The theory of limits is one of the sections of mathematical analysis. The question of the solution of the limits is quite extensive, since there are dozens of decisions of solutions of various species. There are dozens of nuances and tricks that allow you to solve one or another limit. However, we still try to understand the main types of limits that are most common in practice.

Let's start with the very concept of limit. But first a brief historical reference. He lived in the 19th century, French Augusten Louis Cauch, who laid the foundations of mathematical analysis and gave strict definitions, determination of the limit, in particular. I must say, this very Cauchy dreamed, dream and will decline in nightmarish dreams to all students of physical and mathematical faculties, as it has proved a huge number of the theorems of mathematical analysis, and one theorem is disgusting other. In this regard, we will not consider strict determination of the limit, and try to make two things:

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

I apologize for some unsuitable explanation, it is important that the material is understandable even to the teapot, which, in fact, is the task of the project.

So what is the limit?

And immediately an example, which grandmother shake ....

Any limit consists of three parts:

1) All the known limit badge.
2) records under the limit icon, in this case. The record read "X is striving for one." Most often - precisely, although instead of "IKSA" in practice there are other variables. In practical tasks at the site of the unit there may be a completely any number, as well as infinity ().
3) functions under the sign of the limit, in this case.

Record itself It is read like this: "The limit of the function at X is seeking to one."

We will analyze the next important question - what does the expression "X strive to unit "? And what kind of "seeks"?
The concept of limit is the concept, if you can say so dynamic. We construct the sequence: first, then,, ..., , ….
That is the expression "X strive to unity "should be understood as" X "consistently takes values, which are infinitely close to one and virtually coincide with it.

How to solve the above example? Based on the foregoing, it is necessary to simply substitute a unit into a function standing under the sign of the limit:

So, the first rule: When any limit is given, you just try to substitute the number in the function.

We looked at the simplest limit, but these are found in practice, and not so rarely!

Example with infinity:

We understand what? This is the case when it is indefinitely increasing, that is, first, then, then, then, and so on to infinity.

And what happens at this time with the function?
, , , …

So: if, the function seeks to minus infinity:

Roughly speaking, according to our first rule, we instead of "IKSA" we substitute infinity and get the answer.

Another example with infinity:

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

Conclusion: when the function increases indefinitely:

And another series of examples:

Please try to independently analyze the following and remember the simplest types of limits:

, , , , , , , , ,
If somewhere there are doubts, you can take a calculator into the hands and take a little bit.
In the event that, try to build a sequence ,, If, then,.

Note: Strictly speaking, this approach with the construction of sequences from several numbers is incorrect, but it is quite suitable for understanding the simplest examples.

Also pay attention to the next thing. Even if a limit is given with a large number at the top, yes, even with a million: it's all the same Since sooner or later "X" will take such gigantic meanings that a million compared to them will be the most real microbe.

What needs to be remembered and understand from the foregoing?

1) When any limit is given, you just try to substitute the number into the function.

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

Now we will look at the group of limits when, and the function is a fraction, in the numerator and the denominator of which are polynomials

Example:

Calculate the limit

According to our rule, try to substitute infinity into the function. What do we get at the top? Infinity. And what happens at the bottom? Also infinity. Thus, we have the so-called uncertainty of the species. It would be possible to think that, and the answer is ready, but in general it is not at all, and you need to apply some decisions that we now consider.

How to solve the limits of this type?

First we look at the numerator and find to the high degree:

The older degree in the numerator is two.

Now we look at the denominator and also find to the high degree:

The older degree of denominator is equal to two.

Then we choose the most eldest degree of the numerator and the denominator: in this example they coincide and equal to twice.

So, the solution method is as follows: In order to disclose uncertainty, it is necessary to divide the numerator and the denominator to the senior degree.

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

What is fundamentally important in deciding a solution?

First, we indicate uncertainty if it is.

Secondly, it is desirable to interrupt the solution for intermediate explanations. I usually use a sign, it does not bear any mathematical meaning, but denotes that the solution is interrupted for an intermediate explanation.

Thirdly, in the limit, it is advisable to marry what and where to seek. When the work is made from hand, it is more convenient to do this:

For marks, it is better to use a simple pencil.

Of course, you can not do anything, but then, perhaps, the teacher will notice the shortcomings in the decision or starts asking additional questions on the task. Do you need it?

Example 2.

Find a limit
Again in the numerator and denominator we find to the top degree:

Maximum degree in numeric: 3
Maximum degree in denominator: 4
Choose most The value in this case is four.
According to our algorithm, to disclose uncertainty divide the numerator and denominator on.
Full design task may look like this:

We divide the numerator and denominator on

Example 3.

Find a limit
Maximum degree of "IKSA" in the Numerator: 2
Maximum degree "IKSA" in the denominator: 1 (can be written as)
To disclose uncertainty, it is necessary to divide the numerator and denominator on. The finishing solution may look like this:

We divide the numerator and denominator on

Under the record, it is implied not to divide on zero (to divide on zero it is impossible), and dividing to an infinitely small number.

Thus, when disclosing uncertainty, we can get final, zero or infinity.

Limits with uncertainty of the type and method of solving them

The next group of limits is something similar to the only considered limits: in the numerator and denominator are polynomials, but "X" seeks not to infinity, but to finite number.

Example 4.

Solve the limit
First, try to substitute -1 in the fraction:

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

General rule: If there are polynomials in the numerator and denominator, and there is uncertainty of the view, then for its disclosure you need to decompose the numerator and denominator for multipliers..

For this, it is most often necessary to solve the square equation and (or) to use the formulas of abbreviated multiplication. If these things have forgotten, then visit the page Mathematical formulas and tables and familiarize yourself with the methodological material Hot Mathematics School Course Formulas. By the way, it is best to print it, it is necessary very often, and information from paper is absorbed better.

So, we solve our limit

Spread the numerator and denominator for multipliers

In order to expand the numerator on multipliers, you need to solve the square equation:

First find discriminant:

And square root of it :.

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

! If the root is not retrieved aiming (a fractional semicolon is obtained), it is very likely that the discriminant is calculated incorrectly either in the task of typo.

Next, we find the roots:

In this way:

Everything. The numerator on the factors is decomposed.

Denominator. The denominator is already the simplest multiplier, and it is impossible to simplify it.

Obviously, you can cut on:

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

Naturally, in the control work, on the standings, the exam is so detailed, the decision never paint. In the finishing version, the design should look something like this:

Spread the numerator on multipliers.

Example 5.

Calculate the limit

First "finishing" solution

Spread the numerator and denominator for multipliers.

Numerator:
Denominator:



,

What is important in this example?
First, you must understand how the numerator is revealed, first we carried out for a bracket 2, and then used the formula of the square difference. This formula needs to know and see.

Methods for solving limits. Uncertainty.
The procedure for growth of the function. Replacement method

Example 4.

Find a limit

This is a simpler example for an independent solution. In the proposed example, uncertainty (higher growth in growth than root).

If "X" seeks to "minus infinity"

The ghost "minus infinity" has long been vital in this article. Consider the limits with polynomials in which. The principles and methods of solutions will be exactly the same as in the first part of the lesson, with the exception of a number of nuances.

Consider 4 chips that will be required to solve practical tasks:

1) Calculate the limit

The limit value depends only on the foundation, since it has the highest growth procedure. If, then infinitely large modulo negativeIn this case, in the fourth, equal to the "plus infinity" :. Constant ("Two") positive, so:

2) Calculate the limit

Here is the eldest degree again thought, so: . But before setting up "minus" ( negative Constant -1), therefore:

3) Calculate the limit

The limit value depends only from. How do you remember from school, "minus" "pops up" from under the odd degree, so infinitely large modulo negative number in odd degree Equally "minus infinity", in this case :.
Constant ("Four") positiveSo:

4) Calculate the limit

The first guy on the village again possesses odd degree, in addition, for the sinus negative Constant, and therefore: in this way:
.

Example 5.

Find a limit

Using the above items, we conclude that there is uncertainty. The numerator and the denominator of one order of growth, which means that the final number will be obtained. We learn the answer, throwing all the fry:

Trivial solution:

Example 6.

Find a limit

This is an example for an independent solution. Complete solution and answer at the end of the lesson.

