Constant number A called limit sequences(x n ), if for any arbitrarily small positive numberε > 0 there is a number N that has all the values x n, for which n>N, satisfy the inequality

|x n - a|< ε. (6.1)

Write it down as follows: or x n → a.

Inequality (6.1) is equivalent double inequality

a- ε< x n < a + ε, (6.2)

which means that the points x n, starting from some number n>N, lie inside the interval (a-ε, a+ ε ), i.e. fall into any smallε -neighborhood of a point A.

A sequence having a limit is called convergent, otherwise - divergent.

The concept of a function limit is a generalization of the concept of a sequence limit, since the limit of a sequence can be considered as the limit of a function x n = f(n) of an integer argument n.

Let the function f(x) be given and let a - limit point domain of definition of this function D(f), i.e. such a point, any neighborhood of which contains points of the set D(f) other than a. Dot a may or may not belong to the set D(f).

Definition 1.The constant number A is called limit functions f(x) at x→a, if for any sequence (x n ) of argument values ​​tending to A, the corresponding sequences (f(x n)) have the same limit A.

This definition is called by defining the limit of a function according to Heine, or " in sequence language”.

Definition 2. The constant number A is called limit functions f(x) at x→a, if, by specifying an arbitrary arbitrarily small positive number ε , one can find such δ>0 (depending on ε), which is for everyone x, lying inε-neighborhoods of the number A, i.e. For x, satisfying the inequality
0 < x-a< ε , the values ​​of the function f(x) will lie inε-neighborhood of the number A, i.e.|f(x)-A|< ε.

This definition is called by defining the limit of a function according to Cauchy, or “in the language ε - δ “.

Definitions 1 and 2 are equivalent. If the function f(x) as x →a has limit, equal to A, this is written in the form

. (6.3)

In the event that the sequence (f(x n)) increases (or decreases) without limit for any method of approximation x to your limit A, then we will say that the function f(x) has infinite limit, and write it in the form:

A variable quantity (i.e. a sequence or function) whose limit equal to zero, called infinitely small.

Variable quantity whose limit equals infinity, called infinitely large.

To find the limit in practice, the following theorems are used.

Theorem 1 . If every limit exists

(6.4)

(6.5)

(6.6)

Comment. Expressions like 0/0, ∞/∞, ∞-∞ , 0*∞ , - are uncertain, for example, the ratio of two infinitesimal or infinitely large quantities, and finding a limit of this type is called “uncovering uncertainties.”

Theorem 2. (6.7)

those. one can go to the limit based on the power with a constant exponent, in particular, ;

(6.8)

(6.9)

Theorem 3.

(6.10)

(6.11)

Where e » 2.7 - base of natural logarithm. Formulas (6.10) and (6.11) are called the first wonderful limit and the second remarkable limit.

The consequences of formula (6.11) are also used in practice:

(6.12)

(6.13)

(6.14)

in particular the limit,

If x → a and at the same time x > a, then write x→a + 0. If, in particular, a = 0, then instead of the symbol 0+0 write +0. Similarly if x→a and at the same time x a-0. Numbers and are called accordingly right limit And left limit functions f(x) at the point A. For there to be a limit of the function f(x) as x→a is necessary and sufficient so that . The function f(x) is called continuous at the point x 0 if limit

. (6.15)

Condition (6.15) can be rewritten as:

,

that is, passage to the limit under the sign of a function is possible if it is continuous at a given point.

If equality (6.15) is violated, then we say that at x = xo function f(x) It has gap Consider the function y = 1/x. The domain of definition of this function is the set R, except for x = 0. The point x = 0 is a limit point of the set D(f), since in any neighborhood of it, i.e. in any open interval containing the point 0, there are points from D(f), but it itself does not belong to this set. The value f(x o)= f(0) is undefined, so at the point x o = 0 the function has a discontinuity.

The function f(x) is called continuous on the right at the point x o if the limit

,

And continuous on the left at the point x o, if the limit

.

Continuity of a function at a point xo is equivalent to its continuity at this point both to the right and to the left.

In order for the function to be continuous at a point xo, for example, on the right, it is necessary, firstly, that there be a finite limit, and secondly, that this limit be equal to f(x o). Therefore, if at least one of these two conditions is not met, then the function will have a discontinuity.

1. If the limit exists and is not equal to f(x o), then they say that function f(x) at the point x o has rupture of the first kind, or leap.

2. If the limit is+∞ or -∞ or does not exist, then they say that in point xo the function has a discontinuity second kind.

For example, function y = cot x at x→ +0 has a limit equal to +∞, which means that at the point x=0 it has a discontinuity of the second kind. Function y = E(x) (integer part of x) at points with whole abscissas has discontinuities of the first kind, or jumps.

A function that is continuous at every point in the interval is called continuous V . A continuous function is represented by a solid curve.

Many problems associated with the continuous growth of some quantity lead to the second remarkable limit. Such tasks, for example, include: growth of deposits according to the law of compound interest, growth of the country's population, decay of radioactive substances, proliferation of bacteria, etc.

Let's consider example of Ya. I. Perelman, giving an interpretation of the number e in the compound interest problem. Number e there is a limit . In savings banks, interest money is added to the fixed capital annually. If the accession is made more often, then the capital grows faster, since a larger amount is involved in the formation of interest. Let's take a purely theoretical, very simplified example. Let 100 deniers be deposited in the bank. units based on 100% per annum. If interest money is added to the fixed capital only after a year, then by this period 100 den. units will turn into 200 monetary units. Now let's see what 100 denize will turn into. units, if interest money is added to fixed capital every six months. After six months, 100 den. units will grow to 100× 1.5 = 150, and after another six months - at 150× 1.5 = 225 (den. units). If the accession is done every 1/3 of the year, then after a year 100 den. units will turn into 100× (1 +1/3) 3 " 237 (den. units). We will increase the terms for adding interest money to 0.1 year, to 0.01 year, to 0.001 year, etc. Then out of 100 den. units after a year it will be:

100 × (1 +1/10) 10 » 259 (den. units),

100 × (1+1/100) 100 » 270 (den. units),

100 × (1+1/1000) 1000 » 271 (den. units).

With an unlimited reduction in the terms for adding interest, the accumulated capital does not grow indefinitely, but approaches a certain limit equal to approximately 271. The capital deposited at 100% per annum cannot increase by more than 2.71 times, even if the accrued interest were added to the capital every second because the limit

Example 3.1.Using the definition of the limit of a number sequence, prove that the sequence x n =(n-1)/n has a limit equal to 1.

Solution.We need to prove that, no matter whatε > 0, no matter what we take, for it there is a natural number N such that for all n N the inequality holds|x n -1|< ε.

Let's take any e > 0. Since ; x n -1 =(n+1)/n - 1= 1/n, then to find N it is enough to solve the inequality 1/n< e. Hence n>1/ e and, therefore, N can be taken as an integer part of 1/ e , N = E(1/ e ). We have thereby proven that the limit .

Example 3.2 . Find the limit of a sequence given by a common term .

Solution.Let's apply the limit of the sum theorem and find the limit of each term. When n→ ∞ the numerator and denominator of each term tend to infinity, and we cannot directly apply the quotient limit theorem. Therefore, first we transform x n, dividing the numerator and denominator of the first term by n 2, and the second on n. Then, applying the limit of the quotient and the limit of the sum theorem, we find:

.

Example 3.3. . Find .

Solution. .

Here we used the limit of degree theorem: the limit of a degree is equal to the degree of the limit of the base.

Example 3.4 . Find ( ).

Solution.It is impossible to apply the limit of difference theorem, since we have an uncertainty of the form ∞-∞ . Let's transform the general term formula:

.

Example 3.5 . The function f(x)=2 1/x is given. Prove that there is no limit.

Solution.Let's use definition 1 of the limit of a function through a sequence. Let us take a sequence ( x n ) converging to 0, i.e. Let us show that the value f(x n)= behaves differently for different sequences. Let x n = 1/n. Obviously, then the limit Let us now choose as x n a sequence with a common term x n = -1/n, also tending to zero. Therefore there is no limit.

