The presentation “The Limit of a Function” is a visual aid that helps in studying material on this topic in algebra. The manual contains a detailed, understandable description of theoretical material that reveals the concept of the limit of a function, its graphical representation, the rules for calculating the limit of a function, and the connection between the properties of a function and its limit. All the theoretical foundations outlined in the presentation are supported during the demonstration by a description of the solution to the corresponding tasks.

Presenting the material in the form of a presentation makes it possible to present the concepts being studied in a more convenient way for understanding. Use effective tools to memorize material.

The presentation begins with a reminder of the type of functional dependence y=f(n), nϵN. The meaning of the limit of a function is revealed when constructing a graph of this function. It is noted that the equality limf(n)=bat n→∞ means that the straight line y=b drawn on the coordinate plane represents a horizontal asymptote to which the graph of the function tends as n→∞. The second slide shows a graph of the function y=f(x) on the coordinate plane, the domain of which lies on the interval D(f)=. If there is a horizontal asymptote y=b in the domain of definition, the function tends to the limit value limf(x)=b as x→-∞. The approach of the function to the asymptote is demonstrated in the corresponding figure presented on the slide.

Slide 4 describes the case of the graph of a function approaching a horizontal asymptote when its argument tends to both +∞ and -∞. This means the simultaneous fulfillment of the conditions limf(x)=b for x→-∞ and limf(x)=b for x→+∞. Otherwise, we can write limf(x)=b for x→∞. The figure shows an example of such a function and the behavior of its graph on the coordinate plane.

The following demonstrates the rules for calculating the limit of a function. Property 1 notes that for the function k/x m with natural m the equality lim(k/x m)=0 for x→∞ will be true. The second paragraph states that for the limits of two functions limf(x)=b and limg(x)=c, similar properties of sequence limits will be valid. That is, the limit of the sum is determined by the sum of the limits lim(f(x) + g(x))= b+с, the limit of the product is equal to the product of the limits limf(x) g(x)= bс, the limit of the quotient is equal to the quotient of the limits limf(x)/g (x) = b/c for g(x)≠0 and c≠0, and also the constant factor can be taken out of the limit sign limkf(x) = kb.

You can consolidate the knowledge gained by describing the solution to Example 1, in which you need to determine lim(√3 x 5 -17)/(x 5 +9). To obtain a solution, the numerator and denominator of the fraction are divided by the highest power of the variable, that is, x 5. After calculation we get lim(√3-17/ x 5)/(1+9/x 5).

Having assessed the limits and using the property of a quotient limit, we determine that lim(√3 x 5 -17)/(x 5 +9)=√3/1=√3. An important note for this example is that calculating the limits of a function is similar to calculating the limits of sequences, but in this case it is necessary to take into account that x cannot take the value - 5 √9, which turns the denominator to zero.

The next slide considers the case when x→a. The figure clearly shows that for a certain function f(x) when the variable approaches point a, the value of the function approaches the ordinate of the corresponding point on the graph, that is, limf(x)=b as x→a.

Slides 9, 10, 11 contain definitions that reveal the concepts of continuity of a function, continuous function at a point, on an interval. In this case, a function is considered continuous if limf(x)= f(a) as x→a. At point a, a function will be continuous if the relation limf(x)= f(a) is true as x→a, and continuous on the interval X will be a function that is continuous at any point on the interval X.

Examples of estimating the continuity of functions are given. It is noted that the functions y=C, y=kx+m, y=ax 2 +bx+c, y=|x|, y=x n for natural n are continuous on the entire number line, the function y=√x is continuous on the positive semiaxis, and the function y=x n is continuous on the positive semiaxis and negative semiaxis with a discontinuity at point 0, the trigonometric functions y=sinx, y=cosx on the entire straight line, and y=tgx, y=ctgx throughout the entire domain of definition. Also a function consisting of rational or irrational, trigonometric expressions, it is continuous for all points where the function is defined.

In example 2, you need to calculate the limit lim (x 3 +3x 2 -11x-8) for x→-1. At the beginning of the solution, it is noted that this function, consisting of rational expressions, is defined on the entire numerical axis and at the point x = -1. Therefore, the function is continuous at the point x = -1 and when approaching it, the limit receives the value of the function, that is, lim (x 3 +3x 2 -11x-8) = 5 at x→-1.

Example 3 demonstrates the calculation of the limit lim (cosπx/√x+6) for x→1. It is noted that the function is defined on the entire numerical axis, therefore it is continuous at the point x=1, therefore, lim (cosπx/√x+6)=-1/7 at x→1.

In example 4, you need to calculate lim((x 2 -25)/(x-5)) for x→5. This example is special in that for x=5 the denominator of the function goes to zero, which is unacceptable. You can determine the limit by transforming the expression. After reduction we get f(x)=x+5. Only in the search for solutions should one take into account that x≠5. In this case, lim((x 2 -25)/(x-5))= lim(x+5)=10 for x→5.