And now, perhaps, the most subtle of the cases:

Example 7.

Find a limit

Considering the senior terms, we conclude that there is uncertainty. A high-order numerator than a denominator, so it can be immediately said that the limit is equal to infinity. But what infinity, "plus" or "minus"? The reception is the same - in the numerator and denominator, we get rid of the little things:

We decide:

We divide the numerator and denominator on

Example 15.

Find a limit

This is an example for an independent solution. Exemplary sample design at the end of the lesson.

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

Example 16.

Find a limit

When substituting a unit to the limit, uncertainty is obtained. Replacing the variable is already asking for, but first we transform the tangent by the formula. Indeed, why do we need Tangent?

Note that, therefore. If it's not entirely clear, look at the sinus values \u200b\u200bin trigonometric table. Thus, we immediately get rid of the multiplier, in addition, we obtain a more familiar uncertainty of 0: 0. It would be good and the limit of us sought to zero.

We will replace:

If, then

Under the cosine we have "X", which also needs to be expressed through the "TE".
From the replacement, we express :.

Complete the solution:

(1) We carry out the substitution

(2) Reveal brackets under the cosine.

(4) to organize first wonderful limit , artificially dominated the numerator on and the opposity.

Task for an independent solution:

Example 17.

Find a limit

Complete solution and answer at the end of the lesson.

These were uncomplicated tasks in their class, in practice everything is worse, and, in addition to molding formulahave to use a variety of trigonometric formulas, as well as other tricks. In the article, complex limits, I disassembled a couple of these examples \u003d)

On the eve of the holiday, I will finally clarify the situation with one more common uncertainty:

Elimination of the uncertainty "Unit to the degree of infinity"

This uncertainty "serves" the second wonderful limit, and in the second part of the lesson, we considered standard examples of solutions in most cases, which in most cases are found in practice. Now the painting with exhibitors will be completed, in addition, the final tasks of the lesson will be devoted to the limits of "deception", in which it seems that it is necessary to apply the 2nd wonderful limit, although it is not at all.

The lack of two working formulas of the 2nd wonderful limit is that the argument should strive for the "plus infinity" or to zero. But what if the argument strives for another number?

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

Uncertainty can be eliminated by the formula:

Somewhere it seems already explained that they denote square brackets. Nothing special, brackets as brackets. Usually they are used to clearly select the mathematical record.

We highlight the essential moments of the formula:

1) we are talking Only about uncertainty and no other.

2) the argument "X" may strive for arbitrary meaning (and not only to zero or), in particular, to "minus infinity" or to lOVE finite number.

With this formula, you can solve all examples of the lesson Wonderful limitswho belong to the 2nd wonderful limit. For example, we calculate the limit:

In this case , and by the formula :

True, I do not advise doing so, in the traditions still apply "usual" decisions of the solution if it can be applied. but with the help of the formula, it is very convenient to check "Classic" examples on the 2nd wonderful limit.

For those who want to learn to find the limits in this article we will tell about it. We will not delve into the theory, usually give teachers at lectures. So the "boring theory" should be done in you in notebooks. If this is not, then you can read the textbooks taken in the library of an educational institution or on other Internet resources.

So, the concept of limit is quite important in learning the course of higher mathematics, especially when you encounter an integral calculus and understand the link between the limit and integral. In the current material, simple examples will be considered, 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) $

Decision

a) $$ \\ l \\ limits_ (x \\ to 0) \\ FRAC (1) (x) \u003d \\ infty $$ b) $$ \\ lim_ (x \\ to \\ infty) \\ FRAC (1) (x) \u003d 0 $$ We often send these limits with a request to help solve. We decided to highlight them with a separate example and clarify that these limits must be simply remembered, as a rule. If it is impossible to solve your task, then send it to us. We will provide a detailed decision. You can familiarize yourself with the course of calculation and learn information. This will help in a timely manner at the teacher!

Answer

$$ \\ TEXT (A)) \\ LIM \\ Limits_ (X \\ To 0) \\ FRAC (1) (X) \u003d \\ INFTY \\ Text (b)) \\ \u200b\u200bLim \\ Limits_ (X \\ To \\ Infty) \\ FRAC (1 ) (x) \u003d 0 $$

What to do with the uncertainty of the type: $ \\ Bigg [\\ FRAC (0) (0) \\ Bigg] $

Example 3.

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

Decision

As always, we start with the substitution of the value of $ x $ in the expression that stands under the sign of the limit. $$ \\ LIM \\ Limits_ (x \\ to -1) \\ FRAC (X ^ 2-1) (X + 1) \u003d \\ FRAC ((- 1) ^ 2-1) (- 1 + 1) \u003d \\ FRAC ( 0) (0) $$ What's next? What should happen in the end? Since this is uncertainty, this is not a response and continue the calculation. Since we have a polynomial in the numerals, then we will expand it on factors, the help of a familiar formula since the school bench $$ a ^ 2-b ^ 2 \u003d (a - b) (a + b) $$$. Remembered? Excellent! Now forward and use it with the song :) We obtain that the numerator $ x ^ 2-1 \u003d (x - 1) (x + 1) $ We continue to solve considering the above transformation: $$ \\ LIM \\ Limits_ (x \\ to -1) \\ FRAC (x ^ 2-1) (x + 1) \u003d \\ Limits_ (X \\ TO -1) \\ FRAC ((X-1) (X + 1)) (x + 1) \u003d $$ $$ \u003d \\ LIM \\ Limits_ (x \\ to -1) (x-1) \u003d - 1-1 \u003d -2 $$

Answer

$$ \\ LIM \\ Limits_ (x \\ to -1) \\ FRAC (X ^ 2-1) (x + 1) \u003d -2 $$

We will fix the limit in the last two examples to infinity and consider uncertainty: $ \\ Bigg [\\ FRAC (\\ INFTY) (\\ INFTY) \\ Bigg] $

Example 5.

Calculate $ \\ l \\ limits_ (x \\ to \\ infty) \\ FRAC (x ^ 2-1) (x + 1) $

Decision

$ \\ limits_ (x \\ to \\ infty) \\ FRAC (x ^ 2-1) (x + 1) \u003d \\ FRAC (\\ INFTY) (\\ INFTY) $ What to do? How to be? Do not panic, because the impossible is possible. It is necessary to bear behind the brackets and in the numerator and in the indicator of X, and then cut it. After that, the limit to try to calculate. We try ... $$ \\ LIM \\ Limits_ (X \\ To \\ infty) \\ FRAC (X ^ 2-1) (X + 1) \u003d \\ LIM \\ LIMITS_ (X \\ To \\ INFTY) \\ FRAC (X ^ 2 (1- \\ FRAC (1) (x ^ 2))) (x (1+ \\ FRAC (1) (x))) \u003d $$ $$ \u003d \\ LIM \\ Limits_ (x \\ To \\ infty) \\ FRAC (X (1- \\ FRAC (1) (X ^ 2)) ((1+ \\ FRAC (1) (X))) \u003d $$ Using the definition of Example 2 and substituting infinity to the place: $$ \u003d \\ FRAC (\\ INFTY (1- \\ FRAC (1) (\\ INFTY)) ((1+ \\ FRAC (1) (\\ INFTY))) \u003d \\ FRAC (\\ INFTY \\ CDOT 1) (1+ 0) \u003d \\ FRAC (\\ INFTY) (1) \u003d \\ INFTY $$

Answer

$$ \\ LIM \\ Limits_ (x \\ to \\ infty) \\ FRAC (x ^ 2-1) (x + 1) \u003d \\ infty $$

Algorithm for calculating Limit

So, let's briefly summarize the disassembled examples and make the algorithm for the solution of the limits:

1. Substitute the point x in the expression following the limit after the sign. If a certain number is obtained or infinity, the limit is completely resolved. Otherwise, we have uncertainty: "zero divide to zero" or "Infinity to share on infinity" and go to the following instructions.
2. To eliminate the uncertainty "zero divide on zero" you need to decompose the numerator and denominator for multipliers. Reduce the like. Substitute the point x in the expression standing under the sign of the limit.
3. If the uncertainty "Infinity to share in infinity", then we endure both in the numerator, and in the denominator x the greatest extent. Reducing the Xers. We substitute the value of the ICA from under the limit into the remaining expression.

In this article, you got acquainted with the basics of solving the limits often used in the course of mathematical analysis. Of course, these are not all types of tasks offered by examiners, but only the simplest limits. In the following articles, let's talk about other types of tasks, but first need to learn this lesson to move further. Let us discuss what to do if there are roots, degrees, we will study infinitely small equivalent functions, wonderful limits, Lopital rule.

If you do not work yourself to solve the limits, then do not panic. We are always happy to help!

The main elementary functions figured out.

When moving to the functions of a more complex species, we will be faced with the appearance of expressions whose value is not defined. Such expressions are called uncertainty.

List everyone main types of uncertainty: Zero to divide on zero (0 to 0), infinity to divide into infinity, the zero multiply into infinity, infinity minus infinity, the unit to the degree of infinity, zero to the degree of zero, infinity to the degree of zero.

All other expressions are not uncertainties and take a completely concrete finite or infinite meaning.

Disclose uncertainty Allows:

• simplification of the type of function (transformation of expression using formulas of abbreviated multiplication, trigonometric formulas, a multiplication of conjugate expressions with a subsequent reduction, etc.);
• use of wonderful limits;
• application of the Lopital rule;
• using the replacement of infinitely small expression to it is equivalent (using the table is equivalent infinitely small).

Grouped uncertainty B. table of uncertainty. Each type of uncertainty, we will put it in line with the method of its disclosure (the method of finding the limit).

This table, together with the table of limits of basic elementary functions, will be your main tools while finding any limits.

We give a couple of examples when everything is immediately obtained after the substitution of the value and uncertainty do not occur.

Example.

Calculate the limit

Decision.

We substitute the value:

And immediately got the answer.

Answer:

Example.

Calculate the limit

Decision.

We substitute the value x \u003d 0 in the foundation of our significant power function:

That is, the limit can be rewritten in the form

Now we will deal with the indicator. This is a power function. Turn to the table of limits for power functions with a negative indicator. From there have and therefore can be recorded .

Based on this, our limit will be recorded in the form:

We again appeal to the table of limits, but already for indicative functions with the basis of a large unit, from where we have:

Answer:

We will analyze on examples with detailed solutions disclosure of uncertainties by transformation of expressions.

Very often, the expression under the sign of the limit needs to be converted a little to get rid of uncertainties.

Example.

Calculate the limit

Decision.

We substitute the value:

Came to uncertainty. We look at the uncertainty table to select a solution method. We try to simplify the expression.

Answer:

Example.

Calculate the limit

Decision.

We substitute the value:

Came to uncertainty (0 to 0). We look at the uncertainty table to select a solution method and try to simplify the expression. Doming and numerator and denominator for an expression, conjugate to the denominator.

For the denominator, the conjugate expression will be

The denominator we are dominant in order to apply the formula of abbreviated multiplication - the difference of squares and then reduce the resulting expression.

After a series of transformations, the uncertainty disappeared.

Answer:

COMMENT: For the limits of a similar type, the method of multiplication on the conjugate expressions is typical, so we safely use.

Example.

Calculate the limit

Decision.

We substitute the value:

Came to uncertainty. We look at the uncertainty table to select a solution method and try to simplify the expression. Since the numerator and denominator turns to zero at x \u003d 1, then if these expressions can be reduced (x-1) and the uncertainty will disappear.

Spatulate the numerator for multipliers:

Spread the denominator for multipliers:

Our limit will take the form:

After converting, the uncertainty revealed.

Answer:

Consider the limits on infinity from power expressions. If the indicators of the powerful expression are positive, the limit on infinity is infinite. Moreover, the main value is the greatest degree, the rest can be discarded.

Example.

Example.

If the expression under the sign is aimed with a fraction, and the numerator and the denominator there are powerful expressions (M - the degree of the numerator, and n is the degree of denominator), then the uncertainty of the type of infinity occurs in infinity, in this case uncertainty is disclosed division and numerator and denominator on

Example.

Calculate the limit

When calculating the limits should be considered the following basic rules:

1. The limit of the amount (difference) of functions is equal to the sum (difference) limits of the terms of the terms:

2. The limit of the function of functions is equal to the work of the factors:

3. The limit of two functions is equal to the attitude of these functions:

.