Example 3.6 . Prove that there is no limit.

Solution.Let x 1 , x 2 ,..., x n ,... be a sequence for which
. How does the sequence (f(x n)) = (sin x n) behave for different x n → ∞

If x n = p n, then sin x n = sin p n = 0 for all n and the limit If
x n =2 p n+ p /2, then sin x n = sin(2 p n+ p /2) = sin p /2 = 1 for all n and therefore the limit. So it doesn't exist.

Widget for calculating limits on-line

In the upper window, instead of sin(x)/x, enter the function whose limit you want to find. In the lower window, enter the number to which x tends and click the Calcular button, get the desired limit. And if in the result window you click on Show steps in the upper right corner, you will get a detailed solution.

Rules for entering functions: sqrt(x) - square root, cbrt(x) - cube root, exp(x) - exponent, ln(x) - natural logarithm, sin(x) - sine, cos(x) - cosine, tan (x) - tangent, cot(x) - cotangent, arcsin(x) - arcsine, arccos(x) - arccosine, arctan(x) - arctangent. Signs: * multiplication, / division, ^ exponentiation, instead infinity Infinity. Example: the function is entered as sqrt(tan(x/2)).

Function y = f (x) is a law (rule) according to which each element x of the set X is associated with one and only one element y of the set Y.

Element x ∈ X called function argument or independent variable.
Element y ∈ Y called function value or dependent variable.

The set X is called domain of the function.
Set of elements y ∈ Y, which have preimages in the set X, is called area or set of function values.

The actual function is called limited from above (from below), if there is a number M such that the inequality holds for all:
.
The number function is called limited, if there is a number M such that for all:
.

Top edge or exact upper bound A real function is called the smallest number that limits its range of values ​​from above. That is, this is a number s for which, for everyone and for any, there is an argument whose function value exceeds s′: .
The upper bound of a function can be denoted as follows:
.

Respectively bottom edge or exact lower limit A real function is called the largest number that limits its range of values ​​from below. That is, this is a number i for which, for everyone and for any, there is an argument whose function value is less than i′: .
The infimum of a function can be denoted as follows:
.

Determining the limit of a function

Determination of the limit of a function according to Cauchy

Finite limits of a function at end points

Let the function be defined in some neighborhood of the end point, with the possible exception of the point itself.
.
at a point, if for any there is such a thing, depending on , that for all x for which , the inequality holds
.
The limit of a function is denoted as follows:

Or at .
.

Using the logical symbols of existence and universality, the definition of the limit of a function can be written as follows:
One-sided limits.
.
Left limit at a point (left-sided limit):
.
Right limit at a point (right-hand limit):
; .

The left and right limits are often denoted as follows:

Finite limits of a function at points at infinity
.
.
.
Limits at points at infinity are determined in a similar way.
; ; .

They are often referred to as:

Using the concept of neighborhood of a point
.
If we introduce the concept of a punctured neighborhood of a point, then we can give a unified definition of the finite limit of a function at finite and infinitely distant points:
; ;
.
Here for endpoints
; ; .

Any neighborhood of points at infinity is punctured:

Infinite Function Limits
Definition Let the function be defined in some punctured neighborhood of a point (finite or at infinity). (x) Limit of function f 0 as x → x equals infinity > 0 , if for any arbitrarily large number M > 0 , depending on M, that for all x belonging to the punctured δ M - neighborhood of the point: , the following inequality holds:
.
The infinite limit is denoted as follows:
.
The limit of a function is denoted as follows:

Using the logical symbols of existence and universality, the definition of the infinite limit of a function can be written as follows:
.

You can also introduce definitions of infinite limits of certain signs equal to and :
.
.

Universal definition of the limit of a function

Using the concept of a neighborhood of a point, we can give a universal definition of the finite and infinite limit of a function, applicable for both finite (two-sided and one-sided) and infinitely distant points:
.

