Solution online function limits. Find the limiting value of a function or functional sequence at a point, calculate ultimate the value of the function at infinity. determine the convergence of a number series and much more can be done thanks to our online service- . We allow you to find function limits online quickly and accurately. You enter it yourself function variable and the limit to which it strives, our service carries out all the calculations for you, giving an accurate and simple answer. And for finding the limit online you can enter both numerical series and analytical functions containing constants in literal expression. In this case, the found limit of the function will contain these constants as constant arguments in the expression. Our service solves any complex problems of finding limits online, it is enough to indicate the function and the point at which it is necessary to calculate limit value of function. Calculating online limits, you can use various methods and rules for solving them, while checking the result obtained with solving limits online on the www.site, which will lead to the successful completion of the task - you will avoid own mistakes and typos. Or you can completely trust us and use our result in your work, without spending extra effort and time on independently calculating the limit of the function. We allow the entry of such limit values like infinity. It is necessary to enter a common member of a number sequence and www.site will calculate the value limit online to plus or minus infinity.
One of the basic concepts of mathematical analysis is function limit And sequence limit at a point and at infinity, it is important to be able to solve correctly limits. With our service this will not be difficult. A decision is made limits online within a few seconds, the answer is accurate and complete. The study of mathematical analysis begins with transition to the limit, limits are used in almost all areas of higher mathematics, so it is useful to have a server at hand for online limit solutions, which is the site.
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 short one historical reference. 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 this case . The entry reads “X tends to one.” Most often - exactly, although instead of “X” in practice there are other variables. In practical tasks, the place of one can be absolutely any number, as well as infinity ().
3) Functions under the limit sign, in this case .
The recording itself 
reads like this: “the limit of a function as x tends to unity.”
Let's look at the next important question - what does the expression “x” mean? strives to one"? And what does “strive” even mean?
The concept of a limit is a concept, so to speak, dynamic. Let's build a sequence: first , then , , …, 
, ….
That is, the expression “x strives to one” should be understood as follows: “x” consistently takes on the values which approach unity infinitely close and practically coincide with it.
How to solve the above example? Based on the above, you just need to substitute one into the function under the limit sign:
So, the first rule: When given any limit, first we simply try to plug the number into the function.
We have considered the simplest limit, but these also occur in practice, and not so rarely!
Example with infinity:
Let's figure out what it is? This is the case when it increases without limit, that is: first, then, then, then, and so on ad infinitum.
What happens to the function at this time?
, , 
, …
So: if , then the function tends to minus infinity:
Roughly speaking, according to our first rule, instead of “X” we substitute infinity into the function and get the answer.
Another example with infinity:
Again we begin to increase to infinity, and look at the behavior of the function: 
Conclusion: when the function increases without limit:
And another series of examples:
Please try to mentally analyze the following for yourself and remember the simplest types of limits:
, , , , 
, , , , 
,
If you have doubts anywhere, you can pick up a calculator and practice a little.
In the event that , try to construct the sequence , , . If , then , , .
Note: strictly speaking, this approach to constructing sequences of several numbers is incorrect, but for understanding the simplest examples it is quite suitable.
Also pay attention to the following thing. Even if 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 would think that , and the answer is ready, but general case This is not the case at all, and you need to apply some solution, 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 check out methodological material Hot formulas for school mathematics course. By the way, it is best to print it out; it is required very often, and information is absorbed better from paper.
So, let's solve our limit 
Factor the numerator and denominator
In order to factor the numerator, you need to solve the quadratic equation: 
First we find the discriminant:
And the square root of it: .
If the discriminant is large, for example 361, we use a calculator; the function of extracting the square root is on the simplest calculator.
! If the root is not extracted 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 a test, test, or exam, the solution is never described 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.
The theory of limits is one of the branches of mathematical analysis. The question of solving limits is quite extensive, since there are dozens of methods for solving limits of various types. There are dozens of nuances and tricks that allow you to solve this or that limit. Nevertheless, we will still try to understand the main types of limits that are most often encountered in practice.
Let's start with the very concept of a limit. But first, a brief historical background. There lived 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 a limit is given with a large number at the top, or even with a million: , then it’s all the same , since sooner or later “X” will 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 the teaching 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 a function in general does not 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.
Then we choose the highest power of the numerator and denominator: in this example, they are the same and equal to two.
So, the solution method is as follows: in order to reveal the uncertainty, it is necessary to divide the numerator and denominator by the highest power.
Here it is, the answer, and not infinity at all.
What is fundamentally important in the design of a decision?
First, we indicate uncertainty, if any.
Secondly, it is advisable to interrupt the solution for intermediate explanations. I usually use the sign, it does not have any mathematical meaning, but means that the solution is interrupted for an intermediate explanation.
Thirdly, in the limit it is advisable to mark what is going where. When the work is drawn up by hand, it is more convenient to do it this way: 
It is better to use a simple pencil for notes.
Of course, you don’t have to do any of this, but then, perhaps, the teacher will point out shortcomings in the solution or start asking additional questions about the assignment. Do you need it?
Example 2
Find the limit
Again in the numerator and denominator we find in the highest degree: 
Maximum degree in numerator: 3
Maximum degree in denominator: 4
Choose greatest value, in this case four.
According to our algorithm, to reveal uncertainty, we divide the numerator and denominator by .
