Solving limits how to solve. Wonderful Limits

We continue to analyze ready-made answers to the theory of limits and today we will focus only on the case when a variable in a function or a number in a sequence tends to infinity. Instructions for calculating the limit for a variable tending to infinity were given earlier; here we will only dwell on individual cases that are not obvious and simple to everyone.

Example 35. We have a sequence in the form of a fraction, where the numerator and denominator contain root functions.
We need to find the limit when the number tends to infinity.
Here there is no need to reveal the irrationality in the numerator, but only carefully analyze the roots and find where a higher power of the number is contained.
In the first, the roots of the numerator are multiplier n^4, that is, n^2 can be taken out of brackets.
Let's do the same with the denominator.
Next, we evaluate the meaning of radical expressions when passing to the limit.

We got divisions by zero, which is incorrect in the school course, but in the passage to the limit it is acceptable.
Only with an amendment “to estimate where the function is heading.”
Therefore, not all teachers can interpret the above notation as correct, although they understand that the resulting result will not change.
Let's look at the answer compiled according to the requirements of teachers according to the theory.
To simplify, we will evaluate only the main add-ons under the root

Further, in the numerator the power is equal to 2, in the denominator 2/3, therefore the numerator grows faster, which means the limit tends to infinity.
Its sign depends on the factors of n^2, n^(2/3) , so it is positive.

Example 36. Consider an example of a division limit exponential functions. There are few practical examples of this kind, so not all students easily see how to disclose the uncertainties that arise.
The maximum factor for the numerator and denominator is 8^n, and we simplify by it

Next, we evaluate the contribution of each term
The terms 3/8 tend to zero as the variable goes to infinity, since 3/8<1 (свойство степенно-показательной функции).

Example 37. The limit of a sequence with factorials is revealed by writing down the factorial to the greatest common factor for the numerator and denominator.
Next, we reduce it and evaluate the limit based on the value of the number indicators in the numerator and denominator.
In our example, the denominator grows faster, so the limit is zero.


The following is used here

factorial property.

Example 38. Without applying L'Hopital's rules, we compare the maximum indicators of the variable in the numerator and denominator of the fraction.
Since the denominator contains the highest exponent of the variable 4>2, it grows faster.
From this we conclude that the limit of the function tends to zero.

Example 39. We reveal the peculiarity of the form infinity divided by infinity by removing x^4 from the numerator and denominator of the fraction.
As a result of passing to the limit, we obtain infinity.

Example 40. We have a division of polynomials; we need to determine the limit as the variable tends to infinity.
The highest degree of the variable in the numerator and denominator is equal to 3, which means that the boundary exists and is equal to the current one.
Let's take out x^3 and perform the passage to the limit

Example 41. We have a singularity of type one to the power of infinity.
This means that the expression in brackets and the indicator itself must be brought under the second important boundary.
Let's write down the numerator to highlight the expression in it that is identical to the denominator.
Next, we move on to an expression containing one plus a term.
The degree must be distinguished by the factor 1/(term).
Thus we obtain the exponent to the power of the limit of the fractional function.

To evaluate the singularity, we used the second limit:

Example 42. We have a singularity of type one to the power of infinity.
To reveal it, one should reduce the function to the second remarkable limit.
How to do this is shown in detail in the following formula


You can find a lot of similar problems. Their essence is to obtain the required degree in the indicator, and it is equal to reciprocal value the term in brackets at unity.
Using this method we obtain the exponent. Further calculation is reduced to calculating the limit of the exponent degree.
Here exponential function tends to infinity, since the value is greater than one e=2.72>1.

Example 43 In the denominator of the fraction we have an uncertainty of the type infinity minus infinity, which is actually equal to division by zero.
To get rid of the root, we multiply by the conjugate expression, and then use the formula for the difference of squares to rewrite the denominator.
We get the uncertainty of infinity divided by infinity, so we take out the variable to the greatest extent and reduce it by it.
Next, we evaluate the contribution of each term and find the limit of the function at infinity

The theory of limits is one of the sections 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 short one historical background. There lived a Frenchman, Augustin Louis Cauchy, in the 19th century, who gave strict definitions to many of the concepts of matan and laid its foundations. It must be said that this respected mathematician 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 one theorem is more lethal than the other. In this regard, we will not consider yet determination of the Cauchy limit, but let's 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 this case . The entry reads “X tends to one.” Most often - exactly, although instead of “X” in practice there are other variables. In practical tasks, the place of one can be absolutely any number, as well as infinity ().
3) Functions under the limit sign, in this case .

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

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

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

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

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

Example with infinity:

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

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

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

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

Another example with infinity:

Again we start increasing 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 begin to take on such gigantic values ​​that a million in comparison 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.

Moreover, the limit has a very good geometric meaning. For a better understanding of the topic, I recommend that you read methodological material Graphs and properties of elementary functions. After reading this article, you will not only finally understand what a limit is, but also get acquainted with interesting cases, when the limit of the function is generally doesn't exist!

In practice, unfortunately, there are few gifts. And therefore we move on to consider more complex limits. By the way, on this topic there is intensive course in pdf format, which is especially useful if you have VERY little time to prepare. But the site materials, of course, are no worse:

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.

We then choose the highest power of the numerator and denominator: in 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.

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

What is fundamentally important in the design of a decision?

First, we indicate uncertainty, if any.

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

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

It is better to use a simple pencil for notes.

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

Example 2

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

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

Divide the numerator and denominator by

Example 3

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

Divide the numerator and denominator by

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

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

Limits with uncertainty of type and method for solving them

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

Example 4

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

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

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

To do this, most often you need to solve a quadratic equation and/or use abbreviated multiplication formulas. If these things have been forgotten, then visit the page Mathematical formulas and tables and read the teaching material Hot formulas 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 in its entirety (a fractional number with a comma is obtained), it is very likely that the discriminant was calculated incorrectly or there was a typo in the task.

Next we find the roots:

Thus:

All. The numerator is factorized.

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

Obviously, it can be shortened to:

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

Naturally, in 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.

Recommendation: If in a limit (of almost any type) it is possible to take a number out of brackets, then we always do it.
Moreover, it is advisable to move such numbers beyond the limit icon. For what? Yes, just so that they don’t get in the way. The main thing is not to lose these numbers later during the solution.

Please note that at the final stage of the solution, I took the two out of the limit icon, and then the minus.

! Important
During the solution, the type fragment occurs very often. Reduce this fractionit is forbidden . First you need to change the sign of the numerator or denominator (put -1 out of brackets).
, that is, a minus sign appears, which is taken into account when calculating the limit and there is no need to lose it at all.

In general, I noticed that most often in finding limits of this type we have to solve two quadratic equations, that is, both the numerator and the denominator contain square trinomials.

Method of multiplying the numerator and denominator by the conjugate expression

We continue to consider the uncertainty of the form

The next type of limits is similar to the previous type. The only thing, in addition to polynomials, we will add roots.

Example 6

Find the limit

Let's start deciding.

First we try to substitute 3 into the expression under the limit sign
I repeat once again - this is the first thing you need to do for ANY limit. This action is usually carried out mentally or in draft form.

An uncertainty of the form has been obtained that needs to be eliminated.

As you probably noticed, our numerator contains the difference of the roots. And in mathematics it is customary to get rid of roots, if possible. For what? And life is easier without them.

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

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

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

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

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

For those who want to learn how to find limits, in this article we will tell you about it. 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 the study of higher mathematics, especially when you come across integral calculus and understand the connection 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 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 proceed to the following points 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 Calculus course. 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!

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