Determination of the limit of a function according to Heine

Let the function be defined on some set X:.
The number a is called the limit of the function at point:
,
if for any sequence converging to x 0 :
,
whose elements belong to the set X: ,
.

Let us write this definition using the logical symbols of existence and universality:
.

If we take the left-sided neighborhood of the point x as a set X 0 , then we obtain the definition of the left limit. If it is right-handed, then we get the definition of the right limit. If we take the neighborhood of a point at infinity as a set X, we obtain the definition of the limit of a function at infinity.

Theorem
The Cauchy and Heine definitions of the limit of a function are equivalent.
Proof

Properties and theorems of the limit of a function

Further, we assume that the functions under consideration are defined in the corresponding neighborhood of the point, which is a finite number or one of the symbols: .

It can also be a one-sided limit point, that is, have the form or .

The neighborhood is two-sided for a two-sided limit and one-sided for a one-sided limit. (x) Basic properties If the values ​​of the function f change (or make undefined) a finite number of points x 0 .

1, x 2, x 3, ... x n 0 , then this change will not affect the existence and value of the limit of the function at an arbitrary point x (x) If there is a finite limit, then there is a punctured neighborhood of the point x
.

, on which the function f 0 limited:
.
Let the function have at point x 0 finite non-zero limit:
Then, for any number c from the interval , there is such a punctured neighborhood of the point x
, what for ,

, If ;

, If . 0
,
If, on some punctured neighborhood of the point, , is a constant, then .

If there are finite limits and and on some punctured neighborhood of the point x
,
If, on some punctured neighborhood of the point, , is a constant, then .
That .
,
If , and on some neighborhood of the point
if , then and .

If on some punctured neighborhood of a point x 0 :
,
and there are finite (or infinite of a certain sign) equal limits:
, That
.

Proofs of the main properties are given on the page
"Basic properties of the limits of a function."

Arithmetic properties of the limit of a function

Let the functions and be defined in some punctured neighborhood of the point .
And let there be finite limits:
And .
;
;
;
, what for ,

And let C be a constant, that is, a given number. Then

If, then.
Proofs of arithmetic properties are given on the page

"Arithmetic properties of the limits of a function".

Theorem
Cauchy criterion for the existence of a limit of a function 0 In order for a function defined on some punctured neighborhood of a finite or at infinity point x > 0 , had a finite limit at this point, it is necessary and sufficient that for any ε 0 there was such a punctured neighborhood of the point x
.

, that for any points and from this neighborhood, the following inequality holds:

Limit of a complex function Limit theorem
complex function
Let the function have a limit and map a punctured neighborhood of a point onto a punctured neighborhood of a point.
Let the function be defined on this neighborhood and have a limit on it.
.

Here are the final or infinitely distant points: .
.

Neighborhoods and their corresponding limits can be either two-sided or one-sided. Then there is a limit of a complex function and it is equal to::
.
The limit theorem of a complex function is applied when the function is not defined at a point or has a value different from the limit.

To apply this theorem, there must be a punctured neighborhood of the point where the set of values ​​of the function does not contain the point:
If the function is continuous at the point , then the limit sign can be applied to the argument continuous function The following is a theorem corresponding to this case. 0 Theorem on the limit of a continuous function of a function 0 :
.
Let there be a limit of the function g 0 (t)
as t → t (x), and it is equal to x 0 .
Here is point t can be finite or infinitely distant: . And let the function f is continuous at point x:
.

Then there is a limit of the complex function f
(g(t))

, and it is equal to f

(x0)

Infinite Function Limits
Proofs of the theorems are given on the page
.

"Limit and continuity of a complex function". Infinitesimal and infinitely large functions

Infinitesimal functions on some punctured neighborhood of the point , to an infinitesimal at is an infinitesimal function at .

In order for a function to have a finite limit, it is necessary and sufficient that
,
where is an infinitesimal function at .

"Properties of infinitesimal functions".

Infinitely large functions

Infinite Function Limits
A function is said to be infinitely large if
.

The sum or difference of a bounded function, on some punctured neighborhood of the point , and an infinitely large function at is an infinitely large function at .

