The theory of limits is one of the branches of mathematical analysis. The question of solving limits is quite extensive, since there are dozens of methods for solving limits of various types. There are dozens of nuances and tricks that allow you to solve this or that limit. Nevertheless, we will still try to understand the main types of limits that are most often encountered in practice.

Let's start with the very concept of a limit. But first, a brief historical background. There lived in the 19th century a Frenchman, Augustin Louis Cauchy, who laid the foundations of mathematical analysis and gave strict definitions, the definition of a limit, in particular. It must be said that this same Cauchy was, is, and will be in the nightmares of all students of physics and mathematics, since he proved a huge number of theorems of mathematical analysis, and each theorem is more disgusting than the other. In this regard, we will not consider a strict definition of the limit, but will try to do two things:

1. Understand what a limit is.
2. Learn to solve the main types of limits.

I apologize for some unscientific explanations, it is important that the material is understandable even to a teapot, which, in fact, is the goal of the project.

So what is the limit?

And just an example of why to shaggy grandma....

Any limit consists of three parts:

1) The well-known limit icon.
2) Entries under the limit icon, in this case . The entry reads “X tends to one.” Most often - exactly, although instead of “X” in practice there are other variables. In practical tasks, the place of one can be absolutely any number, as well as infinity ().
3) Functions under the limit sign, in this case .

The recording itself reads like this: “the limit of a function as x tends to unity.”

Let's look at the next important question - what does the expression “x” mean? strives to one"? And what does “strive” even mean?
The concept of a limit is a concept, so to speak, dynamic. Let's build a sequence: first , then , , …, , ….
That is, the expression “x strives to one” should be understood as follows: “x” consistently takes on the values which approach unity infinitely close and practically coincide with it.

How to solve the above example? Based on the above, you just need to substitute one into the function under the limit sign:

So, the first rule: When given any limit, first we simply try to plug the number into the function.

We have considered the simplest limit, but these also occur in practice, and not so rarely!

Example with infinity:

Let's figure out what it is? This is the case when it increases without limit, that is: first, then, then, then, and so on ad infinitum.

What happens to the function at this time?
, , , …

So: if , then the function tends to minus infinity:

Roughly speaking, according to our first rule, instead of “X” we substitute infinity into the function and get the answer.

Another example with infinity:

Again we begin to increase to infinity, and look at the behavior of the function:

Conclusion: when the function increases without limit:

And another series of examples:

Please try to mentally analyze the following for yourself and remember the simplest types of limits:

, , , , , , , , ,
If you have doubts anywhere, you can pick up a calculator and practice a little.
In the event that , try to construct the sequence , , . If , then , , .

Note: strictly speaking, this approach to constructing sequences of several numbers is incorrect, but for understanding the simplest examples it is quite suitable.

Also pay attention to the following thing. Even if a limit is given with a large number at the top, or even with a million: , then it’s all the same , since sooner or later “X” will take on such gigantic values ​​that a million compared to them will be a real microbe.

What do you need to remember and understand from the above?

1) When given any limit, first we simply try to substitute the number into the function.

2) You must understand and immediately solve the simplest limits, such as , , etc.

Now we will consider the group of limits when , and the function is a fraction whose numerator and denominator contain polynomials

Example:

Calculate limit

According to our rule, we will try to substitute infinity into the function. What do we get at the top? Infinity. And what happens below? Also infinity. Thus, we have what is called species uncertainty. One might think that , and the answer is ready, but in the general case this is not at all the case, and it is necessary to apply some solution technique, which we will now consider.

How to solve limits of this type?

First we look at the numerator and find the highest power:

The leading power in the numerator is two.

Now we look at the denominator and also find it to the highest power:

The highest degree of the denominator is two.

Then we choose the highest power of the numerator and denominator: in this example, they are the same and equal to two.

So, the solution method is as follows: in order to reveal the uncertainty, it is necessary to divide the numerator and denominator by the highest power.

Here it is, the answer, and not infinity at all.

What is fundamentally important in the design of a decision?

First, we indicate uncertainty, if any.

Secondly, it is advisable to interrupt the solution for intermediate explanations. I usually use the sign, it does not have any mathematical meaning, but means that the solution is interrupted for an intermediate explanation.

Thirdly, in the limit it is advisable to mark what is going where. When the work is drawn up by hand, it is more convenient to do it this way:

It is better to use a simple pencil for notes.

