Presentation on the topic limit of a function. Limit of a function Limit of a function at a point One-sided limits Limit of a function as x tends to infinity Basic theorems on limits Calculation of limits. Basic properties of limits

Rules for calculating limits If lim f(x) = b and lim g(x) =c, then x 1) The limit of the sum is equal to the sum of the limits: lim (f(x)+ g(x)) = lim f(x)+ lim g(x) = b+ c x x x 2) The limit of the product is equal to the product of the limits: lim f(x) g(x) = lim f(x) * lim g(x) = b c x x x 3) The limit of the quotient is equal to the quotient of the limits: lim f(x):g(x) = lim f(x) : lim g(x)= b:c x x x = k b x x

Plan of abstract Graphs of functions y=1/x and y=1/x 2. Graphs of functions y=1/x m, for m even and odd. The concept of a horizontal asymptote. The concept of the limit of a function on +, -,. The geometric meaning of the limit of a function on +, -,. Rules for calculating the limits of a function on. Formulas for calculating the limit of a function on. Techniques for calculating the limits of a function on.

Lesson summary What does the existence of a limit of a function at infinity mean? What asymptote does the function y=1/ x 4 have? What rules do you know for calculating the limits of a function at infinity? What formulas for calculating limits at infinity did you get acquainted with? How to find lim (5-3x 3) / (6x 3 +2)? x

References: - A.G. Mordkovich. Algebra and early calculus classes. Mnemosyne. M A.G. Mordkovich., P.V. Semenov. Methodological guide for the teacher. Algebra and early calculus class. A basic level of. M. Mnemozina. 2010

Lesson Objectives:

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

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. Acquaintance with the theory of the limit of a function. preparatory exercises. (12 min.)
3. Calculation of the limits of a function (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, mark the absent, check the preparation for the lesson. State the topic and purpose of the lesson. In the future, all tasks are displayed on the interactive whiteboard.

2. Acquaintance with the theory of the limit of a function. preparatory exercises.

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

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

Examples: compute limits

If there is a limit at some point of the function's domain and this limit is equal to the value of the function at the given point, then the function is called continuous (at the 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 value is not defined. Such expressions are called uncertainties.

Main types of uncertainties:

Disclosure of Uncertainties

The following is used to resolve uncertainties:

simplify the expression of the function: factorize, transform the function using abbreviated multiplication formulas, trigonometric formulas, multiply by the conjugate, which allows you to further reduce, etc., etc.;
if there is a limit in the disclosure of 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: calculate the limit.
Let's factorize the numerator

3. Calculation of the limits of a function

Example 1. Calculate the function limit:

With direct substitution, the uncertainty is obtained:

4. Independent exercises

Calculate limits:

5. Summing up the lesson

This lesson is the first

In this project, along with theoretical material, practical material was also considered. In practical application, we considered all kinds of ways to calculate the limits. The study of the second section of higher mathematics is already of great interest, since last year the topic “Matrices. Applying Matrix Properties to Solving Systems of Equations”, which was simple, if only for the reason that the result was controllable. There is no such control here. The study of the Sections of Higher Mathematics gives its positive result. Classes on this course have brought their results: - studied a large amount of theoretical and practical material; - the ability to choose a method for calculating the limit has been developed; - the competent use of each method of calculation has been worked out; - the ability to design a task algorithm is fixed. We will continue to study sections of higher mathematics. The purpose of studying it is that we will be well prepared for the re-study of the course of higher mathematics.

Plan I The concept of the limit of a function II The geometric meaning of the limit III Infinitely small and large functions and their properties IV Calculations of limits: 1) Some of the most commonly used limits; 2) Limits of continuous functions; 3) Limits of complex functions; 4) Uncertainties and methods for their solutions

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Basic limit theorems Theorem 1: In order for the number A to be the limit of the function f (x) at, it is necessary and sufficient that this function be represented in the form, where is infinitesimal. Corollary 1: A function cannot have 2 different limits at one point. Theorem 2: The limit of a constant value is equal to the constant itself Theorem 3: If a function for all x in some neighborhood of the point a, except perhaps for the point a itself, and has a limit at the point a, then

Basic limit theorems (continued) Theorem 4: If the function f 1 (x) and f 2 (x) have limits at, then at, their sum f 1 (x) + f 2 (x), the product f 1 also has limits (x)*f 2 (x), and subject to the quotient f 1 (x)/f 2 (x), and Corollary 2: If the function f(x) has a limit at, then, where n is a natural number. Corollary 3: The constant factor can be taken out of the sign of the limit

Subject:

Development and education of any person cannot be given or communicated. Anyone who wants to join them must to achieve this by one's own activity, one's own strength, one's own exertion. From the outside, he can only receive excitement. A. Diesterweg

Setting the goal and objectives of the lesson:

explore definition of infinity;

Determining the limit of a function at infinity;
Determining 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

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

E . Husserl. Introduced a number of important concepts mathematical analysis, was the forerunner G. Cantor 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 → ∞

is 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 will we study:

What is Infinity?

Limit of a function at infinity

Limit of function at minus infinity .

Properties .

Examples.

The limit of a function at infinity.

Infinity - used to characterize limitless, limitless, inexhaustible objects and phenomena, in our case, characterization 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 infinitely continued to the left or right (down or up).

The limit of a function at infinity.

Limit of a function to 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 our function contains a ray , and let the line y=b be the horizontal asymptote of the graph of the function y=f(x), let's write it all in mathematical language:

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

The limit of a function at infinity.

Also, our relations can be performed simultaneously:

Then it is customary to write it as:

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

The limit of a function at infinity.

Example.

Example. Plot the function y=f(x) such that:

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

Solution:

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

The 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 is true:

2) If

That:

a) The sum 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 out of the limit sign:

The 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 .

The limit of a function at infinity.

Example 1

Answer:

Example 2

Answer:

Example 3

Answer:

The limit of a function at infinity.

Construct a graph of a continuous function y=f(x). Such that the limit for x tending to plus infinity is 7, and for x tending to minus infinity 3.
Construct a graph of a continuous function y=f(x). Such that the limit as x tends to plus infinity is 5 and the function is increasing.
Find Limits: