Limit theory is one of the branches of mathematical analysis. The problem of solving the limits is quite extensive, since there are dozens of methods for solving the limits different types... There are dozens of nuances and tricks to solve this or that limit. Nevertheless, we will still try to understand the basic 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 the Frenchman Augustin Louis Cauchy, who laid the foundations of mathematical analysis and gave rigorous definitions, the definition of the limit, in particular. I must say that this same Cauchy dreamed, dreams and will dream in nightmares to all students of physics and mathematics faculties, since he proved a huge number of theorems of mathematical analysis, and one 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 deal with the basic types of limits.

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

So what is the limit?

And right away an example, why shaggy grandmother ....

Any limit has 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 exercises, absolutely any number can be in place of the unit, as well as infinity ().
3) Functions under the limit sign, in this case.

The recording itself reads like this: "the limit of the function when x tends to unity."

Let's analyze the following important question- and what does the expression "x seeks to one "? And what is “striving” anyway?
The concept of a limit is a concept, if I may say so, dynamic... Let's build a sequence: first, then,, ..., , ….
That is, the expression "x seeks to one "should be understood as follows -" x "consistently takes on values, which are infinitely close to unity and practically coincide with it.

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

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

We have considered the simplest limit, but even such ones are found in practice, and, moreover, not so rarely!

Example with infinity:

Let's figure out what is it? This is the case when it increases indefinitely, that is: first, then, then, then and so on to infinity.

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 start increasing to infinity, and we look at the behavior of the function:

Conclusion: when the function increases indefinitely:

And another series of examples:

Please try to mentally analyze the following 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 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 following thing. Even if a limit is given with a large number at the top, but even with a million: then it doesn't matter , because sooner or later "X" will take on such gigantic values ​​that a million in comparison with them will be a real microbe.

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

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

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

Now we will consider a group of limits when, and the function is a fraction, in the numerator and denominator of which there are 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. So we have the so-called uncertainty of the species. One would think that, and the answer is ready, but in the general case this is not at all the case, and you need to apply some solution technique, which we will now consider.

How to solve limits of a given type?

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

The highest degree in the numerator is two.

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

The highest power of the denominator is two.

Then we choose the highest power of the numerator and denominator: in this example they coincide and are 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.

That's how it is, the answer, not infinity.

What is fundamentally important in the design of the solution?

First, we indicate the uncertainty, if any.

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

Thirdly, in the limit it is desirable to mark what is striving and where. When the work is completed by hand, it is more convenient to do it like this:

It is best to use a simple pencil to mark up.

Of course, you can do nothing of this, but then, perhaps, the teacher will note the shortcomings in the solution or start asking additional questions on the assignment. Do you need it?

Example 2

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

Maximum degree in numerator: 3
Maximum degree in denominator: 4
We choose the greatest value, in this case a four.
According to our algorithm, to disclose uncertainty, we divide the numerator and denominator by.
The complete design of the assignment may look like this:

Divide the numerator and denominator by

Example 3

Find the limit
The maximum degree of "x" in the numerator: 2
The maximum degree of "x" in the denominator: 1 (can be written as)
To disclose uncertainty, divide the numerator and denominator by. A clean solution might look like this:

Divide the numerator and denominator by

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

Thus, when disclosing the uncertainty of the species, we can get finite number, zero or infinity.

Limits with Uncertainty of a Type and a Method for Their Solution

The next group of limits is somewhat similar to the limits just considered: there are polynomials in the numerator and denominator, but the "x" no longer tends to infinity, but to finite number.

Example 4

Solve the limit
First, let's 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 are uncertainties of the form, then for its disclosure you need to factor out the numerator and denominator.

To do this, most often you need to decide quadratic equation and / or use abbreviated multiplication formulas. If these things are forgotten, then visit the page Mathematical formulas and tables and check out methodological material Hot Formulas School Mathematics Course... By the way, it is best to print it, it is required very often, and information from paper is assimilated better.

So, we decide our limit

Let us factor out the numerator and denominator

In order to factor out 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 is on the simplest calculator.

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

Next, we find the roots:

Thus:

Everything. The numerator has been expanded.

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

Obviously it can be abbreviated to:

Now we substitute -1 in the expression, which remains under the limit sign:

Naturally, in test work, on the test, on the exam, the solution is never described in such detail. In the final version, the design should look something like this:

Factor out the numerator.

Example 5

Calculate Limit

First, a "clean" solution

Let us factor out the numerator and denominator.

Numerator:
Denominator:



,

What is important in this example?
Firstly, you should understand well how the numerator is disclosed, first we took out 2 outside the bracket, and then we used the formula for the difference of squares. This is the formula you need to know and see.

Methods for solving the limits. Uncertainties.
The growth order 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 ghost of "minus infinity" has been in this article for a long time. Consider the 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.

Consider 4 chips that are required to solve practical assignments:

1) Calculate the limit

The value of the limit depends only on the term, since it has the highest growth order. If, then infinitely large modulo a negative number EVEN degree, in this case - in the fourth, equal to "plus infinity":. Constant ("two") positive, therefore:

2) Calculate the limit

Here again the senior degree even, therefore: . But in front of the "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, therefore infinitely large modulo negative number to 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, moreover, 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. The numerator and denominator are of the same order of growth, which means that in the limit you get a finite number. Let's find out the answer, discarding all the fry:

The solution is trivial:

Example 6

Find the limit

This is an example for a do-it-yourself solution. Complete solution and answer at the end of the tutorial.

And now, perhaps the most subtle of the cases:

Example 7

Find the limit

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

We decide:

Divide the numerator and denominator by

Example 15

Find the limit

This is an example for a do-it-yourself solution. A rough example of finishing at the end of the lesson.

A couple more interesting examples on variable replacement:

Example 16

Find the limit

When substituting a unit in the limit, uncertainty is obtained. Variable replacement is already obvious, but first we transform the tangent using the formula. Indeed, why do we need a tangent?

Note that, therefore. If 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 0: 0 uncertainty. It would be nice if our limit tends to zero.

Let's replace:

If, then

Under the cosine we have "x", which must also be expressed through "te".
From the replacement we express:.

We complete the solution:

(1) We perform the substitution

(2) Expand the brackets under the cosine.

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

Assignment for independent solution:

Example 17

Find the limit

Complete solution and answer at the end of the tutorial.

These were not difficult tasks in their class, in practice everything can be worse, and, in addition to reduction formulas, you have to use a variety of trigonometric formulas, as well as other tricks. In the article Difficult Limits, I took apart a couple of real examples =)

On the eve of the holiday, let us finally clarify the situation with one more widespread uncertainty:

Elimination of uncertainty "one to the degree of infinity"

This uncertainty is "served" second wonderful limit, and in the second part of that lesson, we examined in great detail the standard examples of solutions that in most cases are encountered in practice. Now the picture with the exhibitors will be completed, in addition, the final tasks of the lesson will be devoted to the limits - "trompe l'oeil", 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 of 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 by the formula:

Somewhere like already explained what square brackets mean. Nothing special, brackets are like brackets. They are usually used to make a mathematical notation clearer.

Let's highlight the essential points of the formula:

1) It is only about uncertainty and no other.

2) The "x" argument may tend to arbitrary value(and not only to zero or), in particular, to "minus infinity" or to any a finite number.

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

In this case , and by the formula :

True, I do not advise you to do this, in the tradition it is still to use the "usual" design of the solution, if it can be applied. but using a formula is very convenient to check"Classic" examples at the 2nd remarkable limit.

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

So, the concept of the limit is quite important in studying the course. higher mathematics, especially when you come across integral calculus and understand the relationship between the limit and the integral. The current article 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) $$ \ lim_ (x \ to \ infty) \ frac (1) (x) = 0 $$ We are often sent these limits with a request to help solve. We decided to highlight them as a separate example and explain that these limits must be simply remembered, as a rule. If you can't solve your problem, then send it to us. We will provide a detailed solution. You will be able to familiarize yourself with the course of the calculation and get information. This will help you get credit 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 like: $ \ 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 of $ x $ in 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? What should be the result? Since this is uncertainty, this is not an answer yet and we continue the calculation. Since we have a polynomial in the numerators, we factor it into factors, using the formula familiar to everyone since school $$ a ^ 2-b ^ 2 = (a-b) (a + b) $$. Do you remember? Fine! Now go ahead and apply it with the song :) We get that the numerator $ x ^ 2-1 = (x-1) (x + 1) $ We continue to solve given 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 us push the limit in the last two examples to infinity and consider the uncertainty: $ \ bigg [\ frac (\ infty) (\ infty) \ bigg] $

Example 5

Evaluate $ \ 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? How to be? Don't panic, because the impossible is possible. It is necessary to put the x outside the brackets in both the numerator and the denominator, and then reduce it. Then try to calculate the limit. Trying ... $$ \ lim \ limits_ (x \ to \ infty) \ frac (x ^ 2-1) (x + 1) = \ lim \ limits_ (x \ to \ infty) \ frac (x ^ 2 (1- \ frac (1) (x ^ 2))) (x (1+ \ frac (1) (x))) = $$ $$ = \ lim \ limits_ (x \ to \ infty) \ frac (x (1- \ frac (1) (x ^ 2))) ((1+ \ frac (1) (x))) = $$ 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 analyzed examples and compose an algorithm for solving the limits:

1. Substitute point x in the expression following the limit sign. If you get a certain number, or infinity, then the limit is completely solved. Otherwise, we have ambiguity: "divide zero by zero" or "divide infinity by infinity" and proceed to the next paragraphs of the instruction.
2. To eliminate the ambiguity "zero divided by zero" you need to factor the numerator and denominator into factors. Reduce similar ones. Substitute point x in the expression under the limit sign.
3. If the uncertainty is "infinity divided by infinity", then we take out both in the numerator and in the denominator of x to the greatest degree. Reducing the x's. Substitute the x values ​​from under the limit into the remaining expression.

In this article, you learned the basics of solving limits commonly used in the Math Analysis course. Of course, these are not all types of problems offered by examiners, but only the simplest limits. In the following articles we will talk about other types of tasks, but first you need to learn this lesson in order to move on. We will discuss what to do if there are roots, degrees, we will study infinitesimal equivalent functions, wonderful limits, L'Hôpital's rule.

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

Of the main elementary functions sorted it out.

When moving to functions of a more complex kind, we will certainly encounter the appearance of expressions, the meaning of which is not defined. Such expressions are called uncertainties.

We list all main types of uncertainties: zero divided by zero (0 by 0), infinity divided by infinity, zero multiplied by infinity, infinity minus infinity, one to the power of infinity, zero to the power of zero, infinity to the power of zero.

ALL OTHER UNCERTAINTY STATEMENTS ARE NOT AND TAKE IN FULL SPECIFIC FINAL OR INFINITE VALUES.

Uncover uncertainties allows:

• simplification of the type of function (transformation of an expression using abbreviated multiplication formulas, trigonometric formulas, multiplication by conjugate expressions followed by abbreviation, etc.);
• using wonderful limits;
• application of the L'Hôpital rule;
• using the replacement of an infinitesimal expression with its equivalent (using a table of equivalent infinitesimal).

Let's group the uncertainties into uncertainty table... Each type of uncertainty is associated with the method of its disclosure (the method of finding the limit).

This table, together with the basic elementary function limits table, will be your main tools in finding any limits.

Let's give a couple of examples when everything is immediately obtained after substitution of the value and uncertainties do not arise.

Example.

Calculate Limit

Solution.

Substitute the value:

And immediately received an answer.

Answer:

Example.

Calculate Limit

Solution.

Substitute the value x = 0 at the base of our exponential function:

That is, the limit can be rewritten as

Now let's look at the indicator. This is a power function. Refer to the table of limits for power functions with negative indicator... From there we have and , therefore, we can write .

Based on this, our limit will be written as:

Again we turn to the table of limits, but already for exponential functions with a base greater than one, whence we have:

Answer:

Let's look at examples with detailed solutions disclosure of ambiguities by transformation of expressions.

Very often, the expression under the limit sign needs to be slightly transformed to get rid of ambiguities.

Example.

Calculate Limit

Solution.

Substitute the value:

Came to uncertainty. We look at the table of uncertainties to choose a solution method. Trying to simplify the expression.

Answer:

Example.

Calculate Limit

Solution.

Substitute the value:

We came to uncertainty (0 to 0). We look at the table of uncertainties to choose a solution method and try to simplify the expression. Let's multiply both the numerator and the denominator by the expression conjugated to the denominator.

For the denominator, the conjugate expression is

We multiplied the denominator in order to be able to apply the formula for abbreviated multiplication - the difference of squares and then reduce the resulting expression.

After a series of transformations, the uncertainty disappeared.

Answer:

COMMENT: for limits of this kind, the method of multiplying by conjugate expressions is typical, so feel free to use.

Example.

Calculate Limit

Solution.

Substitute the value:

Came to uncertainty. We look at the table of uncertainties to choose a solution method and try to simplify the expression. Since both the numerator and denominator vanish at x = 1, then if these expressions, it will be possible to reduce (x-1) and the uncertainty will disappear.

Let's factorize the numerator:

Let's factorize the denominator:

Our limit will take the form:

After the transformation, the uncertainty was revealed.