If the function is infinitely large for , and the function is bounded on some punctured neighborhood of the point , then
.

If the function , on some punctured neighborhood of the point , satisfies the inequality:
,
and the function is infinitesimal at:
, and (on some punctured neighborhood of the point), then
.

Proofs of the properties are presented in section
"Properties of infinitely large functions".

Relationship between infinitely large and infinitesimal functions

From the two previous properties follows the connection between infinitely large and infinitesimal functions.

If a function is infinitely large at , then the function is infinitesimal at .

If a function is infinitesimal for , and , then the function is infinitely large for .

The relationship between an infinitesimal and an infinitely large function can be expressed symbolically:
, .

If an infinitesimal function has a certain sign at , that is, it is positive (or negative) on some punctured neighborhood of the point , then this fact can be expressed as follows:
.
In the same way, if an infinitely large function has a certain sign at , then they write:
.

Then the symbolic connection between infinitely small and infinitely large functions can be supplemented with the following relations:
, ,
, .

Additional formulas, linking infinity symbols can be found on the page
"Points at infinity and their properties."

Limits of monotonic functions

Infinite Function Limits
A function defined on some set of real numbers X is called strictly increasing, if for all such that the following inequality holds:
.
Accordingly, for strictly decreasing function the following inequality holds:
.
For non-decreasing:
.
For non-increasing:
.

It follows that a strictly increasing function is also non-decreasing. A strictly decreasing function is also non-increasing.

The function is called monotonous, if it is non-decreasing or non-increasing.

Theorem
Let the function not decrease on the interval where .
If it is bounded above by the number M: then there is a finite limit.
If it is limited from below by the number m: then there is a finite limit.

If not limited from below, then .
If points a and b are at infinity, then in the expressions the limit signs mean that .

This theorem can be formulated more compactly.
;
.

Let the function not decrease on the interval where .

Then there are one-sided limits at points a and b:
;
.

A similar theorem for a non-increasing function.
Let the function not increase on the interval where .

Then there are one-sided limits:
The proof of the theorem is presented on the page
"Limits of monotonic functions".

References: L.D. Kudryavtsev. Course of mathematical analysis. Volume 1. Moscow, 2003. CM. Nikolsky. Course of mathematical analysis. Volume 1. Moscow, 1983.

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

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 departments, 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 task 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

in this case

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 given a limit with a large number at the top, even with a million: 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 check out methodological material Hot formulas school course mathematicians. 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 extraction function square root available on the simplest calculator.

! If the root is not extracted completely (it turns out a fractional number with a comma), 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 test work, during a 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's 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.

Methods for solving limits. Uncertainties.
The order of growth of the function. Replacement method

Example 4

Find the limit

This is a simpler example for independent decision. In the proposed example there is again uncertainty (of a higher order of growth than the root).

If "x" tends to "minus infinity"

The specter of “minus infinity” has been hovering in this article for a long time. Let us consider limits with polynomials in which . The principles and methods of solution will be exactly the same as in the first part of the lesson, with the exception of a number of nuances.

Let's look at 4 tricks that will be required to solve practical tasks:

1) Calculate the limit

The value of the limit depends only on the term, since it has the most high order growth. If , then infinitely large in modulus a negative number to an EVEN degree, in this case – in the fourth, is equal to “plus infinity”: . Constant (“two”) positive, That's why:

2) Calculate the limit

Here is the senior degree again even, That's why: . But in front of it there is a “minus” ( negative constant –1), therefore:

3) Calculate the limit

The limit value depends only on . As you remember from school, the “minus” “jumps out” from under the odd degree, so infinitely large in modulus negative number to an ODD power equals “minus infinity”, in this case: .
Constant (“four”) positive, Means:

4) Calculate the limit

The first guy in the village has again odd degree, in addition, in the bosom negative constant, which means: Thus:
.

Example 5

Find the limit

Using the above points, we come to the conclusion that there is uncertainty here. The numerator and denominator are of the same order of growth, which means that in the limit the result will be a finite number. Let's find out the answer by discarding all the fry:

The solution is trivial:

Example 6

Find the limit

This is an example for you to solve on your own. Full solution and answer at the end of the lesson.

And now, perhaps, the most subtle of cases:

Example 7

Find the limit

Considering the leading terms, we come to the conclusion that there is uncertainty here. The numerator is of a higher order of growth than the denominator, so we can immediately say that the limit is equal to infinity. But what kind of infinity, “plus” or “minus”? The technique is the same - let’s get rid of the little things in the numerator and denominator:

We decide:

Divide the numerator and denominator by

Example 15

Find the limit

This is an example for you to solve on your own. An approximate sample of the final design at the end of the lesson.

A couple more interesting examples on the topic of variable replacement:

Example 16

Find the limit

When substituting unity into the limit, uncertainty is obtained. Changing the variable already suggests itself, but first we transform the tangent using the formula. Indeed, why do we need a tangent?

Note that , therefore . If it’s not entirely clear, look at the sine values ​​in trigonometric table. Thus, we immediately get rid of the multiplier, in addition, we get the more familiar uncertainty of 0:0. It would be nice if our limit tended to zero.

Let's replace:

If , then

Under the cosine we have “x”, which also needs to be expressed through “te”.
From the replacement we express: .

We complete the solution:

(1) We carry out the substitution

(2) Open the parentheses under the cosine.

(4) To organize first wonderful limit, artificially multiply the numerator by and the reciprocal number.

Task for independent solution:

Example 17

Find the limit

Full solution and answer at the end of the lesson.

These were simple tasks in their class, in practice everything can be worse, and, in addition reduction formulas, you have to use a variety of trigonometric formulas, as well as other tricks. In the article Complex Limits I looked at a couple of real examples =)

On the eve of the holiday, we will finally clarify the situation with another common uncertainty:

Elimination of uncertainty “one to the power of infinity”

This uncertainty is “served” second wonderful limit, and in the second part of that lesson we looked in great detail at standard examples of solutions that are found in practice in most cases. Now the picture with the exponents will be completed, in addition, the final tasks of the lesson will be devoted to “false” limits, in which it SEEMS that it is necessary to apply the 2nd wonderful limit, although this is not at all the case.

The disadvantage of the two working formulas for the 2nd remarkable limit is that the argument must tend to “plus infinity” or to zero. But what if the argument tends to a different number?

A universal formula comes to the rescue (which is actually a consequence of the second remarkable limit):

Uncertainty can be eliminated using the formula:

Somewhere I think I already explained what the square brackets mean. Nothing special, brackets are just brackets. They are usually used to highlight mathematical notation more clearly.

Let us highlight the essential points of the formula:

1) It's about only about uncertainty and nothing else.

2) The “x” argument can tend to arbitrary value(and not just to zero or), in particular, to “minus infinity” or to anyone finite number.

Using this formula you can solve all the examples in the lesson. Wonderful Limits, which belong to the 2nd remarkable limit. For example, let's calculate the limit:

In this case , and according to the formula :

True, I don’t recommend doing this; the tradition is to still use the “usual” design of the solution, if it can be applied. However using the formula it is very convenient to check"classical" examples to the 2nd remarkable limit.

For those who want to learn how to find limits, in this article we will talk about this. We won’t delve into the theory; teachers usually give it at lectures. So the “boring theory” should be jotted down in your notebooks. If this is not the case, then you can read textbooks borrowed from the library. educational institution or on other Internet resources.

So, the concept of limit is quite important in studying the course higher mathematics, especially once you encounter integral calculus and understand the relationship between limit and integral. In the current material we will consider simple examples, as well as ways to solve them.

Examples of solutions

Example 1

Calculate a) $ \lim_(x \to 0) \frac(1)(x) $; b)$ \lim_(x \to \infty) \frac(1)(x) $

Solution

a) $$ \lim \limits_(x \to 0) \frac(1)(x) = \infty $$ b)$$ \lim_(x \to \infty) \frac(1)(x) = 0 $$ People often send us these limits with a request to help solve them. We decided to highlight them as a separate example and explain that these limits just need to be remembered, as a rule. If you cannot solve your problem, then send it to us. We will provide detailed solution. You will be able to view the progress of the calculation and gain information. This will help you get your grade from your teacher in a timely manner!

Answer

$$ \text(a)) \lim \limits_(x \to 0) \frac(1)(x) = \infty \text( b))\lim \limits_(x \to \infty) \frac(1 )(x) = 0 $$

What to do with uncertainty of the form: $ \bigg [\frac(0)(0) \bigg ] $

Example 3

Solve $ \lim \limits_(x \to -1) \frac(x^2-1)(x+1) $

Solution

As always, we start by substituting the value $ x $ into the expression under the limit sign. $$ \lim \limits_(x \to -1) \frac(x^2-1)(x+1) = \frac((-1)^2-1)(-1+1)=\frac( 0)(0)$$ What's next now? What should happen in the end? Since this is uncertainty, this is not an answer yet and we continue the calculation. Since we have a polynomial in the numerators, we will factorize it using the formula familiar to everyone from school $$ a^2-b^2=(a-b)(a+b) $$. Do you remember? Great! Now go ahead and use it with the song :) We find that the numerator $ x^2-1=(x-1)(x+1) $ We continue to solve taking into account the above transformation: $$ \lim \limits_(x \to -1)\frac(x^2-1)(x+1) = \lim \limits_(x \to -1)\frac((x-1)(x+ 1))(x+1) = $$ $$ = \lim \limits_(x \to -1)(x-1)=-1-1=-2 $$

Answer

$$ \lim \limits_(x \to -1) \frac(x^2-1)(x+1) = -2 $$

Let's push the limit in the last two examples to infinity and consider the uncertainty: $ \bigg [\frac(\infty)(\infty) \bigg ] $

Example 5

Calculate $ \lim \limits_(x \to \infty) \frac(x^2-1)(x+1) $

Solution

$ \lim \limits_(x \to \infty) \frac(x^2-1)(x+1) = \frac(\infty)(\infty) $ What to do? What should I do? Don't panic, because the impossible is possible. It is necessary to take out the x in both the numerator and the denominator, and then reduce it. After this, try to calculate the limit. Let's try... $$ \lim \limits_(x \to \infty) \frac(x^2-1)(x+1) =\lim \limits_(x \to \infty) \frac(x^2(1-\frac (1)(x^2)))(x(1+\frac(1)(x))) = $$ $$ = \lim \limits_(x \to \infty) \frac(x(1-\frac(1)(x^2)))((1+\frac(1)(x))) = $$ Using the definition from Example 2 and substituting infinity for x, we get: $$ = \frac(\infty(1-\frac(1)(\infty)))((1+\frac(1)(\infty))) = \frac(\infty \cdot 1)(1+ 0) = \frac(\infty)(1) = \infty $$

Answer

$$ \lim \limits_(x \to \infty) \frac(x^2-1)(x+1) = \infty $$

Algorithm for calculating limits

So, let's briefly summarize the examples and create an algorithm for solving the limits:

1. Substitute point x into the expression following the limit sign. If a certain number or infinity is obtained, then the limit is completely solved. Otherwise, we have uncertainty: “zero divided by zero” or “infinity divided by infinity” and move on to the next steps of the instructions.
2. To eliminate the uncertainty of “zero divided by zero,” you need to factor the numerator and denominator. Reduce similar ones. Substitute point x into the expression under the limit sign.
3. If the uncertainty is “infinity divided by infinity,” then we take out both the numerator and the denominator x to the greatest degree. We shorten the X's. We substitute the values ​​of x from under the limit into the remaining expression.

In this article you learned the basics of solving limits often used in the course. Mathematical analysis. Of course, these are not all types of problems offered by examiners, but only the simplest limits. We'll talk about other types of assignments in future articles, but first you need to learn this lesson in order to move forward. Let's discuss what to do if there are roots, degrees, study infinitesimal equivalent functions, remarkable limits, L'Hopital's rule.

If you can't figure out the limits yourself, don't panic. We are always happy to help!