Function limit- number a will be the limit of some variable value if, in the process of its change, this variable quantity approaches indefinitely a.
Or in other words, the number A is the limit of the function y = f(x) at the point x 0, if for any sequence of points from the domain of definition of the function , not equal x 0, and which converges to the point x 0 (lim x n = x0), the sequence of corresponding function values converges to the number A.
The graph of a function whose limit, given an argument that tends to infinity, is equal to L:
Meaning A is limit (limit value) of the function f(x) at the point x 0 in case for any sequence of points 
, which converges to x 0, but which does not contain x 0 as one of its elements (i.e. in the punctured vicinity x 0), sequence of function values 
converges to A.
Limit of a Cauchy function.
Meaning A will be limit of the function f(x) at the point x 0 if for any non-negative number taken in advance ε the corresponding non-negative number will be found δ = δ(ε) such that for each argument x, satisfying the condition 0 < | x - x0 | < δ , the inequality will be satisfied | f(x)A |< ε .
It will be very simple if you understand the essence of the limit and the basic rules for finding it. What is the limit of the function f (x) at x striving for a equals A, is written like this:
Moreover, the value to which the variable tends x, can be not only a number, but also infinity (∞), sometimes +∞ or -∞, or there may be no limit at all.
To understand how find the limits of a function, it is best to look at examples of solutions.
It is necessary to find the limits of the function f (x) = 1/x at:
x→ 2, x→ 0, x→ ∞.
Let's find a solution to the first limit. To do this, you can simply substitute x the number it tends to, i.e. 2, we get:
Let's find the second limit of the function. Here substitute pure 0 instead x it is impossible, because You cannot divide by 0. But we can take values close to zero, for example, 0.01; 0.001; 0.0001; 0.00001 and so on, and the value of the function f (x) will increase: 100; 1000; 10000; 100,000 and so on. Thus, it can be understood that when x→ 0 the value of the function that is under the limit sign will increase without limit, i.e. strive towards infinity. Which means:
Regarding the third limit. The same situation as in the previous case, it is impossible to substitute ∞ in its purest form. We need to consider the case of unlimited increase x. We substitute 1000 one by one; 10000; 100000 and so on, we have that the value of the function f (x) = 1/x will decrease: 0.001; 0.0001; 0.00001; and so on, tending to zero. That's why:
It is necessary to calculate the limit of the function 
Starting to solve the second example, we see uncertainty. From here we find the highest degree of the numerator and denominator - this is x 3, we take it out of brackets in the numerator and denominator and then reduce it by:
Answer 
The first step in finding this limit, substitute the value 1 instead x, resulting in uncertainty. To solve it, let’s factorize the numerator and do this using the method of finding the roots of a quadratic equation x 2 + 2x - 3:
D = 2 2 - 4*1*(-3) = 4 +12 = 16→ √ D=√16 = 4
x 1.2 = (-2±4)/2→ x 1 = -3;x 2= 1.
So the numerator will be:
Answer 
This is the definition of it specific meaning or a specific area where a function that is limited by a limit falls.
To solve limits, follow the rules:
Having understood the essence and main rules for solving the limit, you will get a basic understanding of how to solve them.
Limit of a function at infinity:
|f(x) - a|< ε
при |x| >N
Determination of the Cauchy limit
Let the function f (x) is defined in a certain neighborhood of the point at infinity, with |x| > The number a is called the limit of the function f (x) as x tends to infinity (), if for any, however small, positive number ε > 0
, there is a number N ε >K, depending on ε, which for all x, |x| > N ε, the function values belong to the ε-neighborhood of point a:
|f (x)-a|< ε
.
The limit of a function at infinity is denoted as follows:
.
Or at .
The following notation is also often used:
.
Let's write this definition using the logical symbols of existence and universality:
.
This assumes that the values belong to the domain of the function.
One-sided limits
Left limit of a function at infinity:
|f(x) - a|< ε
при x < -N
There are often cases when the function is defined only for positive or negative values of the variable x (more precisely, in the vicinity of the point or ). Also, the limits at infinity for positive and negative values of x can have different meanings. Then one-sided limits are used.
Left limit at infinity or the limit as x tends to minus infinity () is defined as follows:
.
Right limit at infinity or the limit as x tends to plus infinity ():
.
One-sided limits at infinity are often denoted as follows:
;
.
Infinite limit of a function at infinity
Infinite limit of a function at infinity:
|f(x)| > M for |x| > N
Definition of the infinite limit according to Cauchy
Let the function f (x) is defined in a certain neighborhood of the point at infinity, with |x| > K, where K is a positive number. Limit of function f (x) as x tends to infinity (), is equal to infinity, if for anyone, arbitrarily large number M > 0
, there is such a number N M >K, depending on M, which for all x, |x| > N M , the function values belong to the neighborhood of the point at infinity:
|f (x) | >M.
The infinite limit as x tends to infinity is denoted as follows:
.
Or at .
Using the logical symbols of existence and universality, the definition of the infinite limit of a function can be written as follows:
.
Similarly, definitions of infinite limits of certain signs equal to and are introduced:
.
.
Definitions of one-sided limits at infinity.
Left limits.
.
.
.
Right limits.
.
.
.
Determination of the limit of a function according to Heine
Let the function f (x) defined on some neighborhood of the point x at infinity 0
, where or or .
The number a (finite or at infinity) is called the limit of the function f (x) at point x 0
:
,
if for any sequence (xn), converging to x 0
:
,
whose elements belong to the neighborhood, sequence (f(xn)) converges to a:
.
If we take as a neighborhood the neighborhood of an unsigned point at infinity: , then we obtain the definition of the limit of a function as x tends to infinity, . If we take a left-sided or right-sided neighborhood of the point x at infinity 0 : or , then we obtain the definition of the limit as x tends to minus infinity and plus infinity, respectively.
The Heine and Cauchy definitions of limit are equivalent.
Examples
Example 1
Using Cauchy's definition to show that
.
Let us introduce the following notation:
.
Let's find the domain of definition of the function. Since the numerator and denominator of the fraction are polynomials, the function is defined for all x except the points at which the denominator vanishes. Let's find these points. Solving a quadratic equation. ;
.
Roots of the equation:
;
.
Since , then and .
Therefore the function is defined at . We will use this later.
Let us write down the definition of the finite limit of a function at infinity according to Cauchy:
.
Let's transform the difference:
.
Divide the numerator and denominator by and multiply by -1
:
.
Let .
Then
;
;
;
.
So, we found that when ,
.
.
It follows that
at , and .
Since you can always increase it, let's take . Then for anyone,
at .
It means that .
Example 2
Let .
Using the Cauchy definition of a limit, show that:
1)
;
2)
.
1) Solution as x tends to minus infinity
Since , the function is defined for all x.
Let us write down the definition of the limit of a function at equal to minus infinity:
.
Let . Then
;
.
So, we found that when ,
.
Enter positive numbers and :
.
It follows that for any positive number M, there is a number, so that for ,
.
It means that .
2) Solution as x tends to plus infinity
Let's transform the original function. Multiply the numerator and denominator of the fraction by and apply the difference of squares formula:
.
We have:
.
Let us write down the definition of the right limit of the function at:
.
Let us introduce the notation: .
Let's transform the difference:
.
Multiply the numerator and denominator by:
.
Let
.
Then
;
.
So, we found that when ,
.
Enter positive numbers and :
.
It follows that
at and .
Since this holds for any positive number, then
.
References:
CM. Nikolsky. Well mathematical analysis. Volume 1. Moscow, 1983.
Limits give all mathematics students a lot of trouble. To solve a limit, sometimes you have to use a lot of tricks and choose from a variety of solution methods exactly the one that is suitable for a particular example.
In this article we will not help you understand the limits of your capabilities or comprehend the limits of control, but we will try to answer the question: how to understand limits in higher mathematics? Understanding comes with experience, so at the same time we will give several detailed examples of solving limits with explanations.
The concept of limit in mathematics
The first question is: what is this limit and the limit of what? We can talk about the limits of numerical sequences and functions. We are interested in the concept of the limit of a function, since this is what students most often encounter. But first - the most general definition limit:
Let's say there is some variable value. If this value in the process of change unlimitedly approaches a certain number a , That a – the limit of this value.
For a function defined in a certain interval f(x)=y such a number is called a limit A , which the function tends to when X , tending to a certain point A . Dot A belongs to the interval on which the function is defined.
It sounds cumbersome, but it is written very simply:
Lim- from English limit- limit.
There is also a geometric explanation for determining the limit, but here we will not delve into the theory, since we are more interested in the practical rather than the theoretical side of the issue. When we say that X tends to some value, this means that the variable does not take on the value of a number, but approaches it infinitely close.
Let's give specific example. The task is to find the limit.
To solve this example, we substitute the value x=3 into a function. We get:
By the way, if you are interested, read a separate article on this topic.
In examples X can tend to any value. It can be any number or infinity. Here's an example when X tends to infinity:
Intuitively, the larger the number in the denominator, the smaller the value the function will take. So, with unlimited growth X meaning 1/x will decrease and approach zero.
As you can see, to solve the limit, you just need to substitute the value to strive for into the function X . However, this is the simplest case. Often finding the limit is not so obvious. Within the limits there are uncertainties of the type 0/0 or infinity/infinity . What to do in such cases? Resort to tricks!
Uncertainties within
Uncertainty of the form infinity/infinity
Let there be a limit:
If we try to substitute infinity into the function, we will get infinity in both the numerator and the denominator. In general, it is worth saying that there is a certain element of art in resolving such uncertainties: you need to notice how you can transform the function in such a way that the uncertainty goes away. In our case, we divide the numerator and denominator by X in the senior degree. What will happen?
From the example already discussed above, we know that terms containing x in the denominator will tend to zero. Then the solution to the limit is:
To resolve type uncertainties infinity/infinity divide the numerator and denominator by X to the highest degree.
By the way! For our readers there is now a 10% discount on
Another type of uncertainty: 0/0
As always, substituting values into the function x=-1 gives 0 in the numerator and denominator. Look a little more closely and you will notice that in our numerator quadratic equation. Let's find the roots and write:
Let's reduce and get:
So, if you are faced with type uncertainty 0/0 – factor the numerator and denominator.
To make it easier for you to solve examples, we present a table with the limits of some functions:
L'Hopital's rule within
Another powerful way to eliminate both types of uncertainty. What is the essence of the method?
If there is uncertainty in the limit, take the derivative of the numerator and denominator until the uncertainty disappears.
L'Hopital's rule looks like this:
Important point : the limit in which the derivatives of the numerator and denominator stand instead of the numerator and denominator must exist.
And now - a real example:
There is typical uncertainty 0/0 . Let's take the derivatives of the numerator and denominator:
Voila, uncertainty is resolved quickly and elegantly.
We hope that you will be able to usefully apply this information in practice and find the answer to the question “how to solve limits in higher mathematics.” If you need to calculate the limit of a sequence or the limit of a function at a point, and there is absolutely no time for this work, contact a professional student service for a quick and detailed solution.
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:
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