The complete assignment might look like this:
Divide the numerator and denominator by
Example 3
Find the limit
Maximum degree of “X” in the numerator: 2
Maximum degree of “X” in the denominator: 1 (can be written as)
To reveal the uncertainty, it is necessary to divide the numerator and denominator by . The final solution might look like this:
Divide the numerator and denominator by
Notation does not mean division by zero (you cannot divide by zero), but division by an infinitesimal number.
Thus, by uncovering species uncertainty, we may be able to final number, zero or infinity.
Limits with uncertainty of type and method for solving them
The next group of limits is somewhat similar to the limits just considered: the numerator and denominator contain polynomials, but “x” no longer tends to infinity, but to finite number.
Example 4
Solve limit 
First, let's try to substitute -1 into the fraction: 
In this case, the so-called uncertainty is obtained.
General rule: if the numerator and denominator contain polynomials, and there is uncertainty of the form , then to disclose it you need to factor the numerator and denominator.
To do this, most often you need to solve a quadratic equation and/or use abbreviated multiplication formulas. If these things have been forgotten, then visit the page Mathematical formulas and tables and read the teaching material Hot formulas for school mathematics course. By the way, it is best to print it out; it is required very often, and information is absorbed better from paper.
So, let's solve our limit 
Factor the numerator and denominator
In order to factor the numerator, you need to solve the quadratic equation: 
First we find the discriminant:
And the square root of it: .
If the discriminant is large, for example 361, we use a calculator; the function of extracting the square root is on the simplest calculator.
! If the root is not extracted in its entirety (a fractional number with a comma is obtained), it is very likely that the discriminant was calculated incorrectly or there was a typo in the task.
Next we find the roots: 
Thus:
All. The numerator is factorized.
Denominator. The denominator is already the simplest factor, and there is no way to simplify it.
Obviously, it can be shortened to:
Now we substitute -1 into the expression that remains under the limit sign:
Naturally, in a test, test, or exam, the solution is never described 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 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.
This online math calculator will help you if you need it calculate the limit of a function. Program solution limits not only gives the answer to the problem, it leads detailed solution with explanations, i.e. displays the limit calculation process.
This program may be useful for high school students secondary schools in preparation for tests and exams, when testing knowledge before the Unified State Exam, for parents to control the solution of many problems in mathematics and algebra. Or maybe it’s too expensive for you to hire a tutor or buy new textbooks? Or do you just want to get it done as quickly as possible? homework in mathematics or algebra? In this case, you can also use our programs with detailed solutions.
In this way, you can conduct your own training and/or training of your younger brothers or sisters, while the level of education in the field of solving problems increases.
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A little theory.
Limit of the function at x->x 0
Let the function f(x) be defined on some set X and let the point \(x_0 \in X\) or \(x_0 \notin X\)
Let us take from X a sequence of points different from x 0:
x 1 , x 2 , x 3 , ..., x n , ... (1)
converging to x*. The function values at the points of this sequence also form a numerical sequence
f(x 1), f(x 2), f(x 3), ..., f(x n), ... (2)
and one can raise the question of the existence of its limit.
Definition. The number A is called the limit of the function f(x) at the point x = x 0 (or at x -> x 0), if for any sequence (1) of values of the argument x different from x 0 converging to x 0, the corresponding sequence (2) of values function converges to number A.
$$ \lim_(x\to x_0)( f(x)) = A $$
The function f(x) can have only one limit at the point x 0. This follows from the fact that the sequence
(f(x n)) has only one limit.
There is another definition of the limit of a function.
Definition The number A is called the limit of the function f(x) at the point x = x 0 if for any number \(\varepsilon > 0\) there is a number \(\delta > 0\) such that for all \(x \in X, \; x \neq x_0 \), satisfying the inequality \(|x-x_0| Using logical symbols, this definition can be written as
\((\forall \varepsilon > 0) (\exists \delta > 0) (\forall x \in X, \; x \neq x_0, \; |x-x_0| Note that the inequalities \(x \neq x_0 , \; |x-x_0| The first definition is based on the concept of the limit of a number sequence, so it is often called the definition “in the language of sequences.” The second definition is called the definition “in the language \(\varepsilon - \delta \)”.
These two definitions of the limit of a function are equivalent and you can use either of them depending on which is more convenient for solving a particular problem.
Note that the definition of the limit of a function “in the language of sequences” is also called the definition of the limit of a function according to Heine, and the definition of the limit of a function “in the language \(\varepsilon - \delta \)” is also called the definition of the limit of a function according to Cauchy.
Limit of the function at x->x 0 - and at x->x 0 +
In what follows, we will use the concepts of one-sided limits of a function, which are defined as follows.
Definition The number A is called the right (left) limit of the function f(x) at the point x 0 if for any sequence (1) converging to x 0, the elements x n of which are greater (less than) x 0, the corresponding sequence (2) converges to A.
Symbolically it is written like this:
$$ \lim_(x \to x_0+) f(x) = A \; \left(\lim_(x \to x_0-) f(x) = A \right) $$
We can give an equivalent definition of one-sided limits of a function “in the language \(\varepsilon - \delta \)”:
Definition a number A is called the right (left) limit of the function f(x) at the point x 0 if for any \(\varepsilon > 0\) there is a \(\delta > 0\) such that for all x satisfying the inequalities \(x_0 Symbolic entries: