0, not resolved with respect to y.

Any explicitly given function y \u003d ƒ (x) can be written as implicitly given by the equation ƒ (x) -y \u003d 0, but not vice versa.

It is not always easy, and sometimes impossible, to solve the equation for y (for example, y + 2x + cozy-1 \u003d 0 or 2 y -x + y \u003d 0).

If the implicit function is given by the equation F (x; y) \u003d 0, then to find the derivative of y with respect to x, there is no need to solve the equation for y: it is enough to differentiate this equation with respect to x, considering y as a function of x, and then solve the equation obtained for y ".

The derivative of an implicit function is expressed in terms of the argument x and the function y.

Example:

Find the derivative of the function y given by the equation x 3 + y 3 -3xy \u003d 0.

Solution: Function y is set implicitly. We differentiate the equality x 3 + y 3 -3xy \u003d 0 with respect to x. From the obtained relation

3x 2 + 3y 2 y "-3 (1 y + x y") \u003d 0

it follows that y 2 y "-xy" \u003d y-x 2, that is, y "\u003d (y-x 2) / (y 2 -x).

Higher order derivatives

It is clear that the derivative

function y \u003d f (x)there is also a function from x:

y "\u003d f" (x)

If the function f "(x) is differentiable, then its derivative is denoted by the symbol y "" \u003d f "" (x) xtwice.
Derivative of the second derivative, i.e. function y "" \u003d f "" (x)is called the third derivative of the function y \u003d f (x)or derivative of the function f (x) of the third order and denoted by symbols

Generally n-th derivative or derivative n-th order function y \u003d f (x)denoted by symbols

F-la Leibniz:

Suppose that the functions and are differentiable together with their derivatives up to the nth order inclusive. Applying the rule of differentiation of the product of two functions, we obtain

Let's compare these expressions with the powers of the binomial:

The rule of correspondence is striking: to obtain a formula for the derivative of the 1st, 2nd or 3rd orders of the product of functions and, you need to replace the degrees and in the expression for (where n \u003d 1,2,3) derivatives of the corresponding orders. In addition, the zero degrees of the quantities and should be replaced by derivatives of the zero order, meaning the functions and:

Generalizing this rule to the case of a derivative of arbitrary order n, we get leibniz's formula,

where are binomial coefficients:

Rolle's theorem.

This theorem makes it possible to find critical points, and then, with the help of sufficient conditions, investigate the function for extrema.

Let 1) the fth f (x) be defined and continuous on some closed interval; 2) there is a finite derivative, at least in the open interval (a; b); 3) at the ends of the interval, f-i takes equal values \u200b\u200bf (a) \u003d f (b). Then between the points a and b there is a point c such that the derivative at this point will be \u003d 0.

By the theorem on the property of f-th, continuous on a segment, f-th f (x) takes on this segment its max and min values.

f (x 1) \u003d M - max, f (x 2) \u003d m - min; x 1; x 2 Î

1) Let M \u003d m, i.e. m £ f (x) £ M

Þ f-i f (x) will take constant values \u200b\u200bon the interval from a to b, and Þ its derivative will be equal to zero. f '(x) \u003d 0

2) Let M\u003e m

Because by the conditions of the theorem, f (a) \u003d f (b) Þ its smallest or largest value of φ will take not at the ends of the segment, but Þ will take M or m at the interior point of this segment. Then by Fermat's theorem f '(c) \u003d 0.

Lagrange's theorem.

Finite increment formula or lagrangian mean value theorem asserts that if the function f is continuous on the segment [ a;b] and differentiable in the interval ( a;b), then there is a point such that

Cauchy's theorem.

If functions f (x) and g (x) are continuous on an interval and differentiable on an interval (a, b) and g ¢ (x) ¹ 0 on an interval (a, b), then there is at least one point e, a< e < b, такая, что

Those. the ratio of the increments of functions on a given segment is equal to the ratio of derivatives at the point e. Examples of problem solving course of lectures Calculating the volume of a body from the known areas of its parallel sections Integral calculus

Examples of course work Electrical engineering

At first glance, it is very convenient to use Lagrange's theorem to prove this theorem. Write down the finite difference formula for each function and then divide them by each other. However, this view is erroneous, since the point e for each of the functions is generally different. Of course, in some special cases this point of the interval may turn out to be the same for both functions, but this is a very rare coincidence, and not a rule, therefore it cannot be used to prove the theorem.