Of course, you don’t have to do any of this, but then, perhaps, the teacher will point out shortcomings in the solution or start asking additional questions about the assignment. Do you need it?

Example 2

Find the limit
Again in the numerator and denominator we find in the highest degree:

Maximum degree in numerator: 3
Maximum degree in denominator: 4
Choose greatest value, in this case four.
According to our algorithm, to reveal uncertainty, we divide the numerator and denominator by .
The complete assignment might look like this:

Divide the numerator and denominator by

Example 3

Find the limit
Maximum degree of “X” in the numerator: 2
Maximum degree of “X” in the denominator: 1 (can be written as)
To reveal the uncertainty, it is necessary to divide the numerator and denominator by . The final solution might look like this:

Divide the numerator and denominator by

Notation does not mean division by zero (you cannot divide by zero), but division by an infinitesimal number.

Thus, by uncovering species uncertainty, we may be able to final number, zero or infinity.

Limits with uncertainty of type and method for solving them

The next group of limits is somewhat similar to the limits just considered: the numerator and denominator contain polynomials, but “x” no longer tends to infinity, but to finite number.

Example 4

Solve limit
First, let's try to substitute -1 into the fraction:

In this case, the so-called uncertainty is obtained.

General rule: if the numerator and denominator contain polynomials, and there is uncertainty of the form , then to disclose it you need to factor the numerator and denominator.

To do this, most often you need to solve a quadratic equation and/or use abbreviated multiplication formulas. If these things have been forgotten, then visit the page Mathematical formulas and tables and read the teaching material Hot formulas for school mathematics course. By the way, it is best to print it out; it is required very often, and information is absorbed better from paper.

So, let's solve our limit

Factor the numerator and denominator

In order to factor the numerator, you need to solve the quadratic equation:

First we find the discriminant:

And the square root of it: .

If the discriminant is large, for example 361, we use a calculator; the function of extracting the square root is on the simplest calculator.

! If the root is not extracted in its entirety (a fractional number with a comma is obtained), it is very likely that the discriminant was calculated incorrectly or there was a typo in the task.

Next we find the roots:

Thus:

All. The numerator is factorized.

Denominator. The denominator is already the simplest factor, and there is no way to simplify it.

Obviously, it can be shortened to:

Now we substitute -1 into the expression that remains under the limit sign:

Naturally, in a test, test, or exam, the solution is never written out in such detail. In the final version, the design should look something like this:

Let's factorize the numerator.

Example 5

Calculate limit

First, the “finish” version of the solution

Let's factor the numerator and denominator.

Numerator:
Denominator:



,

What is important in this example?
Firstly, you must have a good understanding of how the numerator is revealed, first we took 2 out of brackets, and then used the formula for the difference of squares. This is the formula you need to know and see.

This online math calculator will help you if you need it calculate the limit of a function. Program solution limits not only gives the answer to the problem, it leads detailed solution with explanations, i.e. displays the limit calculation process.

This program can be useful for high school students in general education schools when preparing for tests and exams, when testing knowledge before the Unified State Exam, and for parents to control the solution of many problems in mathematics and algebra.

In this way, you can conduct your own training and/or training of your younger brothers or sisters, while the level of education in the field of solving problems increases.

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A little theory.

Limit of the function at x->x 0

Let the function f(x) be defined on some set X and let the point \(x_0 \in X\) or \(x_0 \notin X\)

Let us take from X a sequence of points different from x 0:
x 1 , x 2 , x 3 , ..., x n , ... (1)
converging to x*. The function values ​​at the points of this sequence also form a numerical sequence
f(x 1), f(x 2), f(x 3), ..., f(x n), ... (2)
and one can raise the question of the existence of its limit.

Definition. The number A is called the limit of the function f(x) at the point x = x 0 (or at x -> x 0), if for any sequence (1) of values ​​of the argument x different from x 0 converging to x 0, the corresponding sequence (2) of values function converges to number A.

$$ \lim_(x\to x_0)( f(x)) = A $$

The function f(x) can have only one limit at the point x 0. This follows from the fact that the sequence
(f(x n)) has only one limit.

There is another definition of the limit of a function.

Definition The number A is called the limit of the function f(x) at the point x = x 0 if for any number \(\varepsilon > 0\) there is a number \(\delta > 0\) such that for all \(x \in X, \; x \neq x_0 \), satisfying the inequality \(|x-x_0| Using logical symbols, this definition can be written as
\((\forall \varepsilon > 0) (\exists \delta > 0) (\forall x \in X, \; x \neq x_0, \; |x-x_0| Note that the inequalities \(x \neq x_0 , \; |x-x_0| The first definition is based on the concept of the limit of a number sequence, so it is often called the “in the language of sequences” definition. The second definition is called the “in the language of \(\varepsilon - \delta \)”.
These two definitions of the limit of a function are equivalent and you can use either of them depending on which is more convenient for solving a particular problem.

Note that the definition of the limit of a function “in the language of sequences” is also called the definition of the limit of a function according to Heine, and the definition of the limit of a function “in the language \(\varepsilon - \delta \)” is also called the definition of the limit of a function according to Cauchy.

Limit of the function at x->x 0 - and at x->x 0 +

In what follows, we will use the concepts of one-sided limits of a function, which are defined as follows.

Definition The number A is called the right (left) limit of the function f(x) at the point x 0 if for any sequence (1) converging to x 0, the elements x n of which are greater (less than) x 0, the corresponding sequence (2) converges to A.

Symbolically it is written like this:
$$ \lim_(x \to x_0+) f(x) = A \; \left(\lim_(x \to x_0-) f(x) = A \right) $$

We can give an equivalent definition of one-sided limits of a function “in the language \(\varepsilon - \delta \)”:

Definition a number A is called the right (left) limit of the function f(x) at the point x 0 if for any \(\varepsilon > 0\) there exists \(\delta > 0\) such that for all x satisfying the inequalities \(x_0 Symbolic entries:

Topic 4.6. Calculation of limits

The limit of a function does not depend on whether it is defined at the limit point or not. But in the practice of calculating the limits of elementary functions, this circumstance is of significant importance.

1. If the function is elementary and if the limiting value of the argument belongs to its domain of definition, then calculating the limit of the function is reduced to a simple substitution of the limiting value of the argument, because limit of the elementary function f (x) at x striving forA , which is included in the domain of definition, is equal to the partial value of the function at x = A, i.e. lim f(x)=f( a) .

2. If x tends to infinity or the argument tends to a number that does not belong to the domain of definition of the function, then in each such case, finding the limit of the function requires special research.

Below are the simplest limits based on the properties of limits that can be used as formulas:

More complex cases of finding the limit of a function:

each is considered separately.

This section will outline the main ways to disclose uncertainties.

1. The case when x striving forA the function f(x) represents the ratio of two infinitesimal quantities

a) First you need to make sure that the limit of the function cannot be found by direct substitution and, with the indicated change in the argument, it represents the ratio of two infinitesimal quantities. Transformations are made to reduce the fraction by a factor tending to 0. According to the definition of the limit of a function, the argument x tends to its limit value, never coinciding with it.

In general, if we are looking for the limit of a function at x striving forA , then you must remember that x does not take on a value A, i.e. x is not equal to a.

b) Bezout's theorem is applied. If you are looking for the limit of a fraction whose numerator and denominator are polynomials that vanish at the limit point x = A, then according to the above theorem both polynomials are divisible by x- A.

c) Irrationality in the numerator or denominator is destroyed by multiplying the numerator or denominator by the conjugate to the irrational expression, then after simplifying the fraction is reduced.

d) The 1st remarkable limit (4.1) is used.

e) The theorem on the equivalence of infinitesimals and the following principles are used:

2. The case when x striving forA the function f(x) represents the ratio of two infinitely large quantities

a) Dividing the numerator and denominator of a fraction by the highest power of the unknown.

b) In general, you can use the rule

3. The case when x striving forA the function f (x) represents the product of an infinitesimal quantity and an infinitely large one

The fraction is transformed to a form whose numerator and denominator simultaneously tend to 0 or to infinity, i.e. case 3 reduces to case 1 or case 2.

4. The case when x striving forA the function f (x) represents the difference of two positive infinitely large quantities

This case is reduced to type 1 or 2 in one of the following ways:

a) bringing fractions to a common denominator;

b) converting a function to a fraction;

c) getting rid of irrationality.

5. The case when x striving forA the function f(x) represents a power whose base tends to 1 and exponent to infinity.

The function is transformed in such a way as to use the 2nd remarkable limit (4.2).

Example. Find .

Because x tends to 3, then the numerator of the fraction tends to the number 3 2 +3 *3+4=22, and the denominator tends to the number 3+8=11. Hence,

Example

Here the numerator and denominator of the fraction are x tending to 2 tend to 0 (uncertainty of type), we factorize the numerator and denominator, we get lim(x-2)(x+2)/(x-2)(x-5)

Example

Multiplying the numerator and denominator by the expression conjugate to the numerator, we have

Opening the parentheses in the numerator, we get

Example

Level 2. Example. Let us give an example of the application of the concept of the limit of a function in economic calculations. Let's consider an ordinary financial transaction: lending an amount S 0 with the condition that after a period of time T the amount will be refunded S T. Let's determine the value r relative growth formula

r=(S T -S 0)/S 0 (1)

Relative growth can be expressed as a percentage by multiplying the resulting value r by 100.

From formula (1) it is easy to determine the value S T:

S T= S 0 (1 + r)

When calculating long-term loans covering several full years, a compound interest scheme is used. It consists in the fact that if for the 1st year the amount S 0 increases to (1 + r) times, then for the second year in (1 + r) times the sum increases S 1 = S 0 (1 + r), that is S 2 = S 0 (1 + r) 2 . It turns out similarly S 3 = S 0 (1 + r) 3 . From the above examples, we can derive a general formula for calculating the growth of the amount for n years when calculated using the compound interest scheme:

S n= S 0 (1 + r) n.

In financial calculations, schemes are used where compound interest is calculated several times a year. In this case it is stipulated annual rate r And number of accruals per year k. As a rule, accruals are made at equal intervals, that is, the length of each interval Tk forms part of the year. Then for the period in T years (here T not necessarily an integer) amount S T calculated by the formula

(2)

where is the integer part of the number, which coincides with the number itself, if, for example, T? integer.

Let the annual rate be r and is produced n accruals per year at regular intervals. Then for the year the amount S 0 is increased to a value determined by the formula

(3)

In theoretical analysis and in the practice of financial activity, the concept of “continuously accrued interest” is often encountered. To move to continuously accrued interest, you need to increase indefinitely in formulas (2) and (3), respectively, the numbers k And n(that is, to direct k And n to infinity) and calculate to what limit the functions will tend S T And S 1 . Let's apply this procedure to formula (3):

Note that the limit in curly brackets coincides with the second remarkable limit. It follows that at an annual rate r with continuously accrued interest, the amount S 0 in 1 year increases to the value S 1 *, which is determined from the formula

S 1 * = S 0 e r (4)

Let now the sum S 0 is provided as a loan with interest accrued n once a year at regular intervals. Let's denote r e annual rate at which at the end of the year the amount S 0 is increased to the value S 1 * from formula (4). In this case we will say that r e- This annual interest rate n once a year, equivalent to annual interest r with continuous accrual. From formula (3) we obtain

S* 1 =S 0 (1+r e /n) n

Equating the right-hand sides of the last formula and formula (4), assuming in the latter T= 1, we can derive relationships between the quantities r And r e:

These formulas are widely used in financial calculations.

Concepts of limits of sequences and functions. When it is necessary to find the limit of a sequence, it is written as follows: lim xn=a. In such a sequence of sequences, xn tends to a and n tends to infinity. The sequence is usually represented as a series, for example:
x1, x2, x3...,xm,...,xn... .
Sequences are divided into increasing and decreasing. For example:
xn=n^2 - increasing sequence
yn=1/n - sequence
So, for example, the limit of the sequence xn=1/n^ :
lim 1/n^2=0

x→∞
This limit is equal to zero, since n→∞, and the sequence 1/n^2 tends to zero.

Typically, a variable quantity x tends to a finite limit a, and x is constantly approaching a, and the quantity a is constant. This is written as follows: limx =a, while n can also tend to either zero or infinity. There are infinite functions, for which the limit tends to infinity. In other cases, when, for example, the function is slowing down a train, it is possible about the limit tending to zero.
Limits have a number of properties. Typically, any function has only one limit. This is the main property of the limit. Others are listed below:
* The amount limit is equal to the sum of the limits:
lim(x+y)=lim x+lim y
* The product limit is equal to the product of the limits:
lim(xy)=lim x*lim y
* The limit of the quotient is equal to the quotient of the limits:
lim(x/y)=lim x/lim y
* The constant factor is taken outside the limit sign:
lim(Cx)=C lim x
Given a function 1 /x in which x →∞, its limit is zero. If x→0, the limit of such a function is ∞.
For trigonometric functions there are some of these rules. Since the function sin x always tends to unity when it approaches zero, the identity holds for it:
lim sin x/x=1

In a number of functions there are functions, when calculating the limits of which uncertainty arises - a situation in which the limit cannot be calculated. The only way out of this situation is L'Hopital. There are two types of uncertainties:
* uncertainty of the form 0/0
* uncertainty of the form ∞/∞
For example, a limit of the following form is given: lim f(x)/l(x), and f(x0)=l(x0)=0. In this case, an uncertainty of the form 0/0 arises. To solve such a problem, both functions are differentiated, after which the limit of the result is found. For uncertainties of type 0/0, the limit is:
lim f(x)/l(x)=lim f"(x)/l"(x) (at x→0)
The same rule is also true for uncertainties of the ∞/∞ type. But in this case the following equality is true: f(x)=l(x)=∞
Using L'Hopital's rule, you can find the values ​​of any limits in which uncertainties appear. A prerequisite for

volume - no errors when finding derivatives. So, for example, the derivative of the function (x^2)" is equal to 2x. From here we can conclude that:
f"(x)=nx^(n-1)

Function limit- number a will be the limit of some variable quantity if, in the process of its change, this variable quantity indefinitely approaches a.

Or in other words, the number A is the limit of the function y = f(x) at the point x 0, if for any sequence of points from the domain of definition of the function , not equal x 0, and which converges to the point x 0 (lim x n = x0), the sequence of corresponding function values ​​converges to the number A.

The graph of a function whose limit, given an argument that tends to infinity, is equal to L:

Meaning A is limit (limit value) of the function f(x) at the point x 0 in case for any sequence of points , which converges to x 0, but which does not contain x 0 as one of its elements (i.e. in the punctured vicinity x 0), sequence of function values converges to A.

Limit of a Cauchy function.

Meaning A will be limit of the function f(x) at the point x 0 if for any non-negative number taken in advance ε the corresponding non-negative number will be found δ = δ(ε) such that for each argument x, satisfying the condition 0 < | x - x0 | < δ , the inequality will be satisfied | f(x)A |< ε .

It will be very simple if you understand the essence of the limit and the basic rules for finding it. What is the limit of the function f (x) at x striving for a equals A, is written like this:

Moreover, the value to which the variable tends x, can be not only a number, but also infinity (∞), sometimes +∞ or -∞, or there may be no limit at all.

To understand how find the limits of a function, it is best to look at examples of solutions.

It is necessary to find the limits of the function f (x) = 1/x at:

x→ 2, x→ 0, x→ ∞.

Let's find a solution to the first limit. To do this, you can simply substitute x the number it tends to, i.e. 2, we get:

Let's find the second limit of the function. Here substitute pure 0 instead x it is impossible, because You cannot divide by 0. But we can take values ​​close to zero, for example, 0.01; 0.001; 0.0001; 0.00001 and so on, and the value of the function f (x) will increase: 100; 1000; 10000; 100,000 and so on. Thus, it can be understood that when x→ 0 the value of the function that is under the limit sign will increase without limit, i.e. strive towards infinity. Which means:

Regarding the third limit. The same situation as in the previous case, it is impossible to substitute ∞ in its purest form. We need to consider the case of unlimited increase x. We substitute 1000 one by one; 10000; 100000 and so on, we have that the value of the function f (x) = 1/x will decrease: 0.001; 0.0001; 0.00001; and so on, tending to zero. That's why:

It is necessary to calculate the limit of the function

Starting to solve the second example, we see uncertainty. From here we find the highest degree of the numerator and denominator - this is x 3, we take it out of brackets in the numerator and denominator and then reduce it by:

Answer

The first step in finding this limit, substitute the value 1 instead x, resulting in uncertainty. To solve it, let’s factorize the numerator and do this using the method of finding the roots of a quadratic equation x 2 + 2x - 3:

D = 2 2 - 4*1*(-3) = 4 +12 = 16→ √ D=√16 = 4

x 1.2 = (-2±4)/2→ x 1 = -3;x 2= 1.

So the numerator will be:

Answer

This is the definition of its specific value or a certain area where the function falls, which is limited by the limit.

To solve limits, follow the rules:

Having understood the essence and main rules for solving the limit, you will get a basic understanding of how to solve them.