Slide 17 describes a note that demonstrates how to obtain the important limit lim(sint/t)=1 as t →0 using the number circle.

Slide 18 presents the definition of argument increment and function increment. The increment of the argument is represented by the difference between the variables x 1 - x 0 for the function defined at the points x 0 and x 1. In this case, the change in the value of the function f(x 1) - f(x 0) is called the increment of the function. Notations for the increment of the argument Δx and the increment of the function Δf(x) are introduced.

In example 5, the increment of the function y=x 2 is determined when the point x 0 =2 moves to x=2.1 and x=1.98. Solving the example comes down to finding the values ​​at the starting and ending points and their difference. So, in the first case Δу=4.41-4=0.41, and in the second case Δу=3.9204-4=-0.0796.

On slide 21 it is noted that for x→a the notation is valid (x-a)→0, which means Δx→0. Also, as f(x) → f(a) tends, used in the definition of continuity, the notation f(x)-f(a) →0 is valid, that is, Δу→0. Using this notation, a new definition of continuity at the point x=a is given if the function f(x) satisfies the following condition: if Δх→0, then Δу→0.

To consolidate the material, the solution to examples 6 and 7 is described, in which you need to find the increment of a function and the limit of the ratio of the increment of the function to the increment of the argument. In example 6 this needs to be done for the function y=kx+m. The increment of the function when a point moves from x to (x+ Δx) is displayed, demonstrating the changes on the graph. In this case, it turns out Δу= kΔх, and lim(Δу/ Δх)=k for Δх→0. The behavior of the function y = x 3 is analyzed in a similar way. The increment of this function when a point moves from x to (x+ Δx) is equal to Δу=(3x 2 +3x Δx+(Δx) 2) Δx, and the limit of the function is lim(Δу/ Δx)=3x 2.

The presentation “The Limit of a Function” can be used to teach a traditional lesson. It is recommended to use the presentation as a distance learning tool. If the student needs to study the topic independently, the manual is recommended for independent work.

Lesson objectives:

• Educational:
  • introduce the concept of the limit of a number, the limit of a function;
  • give concepts about types of uncertainty;
  • learn to calculate the limits of a function;
  • systematize acquired knowledge, activate self-control, mutual control.
• Educational:
  • be able to apply acquired knowledge to calculate limits.
  • develop mathematical thinking.
• Educational: to cultivate interest in mathematics and the disciplines of mental work.

Lesson type: first lesson

Forms of student work: frontal, individual

Necessary equipment: interactive whiteboard, multimedia projector, cards with oral and preparatory exercises.

Lesson Plan

1. Organizational moment (3 min.)
2. Introduction to the theory of the limit of a function. Preparatory exercises. (12 min.)
3. Calculation of function limits (10 min.)
4. Independent exercises (15 min.)
5. Summing up the lesson (2 min.)
6. Homework (3 min.)

DURING THE CLASSES

1. Organizational moment

Greeting the teacher, marking those absent, checking preparation for the lesson. Inform the topic and purpose of the lesson. Subsequently, all tasks are displayed on the interactive board.

2. Introduction to the theory of the limit of a function. Preparatory exercises.

Function limit (limit value of function) at a given point, limiting the domain of definition of a function, is the value to which the function in question tends as its argument tends to a given point.
The limit is written as follows.

Let's calculate the limit:
We substitute 3 for x.
Note that the limit of a number is equal to the number itself.

Examples: calculate limits

If at some point in the domain of definition of a function there is a limit and this limit is equal to the value of the function at a given point, then the function is called continuous (at a given point).

Let's calculate the value of the function at the point x 0 = 3 and the value of its limit at this point.

The value of the limit and the value of the function at this point coincide, therefore, the function is continuous at the point x 0 = 3.

But when calculating limits, expressions often appear whose meaning is not defined. Such expressions are called uncertainties.

Main types of uncertainties:

Uncovering Uncertainties

To disclose uncertainties, use the following:

• simplify the expression of a function: factor it, transform the function using abbreviated multiplication formulas, trigonometric formulas, multiply by its conjugate, which allows further reduction, etc., etc.;
• if a limit exists when disclosing uncertainties, then the function is said to converge to the specified value; if such a limit does not exist, then the function is said to diverge.

Example: Let's calculate the limit.
Let's factorize the numerator

3. Calculation of function limits

Example 1. Calculate the limit of the function:

With direct substitution, the result is uncertainty:

4. Independent exercises

Calculate limits:

5. Summing up the lesson

This is the first lesson

Subject:

Development and education for not a single person cannot be given or communicated. Anyone who wishes to join them must achieve this through your own activity, your own strength, your own tension. From the outside he can only get excitement. A. Diesterweg

Setting the goal and objectives of the lesson:

study definition of infinity;

• Determining the limit of a function at infinity;
• Determination of the limit of a function at plus infinity;
• Determining the limit of a function at minus infinity;
• Properties of continuous functions;

learn calculate simple limits of functions at infinity.

B. Bolzano

Bernard Bolzano (1781-1848), Czech mathematician and philosopher. He opposed psychologism in logic; He ascribed ideal objective existence to the truths of logic. Influenced

E . Husserl. Introduced a number of important concepts mathematical analysis, was the predecessor G. Cantora in the study of endless sets .

Augustin Louis Cauchy(French Augustin Louis Cauchy; August 21, 1789, Paris - May 23, 1857, Co, France) - great French mathematician and mechanic, member of the Paris Academy of Sciences, Royal Society of London

y =1 /x m

Existence

lim f(x) = b

x → ∞

equivalent to having

horizontal asymptote

the graph of the function y = f(x)

lim f(x) = b x →+∞

lim f(x) = b and lim f(x) = b x →+∞ x→-∞ lim f(x) = b x→∞

What we will study:

What is Infinity?

Limit of a function at infinity

Limit of a function at minus infinity .

Properties .

Examples.

Limit of a function at infinity.

Infinity - used to characterize limitless, boundless, inexhaustible objects and phenomena, in our case the characteristic of numbers.

Infinity is an arbitrarily large (small) unlimited number.

If we consider the coordinate plane, then the abscissa (ordinate) axis goes to infinity if it is continued indefinitely to the left or right (down or up).

Limit of a function at infinity.

Limit of a function at plus infinity.

Now let's move on to the limit of the function at infinity:

Let us have a function y=f(x), the domain of definition of our function contains the ray, and let the straight line y=b be the horizontal asymptote of the graph of the function y=f(x), let's write all this in mathematical language:

the limit of the function y=f(x) as x tends to minus infinity is equal to b

Limit of a function at infinity.

Limit of a function at infinity.

Our relations can also be executed simultaneously:

Then it is customary to write it as:

or

the limit of the function y=f(x) as x tends to infinity is b

Limit of a function at infinity.

Example.

Example. Construct a graph of the function y=f(x), such that:

• The domain of definition is the set of real numbers.
• f(x) is a continuous function

Solution:

We need to construct a continuous function on (-∞; +∞). Let's show a couple of examples of our function.

Limit of a function at infinity.

Basic properties.

To calculate the limit at infinity, several statements are used:

1) For any natural number m the following relation holds:

2) If

That:

a) The amount limit is equal to the sum of the limits:

b) The limit of the product is equal to the product of the limits:

c) The limit of the quotient is equal to the quotient of the limits:

d) The constant factor can be taken beyond the limit sign:

Limit of a function at infinity.

Example 1.

Find

Example 2.

.

Example 3.

Find the limit of the function y=f(x), as x tends to infinity .

Limit of a function at infinity.

Example 1.

Answer:

Example 2.

Answer:

Example 3.

Answer:

Limit of a function at infinity.

.

• Draw a graph of the continuous function y=f(x). Such that the limit as x tends to plus infinity is 7, and as x tends to minus infinity 3.
• Draw a graph of the continuous function y=f(x). Such that the limit as x tends to plus infinity is 5 and the function increases.
• Find limits:

• Find limits:

Limit of a function at infinity.

Problems to solve independently .

Answers:

• What does the existence of a limit of a function mean?

at infinity?

• What asymptote does the graph of the function y=1/x have? 4 ?
• What rules do you know for calculating limits?

functions at infinity?

• What are the formulas for calculating limits?

did you meet at infinity?

• How to find lim (5-3x3) / (6x3 +2)?

• What new did you learn in the lesson?
• What goal did we set at the beginning of the lesson?
• Has our goal been achieved?
• What helped us cope with the difficulty?
• What knowledge was useful to us when

doing assignments in class?

• How can you evaluate your work?

Stages

Theoretical questions

Number of points

Front work

Max-oh

Work at the board

points

The work itself

Reward points

6 points

From 20 points and above the score is “5”

From 15 to 19 points the score is “4”

From 10 to 14 points score – “3”

Homework

§31, paragraph 1, pp. 150-151 - textbook;

№ 669 (c), 670 (c), 671 (c), 672 (c),

673(c), 674(c), 676(c), 700 (d) – problem book.

Today's lesson is over,

You couldn't be more friendly.

But everyone should know:

Knowledge, perseverance, work

They will lead to progress in life.

