The definitions of the limit of a function according to Heine (in terms of sequences) and in terms of Cauchy (in terms of epsilon and delta neighborhoods) are given. The definitions are given in a universal form applicable to both bilateral and one-sided limits at finite and at infinity points. The definition that a point a is not a limit of a function is considered. Proof of the equivalence of the definitions according to Heine and according to Cauchy.

Content

See also: Neighborhood of a point
Determining the limit of a function at the end point
Determining the limit of a function at infinity

First definition of the limit of a function (according to Heine)

(x) at point x 0 :
,
If
1) there is such a punctured neighborhood of the point x 0
2) for any sequence ( x n ), converging to x 0 :
, whose elements belong to the neighborhood ,
subsequence (f(xn)) converges to a :
.

Here x 0 and a can be either finite numbers or points at infinity. The neighborhood can be either two-sided or one-sided.

.

The second definition of the limit of a function (according to Cauchy)

The number a is called the limit of the function f (x) at point x 0 :
,
If
1) there is such a punctured neighborhood of the point x 0 on which the function is defined;
2) for any positive number ε > 0 there exists a number δ ε > 0 , depending on ε, that for all x belonging to a punctured δ ε neighborhood of the point x 0 :
,
function values ​​f (x) belong to ε - neighborhoods of the point a :
.

points x 0 and a can be either finite numbers or points at infinity. The neighborhood can also be both two-sided and one-sided.

We write this definition using the logical symbols of existence and universality:
.

This definition uses neighborhoods with equidistant ends. An equivalent definition can also be given using arbitrary neighborhoods of points.

Definition using arbitrary neighborhoods
The number a is called the limit of the function f (x) at point x 0 :
,
If
1) there is such a punctured neighborhood of the point x 0 on which the function is defined;
2) for any neighborhood U (a) point a there is such a punctured neighborhood of the point x 0 , that for all x that belong to a punctured neighborhood of the point x 0 :
,
function values ​​f (x) belong to the neighborhood U (a) points a :
.

Using the logical symbols of existence and universality, this definition can be written as follows:
.

Unilateral and bilateral limits

The above definitions are universal in the sense that they can be used for any type of neighborhood. If, as we use the left-handed punctured neighborhood of the end point, then we get the definition of the left-handed limit . If we use the neighborhood of a point at infinity as a neighborhood, then we get the definition of the limit at infinity.

To determine the limit according to Heine, this reduces to the fact that an additional restriction is imposed on an arbitrary sequence converging to , that its elements must belong to the corresponding punctured neighborhood of the point .

To determine the Cauchy limit, it is necessary in each case to transform the expressions and into inequalities, using the corresponding definitions of a neighborhood of a point.
See "Neighbourhood of a point".

Determining that a point a is not the limit of a function

Often there is a need to use the condition that the point a is not the limit of the function for . Let us construct negations to the above definitions. In them, we assume that the function f (x) is defined on some punctured neighborhood of the point x 0 . Points a and x 0 can be both finite numbers and infinitely distant. Everything stated below applies to both bilateral and one-sided limits.

According to Heine.
Number a is not limit of the function f (x) at point x 0 : ,
if there is such a sequence ( x n ), converging to x 0 :
,
whose elements belong to the neighborhood ,
what sequence (f(xn)) does not converge to a :
.
.

According to Cauchy.
Number a is not limit of the function f (x) at point x 0 :
,
if there is such a positive number ε > 0 , so that for any positive number δ > 0 , there exists x that belongs to a punctured δ neighborhood of the point x 0 :
,
that the value of the function f (x) does not belong to the ε neighborhood of the point a :
.
.

Of course, if the point a is not the limit of the function at , then this does not mean that it cannot have a limit. Perhaps there is a limit, but it is not equal to a . It is also possible that the function is defined in a punctured neighborhood of the point , but has no limit at .

Function f(x) = sin(1/x) has no limit as x → 0.

For example, the function is defined at , but there is no limit. For proof, we take the sequence . It converges to a point 0 : . Because , then .
Let's take a sequence. It also converges to the point 0 : . But since , then .
Then the limit cannot equal any number a . Indeed, for , there is a sequence with which . Therefore, any non-zero number is not a limit. But it is also not a limit, since there is a sequence with which .

Equivalence of the definitions of the limit according to Heine and according to Cauchy

Theorem
The Heine and Cauchy definitions of the limit of a function are equivalent.

Proof

In the proof, we assume that the function is defined in some punctured neighborhood of the point (finite or at infinity). The point a can also be finite or at infinity.

Heine proof ⇒ Cauchy

Let a function have a limit a at a point according to the first definition (according to Heine). That is, for any sequence that belongs to a punctured neighborhood of the point and has a limit
(1) ,
the limit of the sequence is a :
(2) .

Let us show that the function has a Cauchy limit at a point. That is, for any there exists that for all.

Let's assume the opposite. Let conditions (1) and (2) be satisfied, but the function has no Cauchy limit. That is, there exists such that for any exists , so that
.

Take , where n is a natural number. Then exists and
.
Thus we have constructed a sequence converging to , but the limit of the sequence is not equal to a . This contradicts the condition of the theorem.

The first part is proven.

Cauchy proof ⇒ Heine

Let a function have a limit a at a point according to the second definition (according to Cauchy). That is, for any there exists that
(3) for all .

Let us show that the function has a limit a at a point according to Heine.
Let's take an arbitrary number. According to Cauchy's definition, there exists a number , so (3) holds.

Take an arbitrary sequence belonging to the punctured neighborhood and converging to . By the definition of a convergent sequence, for any there exists such that
at .
Then from (3) it follows that
at .
Since this holds for any , then
.

The theorem has been proven.

References:
L.D. Kudryavtsev. Course of mathematical analysis. Volume 1. Moscow, 2003.

See also:

In proving the properties of the limit of a function, we made sure that nothing really was required from the punctured neighborhoods in which our functions were defined and which arose in the course of proofs, except for the properties indicated in the introduction to the previous paragraph 2. This circumstance serves as a justification for singling out the following mathematical object.

A. Base; definition and main examples

Definition 11. A set B of subsets of a set X will be called a base in a set X if two conditions are met:

In other words, the elements of the collection B are non-empty sets, and the intersection of any two of them contains some element from the same collection.

Let us indicate some of the most commonly used bases in the analysis.

If then instead they write and say that x tends to a from the right or from the side of large values ​​(respectively, from the left or from the side of smaller values). When a short record is accepted instead of

The entry will be used instead of It means that a; tends over the set E to a, remaining greater (less) than a.

then instead they write and say that x tends to plus infinity (respectively, to minus infinity).

The notation will be used instead

When instead of we (if this does not lead to misunderstanding) we will write, as is customary in the theory of the limit of a sequence,

Note that all the listed bases have the feature that the intersection of any two elements of the base is itself an element of this base, and not only contains some element of the base. We will meet with other bases when studying functions that are not given on the real axis.

We also note that the term “base” used here is a short designation of what is called “filter basis” in mathematics, and the base limit introduced below is the most essential part for analysis of the concept of filter limit created by the modern French mathematician A. Cartan

b. Base function limit

Definition 12. Let be a function on the set X; B is a base in X. A number is called the limit of a function with respect to the base B if for any neighborhood of the point A there is an element of the base whose image is contained in the neighborhood

If A is the limit of the function with respect to base B, then we write

Let's repeat the definition of the limit by the base in logical symbolism:

Since we are now considering functions with numeric values, it is useful to keep in mind the following form of this basic definition:

In this formulation, instead of an arbitrary neighborhood V(A), we take a neighborhood that is symmetric (with respect to the point A) (e-neighborhood). The equivalence of these definitions for real-valued functions follows from the fact that, as already mentioned, any neighborhood of a point contains some symmetric neighborhood of the same point (carry out the proof in full!).

We have given a general definition of the limit of a function with respect to the base. Above were considered examples of the most common bases in the analysis. In a specific problem where one or another of these bases appears, it is necessary to be able to decipher the general definition and write it down for a particular base.

Considering examples of bases, we, in particular, introduced the concept of a neighborhood of infinity. If we use this concept, then in accordance with the general definition of the limit, it is reasonable to adopt the following conventions:

or, which is the same,

Usually, by means a small value. In the above definitions, this is, of course, not the case. In accordance with the accepted conventions, for example, we can write

In order to be considered proven in the general case of a limit over an arbitrary base, all those theorems on limits that we proved in Section 2 for a special base , it is necessary to give the appropriate definitions: finally constant, finally bounded, and infinitely small for a given base of functions.

Definition 13. A function is called finally constant at base B if there exists a number and such an element of the base, at any point of which

Definition 14. A function is called bounded at base B or finally bounded at base B if there exists a number c and such an element of the base, at any point of which

Definition 15. A function is called infinitesimal with base B if

After these definitions and the basic observation that only base properties are needed to prove limit theorems, we can assume that all the properties of the limit established in Section 2 are valid for limits over any base.

In particular, we can now talk about the limit of a function at or at or at

In addition, we have secured the possibility of applying the theory of limits even in the case when the functions are not defined on numerical sets; this will prove to be especially valuable in the future. For example, the length of a curve is a numerical function defined on some class of curves. If we know this function on broken lines, then by passing to the limit we determine it for more complex curves, for example, for a circle.

At the moment, the main benefit of the observation made and the concept of base introduced in connection with it is that they save us from checks and formal proofs of limit theorems for each specific type of passage to the limit or, in our current terminology, for each specific type bases

In order to finally get used to the concept of a limit over an arbitrary base, we will prove the further properties of the limit of a function in a general form.

Let the function y=ƒ(x) be defined in some neighborhood of the point x o, except, perhaps, for the point x o itself.

Let us formulate two equivalent definitions of the limit of a function at a point.

Definition 1 (in the "language of sequences", or according to Heine).

The number A is called the limit of the function y \u003d ƒ (x) in the furnace x 0 (or at x® x o), if for any sequence of admissible values ​​of the argument x n, n є N (x n ¹ x 0) converging to x o the sequence of corresponding values ​​of the function ƒ(х n), n є N, converges to the number A

In this case, write
or ƒ(x)->A at x→x o. The geometric meaning of the limit of a function: means that for all points x sufficiently close to the point x o, the corresponding values ​​of the function differ arbitrarily little from the number A.

Definition 2 (in the "language of ε", or after Cauchy).

The number A is called the limit of the function at the point x o (or at x → x o) if for any positive ε there is a positive number δ such that for all x¹ x o satisfying the inequality |x-x o |<δ, выполняется неравенство |ƒ(х)-А|<ε.

The geometric meaning of the function limit:

if for any ε-neighbourhood of the point A there is such a δ-neighborhood of the point x o such that for all x¹ ho from this δ-neighborhood the corresponding values ​​of the function ƒ(x) lie in the ε-neighbourhood of the point A. In other words, the points of the graph of the function y = ƒ(x) lie inside a strip of width 2ε bounded by straight lines y=A+ ε , y=A-ε (see Fig. 110). Obviously, the value of δ depends on the choice of ε, so we write δ=δ(ε).

<< Пример 16.1

Prove that

Solution: Take an arbitrary ε>0, find δ=δ(ε)>0 such that for all x satisfying the inequality |х-3|< δ, выполняется неравенство |(2х-1)-5|<ε, т. е. |х-3|<ε.

Taking δ=ε/2, we see that for all x satisfying the inequality |x-3|< δ, выполняется неравенство |(2х-1)-5|<ε. Следовательно, lim(2x-1)=5 при х –>3.

<< Пример 16.2

16.2. One-sided limits

In the definition of the limit of the function, it is considered that x tends to x 0 in any way: remaining less than x 0 (to the left of x 0), greater than x o (to the right of x o), or fluctuating around the point x 0 .

There are cases when the method of approaching the argument x to xo significantly affects the value of the limit of the function. Therefore, the concept of one-sided limits is introduced.

The number A 1 is called the limit of the function y \u003d ƒ (x) on the left at the point x o, if for any number ε> 0 there is a number δ \u003d δ (ε)> 0 such that for x є (x 0 -δ; x o), the inequality |ƒ(x)-A|<ε. Предел слева записывают так: limƒ(х)=А при х–>x 0 -0 or briefly: ƒ (x o- 0) \u003d A 1 (Dirichlet notation) (see Fig. 111).

The limit of the function on the right is defined similarly, we write it using symbols:

Briefly, the limit on the right is denoted by ƒ(x o +0)=A.

The limits of a function on the left and right are called one-sided limits. Obviously, if exists, then both one-sided limits exist, and A=A 1 =A 2 .

The converse statement is also true: if both limits ƒ(x 0 -0) and ƒ(x 0 +0) exist and they are equal, then there is a limit and A \u003d ƒ(x 0 -0).

If A 1 ¹ A 2, then this aisle does not exist.

16.3. Limit of the function at x ® ∞

Let the function y=ƒ(x) be defined in the interval (-∞;∞). The number A is called function limitƒ(x) at x→ ∞ , if for any positive number ε there is such a number М=М()>0 that for all х satisfying the inequality |х|>М the inequality |ƒ(х)-А|<ε. Коротко это определение можно записать так:

The geometric meaning of this definition is as follows: for "ε>0 $ M>0, that for x є(-∞; -M) or x є(M; +∞) the corresponding values ​​of the function ƒ(x) fall into the ε-neighborhood of the point A , i.e., the points of the graph lie in a strip of width 2ε, bounded by straight lines y \u003d A + ε and y \u003d A-ε (see Fig. 112).

16.4. Infinitely large function (b.b.f.)

The function y=ƒ(x) is called infinitely large for x→x 0 if for any number M>0 there is a number δ=δ(M)>0, which for all x satisfying the inequality 0<|х-хо|<δ, выполняется неравенство |ƒ(х)|>M.

For example, the function y=1/(x-2) is a b.b.f. at x->2.

If ƒ(x) tends to infinity as x→x o and takes only positive values, then we write

if only negative values, then

The function y \u003d ƒ (x), given on the entire number line, called infinite for x→∞, if for any number M>0 there is such a number N=N(M)>0 that for all x satisfying the inequality |x|>N, the inequality |ƒ(x)|>M is satisfied. Short:

For example, y=2x has a b.b.f. at x→∞.

Note that if the argument х, tending to infinity, takes only natural values, i.e., хєN, then the corresponding b.b.f. becomes an infinitely large sequence. For example, the sequence v n =n 2 +1, n є N, is an infinitely large sequence. Obviously, every b.b.f. in a neighborhood of the point x o is unbounded in this neighborhood. The converse is not true: an unbounded function may not be a b.b.f. (For example, y=xsinx.)

However, if limƒ(x)=A for x→x 0 , where A is a finite number, then the function ƒ(x) is bounded in the vicinity of the point x o.

Indeed, from the definition of the limit of the function it follows that for x → x 0 the condition |ƒ(x)-A|<ε. Следовательно, А-ε<ƒ(х)<А+ε при х є (х о -ε; х о +ε), а это и означает, что функция ƒ (х) ограничена.

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 one or another 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. Once upon a time there was a Frenchman Augustin Louis Cauchy in the 19th century, who gave strict definitions to many concepts of matan and laid its foundations. I must say that this respected mathematician dreamed, dreams and will dream in nightmares of all students of physics and mathematics faculties, as he proved a huge number of theorems of mathematical analysis, and one theorem is more killer than the other. For this reason, we will not consider 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 immediately an example of why to shag your grandmother ....

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 unity." Most often - exactly, although instead of "x" in practice there are other variables. In practical tasks, in place of a unit, there can be absolutely any number, as well as infinity ().
3) Functions under the limit sign, in this case .

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

Let's analyze the next important question - what does the expression "x seeks to unity? And what is “strive” anyway?
The concept of a limit is a concept, so to speak, dynamic. Let's construct a sequence: first , then , , …, , ….
That is, the expression "x seeks to one" should be understood as follows - "x" consistently takes the 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 the unit in the function under the limit sign:

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

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

Infinity example:

Understanding what is it? This is the case when it increases indefinitely, that is: first, then, then, then, and so on ad infinitum.

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

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

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

Another example with infinity:

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

Conclusion: for , the function increases indefinitely:

And another series of examples:

Please try to mentally analyze the following for yourself and remember the simplest types of limits:

, , , , , , , , ,
If there is any doubt somewhere, you can pick up a calculator and practice a little.
In the event that , try to build the sequence , , . If , then , , .

! Note: strictly speaking, such an approach with the construction of sequences of 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, or at least with a million: , then all the same , because sooner or later "x" will begin to take on such gigantic values ​​that a million compared to them will be a real microbe.

What should be remembered and understood from the above?

1) When given any limit, first we simply try to substitute a 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 familiarize yourself with the 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 a function is generally does not exist!

In practice, unfortunately, there are few gifts. And so we turn to the consideration of 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 materials of the site, of course, are no worse:

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

Example:

Calculate Limit

According to our rule, we will try to substitute infinity into a function. What do we get at the top? Infinity. And what happens below? Also infinity. Thus, we have the so-called indeterminacy of the form. One might think that , and the answer is ready, but in the general case this is not the case at all, and some solution must be applied, which we will now consider.

How to solve the limits of this type?

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

The highest power in the numerator is two.

Now we look at the denominator and also find the highest degree:

The highest power 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 to the highest degree.

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

What is essential in making a decision?

First, we indicate the uncertainty, if any.

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

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

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

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. And do you need it?

Example 2

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

Maximum degree in the numerator: 3
Maximum degree in the denominator: 4
Choose greatest value, in this case four.
According to our algorithm, to reveal the uncertainty, we divide the numerator and denominator by .
A complete assignment might 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 power of "x" in the denominator: 1 (can be written as)
To reveal the uncertainty, it is necessary to divide the numerator and denominator by . A clean solution might look like this:

Divide the numerator and denominator by

The record does not mean division by zero (it is impossible to divide by zero), but division by an infinitely small number.

Thus, when disclosing the indeterminacy of the form, we can get finite number, zero or infinity.

Limits with type uncertainty 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 “x” no longer tends to infinity, but to final number.

Example 4

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

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

General rule: if there are polynomials in the numerator and denominator, and there is an uncertainty of the form , then for its disclosure factorize 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 are forgotten, then visit the page Mathematical formulas and tables and get acquainted with the methodological material Hot School Mathematics Formulas. By the way, it is best to print it out, it is required very often, and information from paper is absorbed better.

So let's solve our limit

Factoring the numerator and denominator

In order to factorize 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 square root function is on the simplest calculator.

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

Next, we find the roots:

Thus:

All. The numerator is factored.

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 in the expression that remains under the limit sign:

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

Let's factorize the numerator.

Example 5

Calculate Limit

First, a "clean" solution

Let's factorize the numerator and denominator.

Numerator:
Denominator:



,

What is important in this example?
First, you must understand well how the numerator is revealed, first we bracketed 2, and then used the difference of squares formula. This is the formula you need to know and see.

Recommendation: If in the limit (of almost any type) it is possible to take a number out of the bracket, then we always do this.
Moreover, it is advisable to take such numbers beyond the limit sign. For what? Just so they don't get in the way. The main thing is not to lose these numbers in the course of the decision.

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

! Important
In the course of the solution, a type fragment occurs very often. Reduce this fractionit is forbidden . First you need to change the sign of the numerator or the 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 it is necessary to solve two quadratic equations, that is, both in the numerator and in the denominator there are square trinomials.

The method of multiplying the numerator and denominator by the adjoint 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

We start to decide.

First, we try to substitute 3 in the expression under the limit sign
Once again I repeat - this is the first thing to do for ANY limit. This action is usually carried out mentally or on a draft.

An uncertainty of the form , which needs to be eliminated, is obtained.

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

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

So, the concept of the limit is quite important in the study of the course of higher mathematics, especially when you come across the integral calculus and understand the relationship between the limit and the integral. In the current material, simple examples will be considered, as well as ways to solve them.

Solution examples

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 often get these limits sent to us asking for help to solve. We decided to highlight them as a separate example and explain that these limits simply need to be remembered, as a rule. If you cannot solve your problem, then send it to us. We will provide a detailed solution. You will be able to familiarize yourself with the progress of the calculation and gather information. This will help you get a credit from the teacher in a timely manner!

Answer

$$ \text(a)) \lim \limits_(x \to \to 0) \frac(1)(x) = \infty \text( b))\lim \limits_(x \to \infty) \frac(1 )(x) = 0 $$

What to do with the 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 of $ 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? What should be the result? Since this is an uncertainty, this is not yet an answer and we continue the calculation. Since we have a polynomial in the numerators, we decompose it into factors using the familiar formula $$ a^2-b^2=(a-b)(a+b) $$. Remembered? Great! 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's take 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? How to be? Do not panic, because the impossible is possible. It is necessary to take out the brackets in both the numerator and the denominator X, and then reduce it. After that, 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 make an algorithm for solving the limits:

1. Substitute point x in the expression following the limit sign. If a certain number is obtained, or infinity, then the limit is completely solved. Otherwise, we have uncertainty: "zero divided by zero" or "infinity divided by infinity" and proceed to the next paragraphs of the instruction.
2. To eliminate the uncertainty "zero divide by zero" you need to factorize the numerator and denominator. Reduce similar. Substitute the 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 x of the greatest degree. We shorten the x's. We substitute x values ​​from under the limit into the remaining expression.

In this article, you got acquainted with 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 will talk about other types of tasks in future articles, 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'Hopital's rule.

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