Answer:

Consider the limits at infinity from power expressions. If the exponents of the exponential expression are positive, then the limit at infinity is infinite. Moreover, the main importance is the greatest degree, the rest can be discarded.

Example.

Example.

If the expression under the limit sign is a fraction, and both the numerator and the denominator are exponential expressions (m is the degree of the numerator, and n is the degree of the denominator), then when there is an uncertainty of the form infinity to infinity, in this case uncertainty is revealed division and numerator and denominator by

Example.

Calculate Limit

When calculating limits, consider following basic rules:

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

2. Limit of the product of functions is equal to the product limits of factors:

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

.

4. The constant factor can be taken out of the limit sign:

.

5. The limit of the constant is equal to the most constant:

6. For continuous functions limit and function symbols can be swapped:

.

Finding the limit of a function should begin by substituting a value in the expression for the function. Moreover, if a numerical value of 0 or ¥ is obtained, then the sought limit is found.

Example 2.1. Calculate the limit.

Solution.

.

Expressions of the form,,,,, are called uncertainties.

If an uncertainty of the form is obtained, then in order to find the limit, it is necessary to transform the function so as to reveal this uncertainty.

Uncertainty of kind is usually obtained when the limit of the ratio of two polynomials is given. In this case, it is recommended to factor the polynomials and cancel them by a common factor to calculate the limit. This multiplier is zero at the limit value NS .

Example 2.2. Calculate the limit.

Solution.

Substituting, we get the uncertainty:

.

Let's factorize the numerator and denominator:

;

Reduce by a common factor and get

.

The uncertainty of the form is obtained when the limit of the ratio of two polynomials is given at. In this case, for the calculation, it is recommended to divide both polynomials by NS in the senior degree.

Example 2.3. Calculate the limit.

Solution. Substitution ∞ gives an uncertainty of the form, so we divide all terms of the expression by x 3.

.

It is taken into account that.

When calculating the limits of a function containing roots, it is recommended to multiply and divide the function by its conjugate expression.

Example 2.4. Calculate Limit

Solution.

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

Many problems associated with the continuous growth of any quantity lead to the second remarkable limit.

Consider Ya.I. Perelman's example, which gives an interpretation of the number e in the problem of compound interest. In savings banks, interest money is added to the fixed capital annually. If the connection is made more often, then the capital grows faster, since it participates in the formation of interest large sum... Let's take a purely theoretical, highly simplified example.

Let the bank put 100 den. units at the rate of 100% per annum. If interest money will be added to the fixed capital only after a year, then by this date 100 den. units will turn into 200 monetary units.

Now let's see what will turn into 100 den. units, if interest money is added to the fixed capital every six months. After half a year, 100 den. units will grow to 100 × 1.5 = 150, and after another six months - to 150 × 1.5 = 225 (monetary units). If the connection is done every 1/3 of the year, then after a year, 100 den. units will turn into 100 × (1 +1/3) 3 "237 (monetary units).

We will speed up the terms for joining interest-bearing money to 0.1 years, to 0.01 years, to 0.001 years, etc. Then out of 100 den. units after a year it will turn out:

100 × (1 +1/10) 10 "259 (monetary units),

100 × (1 + 1/100) 100 * 270 (monetary units),

100 × (1 + 1/1000) 1000 * 271 (monetary units).

With an unlimited reduction in the terms of interest attachment, the accrued capital does not grow infinitely, but approaches a certain limit, equal to approximately 271. The capital allocated at 100% per annum cannot increase by more than 2.71 times, even if the accrued interest was added to the capital each second because

Example 2.5. Calculate the limit of a function

Solution.

Example 2.6. Calculate the limit of a function .

Solution. Substituting we get the uncertainty:

.

Using the trigonometric formula, convert the numerator to product:

As a result, we get

Here, a second remarkable limit is taken into account.

Example 2.7. Calculate the limit of a function

Solution.

.

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

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

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

Example 2.8. Find

Solution. When substituting, we have an uncertainty of the form. Applying L'Hôpital's rule, we get

Continuity of function

Continuity is an important property of a function.

Definition. The function is considered continuous if a small change in the value of the argument entails a small change in the value of the function.

Mathematically, it is written as follows: for

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

Figure 2.3 - Increment of variables

It follows from the definition of a function continuous at a point that ... This equality means the fulfillment of three conditions:

Solution. For function the point is suspicious of a break, check it, find one-sided limits

Hence, , means - removable discontinuity point

Derivative of a function