Slide 2

Title page Contents Introduction Limit of a variable quantity Basic properties of limits Limit of a function at a point Concept of continuity of a function Limit of a function at infinity Remarkable limits Conclusion

Slide 3

Variable Limit

Limit is one of the basic concepts of mathematical analysis. The concept of a limit was used by Newton in the second half of the 17th century and by 18th-century mathematicians such as Euler and Lagrange, but they understood the limit intuitively. The first rigorous definitions of the limit were given by Bolzano in 1816 and Cauchy in 1821.

Slide 4

1. Variable value limit

Let the variable x in the process of its change unlimitedly approach the number 5, taking on the following values: 4.9; 4.99;4.999;...or 5.1; 5.01; 5.001;... In these cases, the difference modulus tends to zero: = 0.1; 0.01; 0.001;... The number 5 in the given example is called the limit of the variable x and is written lim x = 5. Definition 1. A constant quantity a is called the limit of the variable x if the modulus of the difference as x changes becomes and remains less than any arbitrarily small positive number e.

Slide 5

2. Basic properties of limits

1. The limit of the algebraic sum of a finite number of variables is equal to the algebraic sum of the limits of the terms: lim(x + y + … + t) = lim x + lim y + … + lim t. 2. The limit of the product of a finite number of variables is equal to the product of their limits: lim(x·y…t) = lim x · lim y…lim t. 3. The constant factor can be taken out of the limit sign: lim(cx) = lim c · lim x = c lim x. For example, lim(5x + 3) = lim 5x + lim 3 = 5 lim x + 3. 4. The limit of the ratio of two variables is equal to the ratio of the limits if the limit of the denominator is not zero: lim = lim y 5. The limit of a positive integer power of a variable magnitude is equal to the same power of the limit of the same variable: lim = (lim x)n For example: = = x3 + 3 x2 = (-2)2 + 3·(-2)2 = -8 + 12 = 4 6. If the variables x, y, z satisfy the inequalities x and xzy

Slide 6

3. Limit of a function at a point

Definition 2. The number b is called the limit* of a function at point a if for all values ​​of x sufficiently close to a and different from a, the values ​​of the function differ as little as desired from the number b. 1.Find: (3x2 – 2x). Solution. Using sequentially properties 1,3 and 5 of the limit, we obtain (3x2 – 2x) = (3x2) - (2x) = 3x2 - 2x = 3 - 2x = 3 22 - 2 2 = 8

Slide 7

4. The concept of continuity of function

2. Calculate Solution. When x = 1, the fraction is defined because its denominator is non-zero. Therefore, to calculate the limit, it is enough to replace the argument with its limit value. Then we get The specified rule for calculating limits cannot be applied in the following cases: 1) If the function at x = a is not defined; 2) If the denominator of the fraction when substituting x = a turns out to be equal to zero; 3) If the numerator and denominator of the fraction, when substituting x = a, simultaneously turns out to be zero or infinity. In such cases, the limits of functions are found using various artificial techniques.

Slide 8

5. Limit of a function at infinity

3. Find a Solution. At x, the denominator x + 5 also tends to infinity, and its inverse value is 0. Consequently, the product · 3 = tends to zero if x . So = 0

Slide 9

6. Remarkable Limits

Some limits cannot be found using the methods outlined above. For example, let's say you need to find. Direct substitution of the argument for its limit gives an uncertainty of the form 0/0. It is also impossible to transform the numerator and denominator in such a way as to isolate a common factor whose limit is zero. Let's proceed as follows. Let's take a circle with a radius of 1 and construct a central angle AOB equal to 2 radians. Let us draw the chord AB and the tangents AD and BD to the circle at points A and B. Obviously, |AC| = |CB| = sin x, |AD| = |DB| = tgх = 1 – The first remarkable limit. x = e 2.7182…,. x – The second remarkable limit. Solution. Dividing the numerator and denominator by x, we get x = ()x = = =

Slide 10

7. Limit calculations

1. (x2 – 7x + 4) = 32 – 7 3 + 4 = - 8. Solution. To find the direct limit, we replace the limits of the function at a point. 2. . Solution. Here are the limits of the numerator and denominator with x equal to zero. Multiply the numerator and denominator by the expression conjugate to the numerator, we get = = = = Therefore, = = = =

Slide 11

Conclusion

In this project, along with theoretical material, practical material was also considered. In practical application, we considered all possible ways to calculate limits. The study of the second section of higher mathematics is already of great interest, since last year we considered the topic “Matrixes. Application of matrix properties to solving systems of equations,” which was simple, if only for the reason that the result obtained was controllable. There is no such control here. Studying Sections of Higher Mathematics gives positive results. Classes in this course brought results: - a large amount of theoretical and practical material was studied; - developed the ability to choose a method for calculating the limit; - competent use of each calculation method has been developed; - the ability to design a task algorithm has been consolidated. We will continue to study sections of higher mathematics. The purpose of studying it is that we will be well prepared to re-study the course of higher mathematics.

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