Evidence. Consider the helper function

When x → x 0, the quantity c also tends to x 0; we pass in the previous equality to the limit:

Because then.

therefore

(the limit of the ratio of two infinitesimal is equal to the limit of the ratio of their derivatives, if the latter exists)

L'Hôpital's rule, at ∞ / ∞.

We will call a function y \u003d f (x) BOUNDED UP (BOTTOM) on a set A from the domain D (f) if there is such a number M such that for any x from this set the condition

Using logical symbols, the definition can be written as:

f (x) – bounded above on the set

(f (x) – bounded below on the set

Functions limited in modulus or simply limited are also introduced.

We will call a function Bounded on a set A from the domain if there exists a positive number M such that

In the language of logical symbols

f (x) – bounded on the set

A function that is not limited is called unlimited. We know that the definitions given through negation are of little meaning. To formulate this statement as a definition, we use the properties of quantifier operations (3.6) and (3.7). Then the negation of the boundedness of the function in the language of logical symbols will give:

f (x) – bounded on the set

The result obtained allows us to formulate the following definition.

A function is called UNBOUNDED on a set A belonging to the domain of the function if on this set for any positive number M there is such a value of the argument x , that the value will still exceed the value of M, that is.

As an example, consider the function

It is defined on the entire real axis. If we take the segment [–2; 1] (set A), then on it it will be bounded both from above and from below.

Indeed, to show its boundedness from above, one must consider the predicate

and show that there is (exists) such M such that for all x taken on the interval [–2; 1], it will be true

Finding such an M is not difficult. We can assume M \u003d 7, the existential quantifier implies finding at least one value of M. The presence of such M and confirms the fact that the function on the interval [–2; 1] is bounded from above.

To prove its boundedness from below, it is necessary to consider the predicate

The value of M that ensures the truth of this predicate is, for example, M \u003d –100.

It can be proved that the function will be limited in modulus as well: for all x from the segment [–2; 1], the values \u200b\u200bof the function coincide with the values, so as M we can take, for example, the previous value M \u003d 7.

Let us show that the same function, but on the interval, will be unlimited, that is

To show that such x exist, consider the statement

Looking for the required values \u200b\u200bof x among the positive values \u200b\u200bof the argument, we get

This means that no matter what positive Mm we take, the values \u200b\u200bof x that ensure that the inequality

are obtained from the ratio.

Considering the function on the entire real axis, one can show that it is unlimited in absolute value.

Indeed, from the inequality

That is, no matter how large the positive M, or will ensure the fulfillment of the inequality.

EXTREME FUNCTION.

The function has at the point from local maximum (minimum) if there is a neighborhood of this point such that for x¹ from from this neighborhood the inequality

especially that the extremum point can only be the inner point of the interval and f (x) must be defined there. Possible cases of the absence of an extremum are shown in Fig. 8.8.

If the function increases (decreases) over some interval and decreases (increases) over some interval, then the point from is the point of the local maximum (minimum).

The absence of the maximum of the function f (x) at the point from can be formulated as follows:

_______________________

f (x) has a maximum at point c

This means that if the point c is not a local maximum point, then whatever the neighborhood that includes the point c as an internal one, there will be at least one value x not equal to c, at which. Thus, if there is no maximum at point c, then at this point there may be no extremum at all, or it may be a minimum point (Fig. 8.9).

The concept of an extremum gives a comparative estimate of the value of a function at any point in relation to those nearby. A similar comparison of the values \u200b\u200bof functions can be carried out for all points of a certain interval.

The LARGE (LOWEST) value of a function on a set will mean its value at a point from this set such that - for. The largest value of the function is attained at the inner point of the segment, and the smallest – at its left end.

To determine the largest (smallest) value of a function specified on a segment, it is necessary to choose the largest (smallest) number among all the values \u200b\u200bof its maxima (minima), as well as the values \u200b\u200btaken at the ends of the interval. It will be the largest (smallest) value of the function. This rule will be further refined.

The problem of finding the largest and smallest values \u200b\u200bof a function in an open interval is not always easy to solve. For example, the function

in the interval (Fig. 8.11) does not have them.

Let's make sure, for example, that this function doesn't matter most. In fact, given the monotonicity of the function, it can be argued that no matter how close we set the values \u200b\u200bof x to the left of unity, there will be other x in which the values \u200b\u200bof the function will be greater than its values \u200b\u200bat the fixed points, but still less than one.

1) Domain of function and domain of function values.

Function scope is the set of all valid valid values \u200b\u200bof the argument x (variable x) for which the function y \u003d f (x) defined. The range of values \u200b\u200bof a function is the set of all real values ythat the function accepts.

In elementary mathematics, functions are studied only on the set of real numbers.

2) Function zeros.

Function zero is an argument value at which the function value is equal to zero.

3) Intervals of constancy of function.

The intervals of constant sign of a function are such sets of argument values \u200b\u200bon which the function values \u200b\u200bare only positive or only negative.

4) Monotonicity of function.

An increasing function (in a certain interval) is a function for which a larger value of the argument from this interval corresponds to a larger value of the function.

Decreasing function (in a certain interval) - a function in which the larger value of the argument from this interval corresponds to the smaller value of the function.

5) Parity (odd) function.

An even function is a function whose domain of definition is symmetric about the origin and for any x from the domain, the equality f (-x) \u003d f (x)... The graph of an even function is symmetric about the ordinate axis.

An odd function is a function whose domain of definition is symmetric about the origin and for any x from the domain of definition, the equality f (-x) \u003d - f (x). The graph of an odd function is symmetric about the origin.

6) Limited and unlimited functions.

A function is called bounded if there exists a positive number M such that | f (x) | ≤ M for all values \u200b\u200bof x. If there is no such number, then the function is unlimited.

7) Periodicity of function.

A function f (x) is periodic if there exists a nonzero number T such that for any x from the domain of the function the following holds: f (x + T) \u003d f (x). This smallest number is called the period of the function. All trigonometric functions are periodic. (Trigonometric formulas).

19. Basic elementary functions, their properties and graphics. Application of functions in economics.

Basic elementary functions. Their properties and graphs

1. Linear function.

Linear function called a function of the form, where x is a variable, a and b are real numbers.

Number and called the slope of a straight line, it is equal to the tangent of the angle of inclination of this straight line to the positive direction of the abscissa axis. The graph of a linear function is a straight line. It is defined by two points.

Linear function properties

1. Domain of definition - the set of all real numbers: D (y) \u003d R

2. The set of values \u200b\u200bis the set of all real numbers: E (y) \u003d R

3. The function takes on a zero value for or.

4. The function increases (decreases) over the entire domain of definition.

5. The linear function is continuous on the whole domain of definition, differentiable and.

2. Quadratic function.

A function of the form, where x is a variable, the coefficients a, b, c are real numbers, is called quadratic.

Odds a, b, c determine the location of the graph on the coordinate plane

The coefficient a determines the direction of the branches. The graph of a quadratic function is a parabola. The coordinates of the vertex of the parabola are found by the formulas:

Function properties:

2. The set of values \u200b\u200bof one of the intervals: or.

3. The function takes zero values \u200b\u200bwhen , where the discriminant is calculated by the formula :.

4. The function is continuous on the entire domain and the derivative of the function is equal to.

Lesson and presentation on the topic: "Properties of a function. Increasing and decreasing functions"

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Guys, we continue to study numeric functions. Today we will focus on such a topic as function properties. Functions have many properties. Remember what properties we recently studied. That's right, scope and scope, they are one of the key properties. Never forget about them and remember that a function always has these properties.

In this section, we will define some of the properties of the functions. The order in which we will define them, I recommend that you follow them when solving problems.

Increasing and decreasing functions

The first property we will define is the increase and decrease of the function.

The concepts of "increase" and "decrease" of a function are very easy to understand if you look closely at the function graphs. For an increasing function: we, as it were, go up a hill, for a decreasing function, we go down. The general view of increasing and decreasing functions is presented in the graphs below.

The increase and decrease of a function is generally called monotonicity.That is, our task is to find the intervals of decreasing and increasing of the function. In the general case, this is formulated as follows: find intervals of monotonicity or examine the function for monotonicity.

Investigate the monotonicity of the function $ y \u003d 3x + 2 $.
Solution: Check the function for any x1 and x2 and let x1< x2.
$ f (x1) \u003d 3x1 + 2 $
$ f (x2) \u003d 3x2 + 2 $
Since, x1< x2, то f(x1) < f(x2), т. е. большему значению аргумента, соответствует большее значение функции.

Limited function

A function $ y \u003d f (x) $ is called lower bounded on a set X⊂D (f) if there exists a number a such that for any xϵX the inequality f (x)< a.

A function $ y \u003d f (x) $ is called upper bounded on a set X⊂D (f) if there exists a number a such that for any xϵX the inequality f (x)< a.

If the interval X is not indicated, then the function is considered to be limited over the entire domain. A function bounded both above and below is called bounded.

The limited function is easy to read from the graph. Can you draw some straight
$ y \u003d a $, and if the function is higher than this line, then it is bounded from below. If lower, then respectively above. Below is a graph of a function bounded from below. The graph of the limited function, guys, try to draw it yourself.

Investigate the boundedness of the function $ y \u003d \\ sqrt (16-x ^ 2) $.
Solution: The square root of a number is greater than or equal to zero. Obviously, our function is also greater than or equal to zero, that is, bounded from below.
We can extract the square root only from a non-negative number, then $ 16-x ^ 2≥0 $.
The solution to our inequality will be the interval [-4; 4]. On this segment $ 16-x ^ 2≤16 $ or $ \\ sqrt (16-x ^ 2) ≤4 $, but this means boundedness from above.
Answer: our function is limited to two straight lines $ y \u003d 0 $ and $ y \u003d 4 $.

Highest and lowest value

The smallest value of the function y \u003d f (x) on the set Х⊂D (f), is called some number m, such that:

b) For any xϵX, $ f (x) ≥f (x0) $ holds.

The largest value of the function y \u003d f (x) on the set X⊂D (f) is called some number m such that:
a) There is some х0 such that $ f (x0) \u003d m $.
b) For any хϵХ, $ f (x) ≤f (x0) $ holds.

The highest and lowest values \u200b\u200bare usually denoted by y naib. and y name. ...

The concepts of boundedness and the greatest with the least value of a function are closely related. The following statements are true:
a) If the function has the smallest value, then it is bounded from below.
b) If the function has the greatest value, then it is bounded from above.
c) If the function is not bounded from above, then the maximum value does not exist.
d) If the function is not bounded from below, then the smallest value does not exist.

Find the largest and smallest function value $ y \u003d \\ sqrt (9-4x ^ 2 + 16x) $.
Solution: $ f (x) \u003d y \u003d \\ sqrt (9-4x ^ 2 + 16x) \u003d \\ sqrt (9- (x-4) ^ 2 + 16) \u003d \\ sqrt (25- (x-4) ^ 2 ) ≤5 $.
For $ x \u003d 4 $ $ f (4) \u003d 5 $, for all other values, the function takes smaller values \u200b\u200bor does not exist, that is, this is the largest value of the function.
By definition: $ 9-4x ^ 2 + 16x≥0 $. Find the roots of the square trinomial $ (2x + 1) (2x-9) ≥0 $. For $ x \u003d -0.5 $ and $ x \u003d 4.5 $ the function vanishes, at all other points it is greater than zero. Then, by definition, the smallest value of the function is zero.
Answer: y naib. \u003d 5 and y naim. \u003d 0.

Guys, we have also studied the concept of convexity of a function. When solving some problems, we may need this property. This property is also easily determined using graphs.

The function is convex downward if any two points of the graph of the original function are connected, and the graph of the function is below the line of connection of the points.

The function is convex upward if any two points of the graph of the original function are connected, and the graph of the function is above the line of connection of points.

The function is continuous if the graph of our function has no discontinuities, for example, as the graph of the function above.

If you need to find the properties of a function, then the sequence of searching for properties is as follows:
a) Scope of definition.
b) Monotony.
c) Limitation.
d) Highest and lowest value.
e) Continuity.
f) Range of values.

Find the properties of the function $ y \u003d -2x + 5 $.
Decision.
a) Domain of definition D (y) \u003d (- ∞; + ∞).
b) Monotony. Let us check for any values \u200b\u200bof х1 and х2 and let х1< x2.
$ f (x1) \u003d - 2x1 + 2 $.
$ f (x2) \u003d - 2x2 + 2 $.
Since x1< x2, то f(x1) < f(x2), то есть большему значению аргумента, соответствует меньшее значение функции. Функция убывает.
c) Limitation. Obviously, the function is not limited.
d) Highest and lowest value. Since the function is unlimited, there is no largest or smallest value.
e) Continuity. The graph of our function has no discontinuities, then the function is continuous.
f) Range of values. E (y) \u003d (- ∞; + ∞).

Tasks on the properties of a function for independent solution

The theorem on the limit of a monotone function. The proof of the theorem is given using two methods. Also, definitions of strictly increasing, non-decreasing, strictly decreasing and non-increasing functions are given. Definition of a monotone function.

The function is not limited from above

1.1. Let the number b be finite:.
1.1.2. Let the function be not bounded above.

.

at.

Let us denote. Then exists for any, so that
at.
This means that the left limit at b is equal (see "Definitions of one-sided infinite limits of a function at an end point").

b early plus infinity

Function bounded from above

1. Let the function not decrease on the interval.
1.2.1. Let the function be bounded from above by the number M: for.
Let us prove that in this case there is a limit.

Since the function is bounded from above, there is a finite upper bound
.
According to the definition of the exact top face, the following conditions are met:
;
for any positive there is an argument for which
.

Since the function does not decrease, then at. Then at. Or
at.

So, we found that for anyone there is a number, so
at.
"Definitions of one-sided limits at infinity").

The function is not limited from above

1. Let the function not decrease on the interval.
1.2. Let the number b be equal to plus infinity:.
1.2.2. Let the function be not bounded above.
Let us prove that in this case there is a limit.

Since the function is not bounded from above, for any number M there exists an argument for which
.

Since the function does not decrease, then at. Then at.

So, for anyone there is a number, so
at.
This means that the limit at is equal to (see "Definitions of one-sided infinite limits at infinity").

The function does not increase

Now consider the case where the function does not increase. You can, as above, consider each option separately. But we'll cover them right away. For this we use. Let us prove that in this case there is a limit.

Consider the finite lower bound of the set of values \u200b\u200bof the function:
.
Here B can be either a finite number or an infinitely distant point. According to the definition of the exact bottom edge, the following conditions are met:
;
for any neighborhood of the point B, there is an argument for which
.
By the hypothesis of the theorem,. Therefore .

Since the function does not increase, then at. Since, then
at.
Or
at.
Further, we note that the inequality defines the left punctured neighborhood of the point b.

So, we found that for any neighborhood of the point, there exists a punctured left neighborhood of the point b such that
at.
This means that the limit on the left at point b is:

(see the universal definition of the limit of a function according to Cauchy).

Limit at point a

Now we will show that there is a limit at point a and find its value.

Let's consider a function. By the hypothesis of the theorem, the function is monotone for. Replace x with - x (or substitute and then replace t with x). Then the function is monotonic for. Multiplying inequalities by -1 and changing their order, we come to the conclusion that the function is monotonic for.

In a similar way, it is easy to show that if it does not decrease, then it does not increase. Then, according to what was proved above, there is a limit
.
If it does not increase, then it does not decrease. In this case, there is a limit
.

Now it remains to show that if there is a limit of the function at, then there is a limit of the function at, and these limits are equal:
.

Let's introduce the notation:
(1) .
Let us express f in terms of g:
.
Let's take an arbitrary positive number. Let there be an epsilon neighborhood of point A. Epsilon neighborhood is defined for both finite and infinite values \u200b\u200bof A (see “Point Neighborhood”). Since there is a limit (1), then, according to the definition of the limit, for any there exists such that
at.

Let a be a finite number. Let us express the left punctured neighborhood of the point -a using the inequalities:
at.
Replace x with -x and note that:
at.
The last two inequalities define the punctured right-hand neighborhood of the point a. Then
at.

Let a be an infinite number,. We repeat the reasoning.
at;
at;
at;
at.

So, we found that for anyone there is such that
at.
It means that
.

The theorem is proved.