4. A permanent multiplier can be taken out of the limit:

.

5. The limit is equal to the most permanent:

6. For continuous functions, the limits and function symbols can be changed in places:

.

Finding the limit of the function should be started with the substitution of the value in the expression for the function. In this case, if the numeric value is 0 or ¥, then the desired limit is found.

Example 2.1.Calculate the limit.

Decision.

.

Expressions of the species ,,,, uncertainty.

If the uncertainty of the view is obtained, then to find the limit you need to convert a function so as to disclose this uncertainty.

The uncertainty of the species is usually obtained when the limit of the relation of two polynomials is specified. In this case, it is recommended to decompose the polynomials on multipliers and reduce the total multiplier. This multiplier is zero with the limit value. h. .

Example 2.2.Calculate the limit.

Decision.

Substituting, we get uncertainty:

.

Spread the numerator and denominator for multipliers:

;

Slim on the general multiplier and get

.

The uncertainty of the species is obtained when the limit of the ratio of two polynomials is set. In this case, it is recommended to divide both polynomials to h. in high degree.

Example 2.3. Calculate the limit.

Decision.When substituting ∞, uncertainty of the species is obtained, so we divide all the members of the expression on x 3..

.

It takes into account that.

When calculating the limits of the function containing the roots, it is recommended to multiply and split the function to the conjugate expression.

Example 2.4.Calculate the limit

Decision.

When calculating the limits for the disclosure of the uncertainty of the form or (1) ∞, the first and second wonderful limits are often used:

To the second remarkable limit, many tasks related to the continuous growth of any value are given.

Consider the example of Ya. I. Perelman, who gives the interpretation of the number e. In the task of complex percentages. In Sberbanks, interest money joins the principal capital annually. If the accession is performed more often, the capital grows faster, as a large amount is involved in the education of interest. Take a purely theoretical, very simplified example.

Let 100 den go to the bank. units. At the rate of 100% per annum. If interest money will be attached to the main capital only after the expiration of the year, then to this term 100 den. units. turns into 200 den.

Let's see now, what will turn 100 den. un., if interest money to attach every six months to the main capital. After half of the year 100 den. units. It will grow in 100 × 1.5 \u003d 150, and after half a year - in 150 × 1,5 \u003d 225 (den. units). If the join to do every 1/3 year, then after the year of 100 den. units. turns into 100 × (1 +1/3) 3 "237 (den. units).

We will participate the time to attach interest money to 0.1 years, to 0.01 years, up to 0.001 years, etc. Then out of 100 den. units. A year later, it will turn out:

100 × (1 +1/10) 10 "259 (den. Units),

100 × (1 + 1/100) 100 "270 (den. Units),

100 × (1 + 1/1000) 1000 "271 (den. Units).

In case of limitless reduction in the time of interest, the increasing capital is not increasingly increasing, but approximately 271 is approximately 2.71 times. Capital, laid under 100% per annum, cannot increase second because

Example 2.5.Calculate the limit of the function

Decision.

Example 2.6.Calculate the limit of the function .

Decision.Substituting we get uncertainty:

.

Using a trigonometric formula, we transform the numerator into the work:

As a result, we get

The second wonderful limit is taken into account here.

Example 2.7.Calculate the limit of the function

Decision.

.

To disclose the uncertainty of the form or you can use the Lopital rule, which is based on the following theorem.

Theorem.The limit of the ratio of two infinitely small or infinitely large functions is equal to the limit of the relations of their derivatives

Note that this rule can be applied several times in a row.

Example 2.8. To find

Decision.When substitution, we have uncertainty of the species. Applying the Lopital rule, we get

Continuity function

An important feature of the function is continuity.

Definition.The function is considered continuousIf a small change in the values \u200b\u200bof the argument entails a small change in the function value.

Mathematically, this is written like this: when

Under and understood the increment of variables, that is, the difference between the subsequent and previous values:, (Figure 2.3)

Figure 2.3 - Protecting variables

From the definition of the function, continuous at the point, it follows that . This equality means performing three conditions:

Decision.For function The point is suspicious on the gap, check it, we find one-way limits

Hence, So - disposable break point

Derived function