Limits give all students of mathematics a lot of trouble. To solve the limit, sometimes you have to use a lot of tricks and choose from a variety of solutions exactly the one that is suitable for a particular example.
In this article, we will not help you understand the limits of your abilities or comprehend the limits of control, but we will try to answer the question: how to understand the limits in higher mathematics? Understanding comes with experience, so at the same time we will give some detailed examples of solving limits with explanations.
The concept of a limit in mathematics
The first question is: what is the 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 it is with them that students most often encounter. But first, the most general definition of a limit:
Let's say there is some variable. If this value in the process of change indefinitely approaches a certain number a , then a is the limit of this value.
For a function defined in some interval f(x)=y the limit is the number A , to which the function tends 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 the definition of the limit, but here we will not go into theory, since we are more interested in the practical 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 take a concrete example. The challenge 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 the examples X can tend to any value. It can be any number or infinity. Here is an example when X tends to infinity:
It is intuitively clear that the larger the number in the denominator, the smaller the value will be taken by the function. So, with unlimited growth X meaning 1/x will decrease and approach zero.
As you can see, in order 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 type 0/0 or infinity/infinity . What to do in such cases? Use tricks!
Uncertainties within
Uncertainty of the form infinity/infinity
Let there be a limit:
If we try to substitute infinity into the function, we get infinity both in the numerator and in 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 is gone. In our case, we divide the numerator and denominator by X in senior degree. What will happen?
From the example already considered above, we know that terms containing x in the denominator will tend to zero. Then the solution to the limit is:
To uncover type ambiguities 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, substitution into the value function x=-1 gives 0 in the numerator and denominator. Look a little more closely and you will notice that we have a quadratic equation in the numerator. Let's find the roots and write:
Let's reduce and get:
So, if you encounter type ambiguity 0/0 - factorize the numerator and denominator.
To make it easier for you to solve examples, here is a table with the limits of some functions:
L'Hopital's rule within
Another powerful way to eliminate both types of uncertainties. What is the essence of the method?
If there is uncertainty in the limit, we take the derivative of the numerator and denominator until the uncertainty disappears.
Visually, L'Hopital's rule looks like this:
Important point : the limit, in which the derivatives of the numerator and denominator are instead of the numerator and denominator, must exist.
And now a real example:
There is a typical uncertainty 0/0 . Take the derivatives of the numerator and denominator:
Voila, the uncertainty is eliminated quickly and elegantly.
We hope that you will be able to put this information to good use 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 no time for this work from the word "absolutely", refer to for a quick and detailed solution.
Statements of the main theorems and properties of numerical sequences with limits are given. Contains the definition of a sequence and its limit. Arithmetic operations with sequences, properties related to inequalities, convergence criteria, properties of infinitely small and infinitely large sequences are considered.
Sequences
Numerical sequence called the law (rule), according to which, each natural number is assigned a number.
The number is called the nth member or element of the sequence.
In what follows, we will assume that the elements of the sequence are real numbers.
limited, if there exists a number M such that for all real n .
top face sequences are called the smallest of the numbers that bounds the sequence from above. That is, this is a number s for which for all n and for any , there is such an element of the sequence that exceeds s′ : .
bottom face sequences name the largest of the numbers that bounds the sequence from below. That is, this is a number i for which for all n and for any , there is such an element of the sequence that is less than i : .
The top edge is also called exact upper bound, and the lower bound precise lower bound. The concepts of upper and lower bounds are valid not only for sequences, but also for any sets of real numbers.
Determining the Limit of a Sequence
The number a is called the limit of the sequence, if for any positive number there exists such a natural number N , depending on , that for all natural numbers the inequality
.
The limit of a sequence is denoted as follows:
.
Or at .
Using the logical symbols of existence and universality, the definition of the limit can be written as follows:
.
Open interval (a - ε, a + ε) is called the ε-neighborhood of the point a.
A sequence that has a limit is called convergent sequence. It is also said that the sequence converges to a. A sequence that has no limit is called divergent.
point a is not the limit of the sequence, if there exists such that for any natural n there exists such a natural m >n, what
.
.
This means that you can choose such an ε - neighborhood of the point a , outside of which there will be an infinite number of elements of the sequence.
Properties of finite limits of sequences
Basic properties
A point a is the limit of a sequence if and only if outside any neighborhood of this point is finite number of elements sequences or the empty set.
If the number a is not the limit of the sequence , then there is such a neighborhood of the point a , outside of which there is infinite number of sequence elements.
Uniqueness theorem for the limit of a number sequence. If a sequence has a limit, then it is unique.
If a sequence has a finite limit, then it limited.
If each element of the sequence is equal to the same number C : , then this sequence has a limit equal to the number C .
If the sequence add, drop or change the first m elements, then this will not affect its convergence.
Proofs of basic properties given on the page
Basic properties of finite limits of sequences >>> .
Arithmetic with limits
Let there be finite limits and sequences and . And let C be a constant, that is, a given number. Then
;
;
;
, if .
In the case of the quotient, it is assumed that for all n .
If , then .
Arithmetic property proofs given on the page
Arithmetic properties of finite limits of sequences >>> .
Properties associated with inequalities
If the elements of the sequence, starting from some number, satisfy the inequality , then the limit a of this sequence also satisfies the inequality .
If the elements of the sequence, starting from some number, belong to a closed interval (segment) , then the limit a also belongs to this interval: .
If and and elements of sequences, starting from some number, satisfy the inequality , then .
If and, starting from some number, , then .
In particular, if, starting from some number, , then
if , then ;
if , then .
If and , then .
Let and . If a < b , then there is a natural number N such that for all n > N the inequality is satisfied.
Proofs of properties related to inequalities given on the page
Properties of sequence limits related to >>> inequalities.
Infinitesimal and infinitesimal sequences
Infinitesimal sequence
Sequence is called an infinitesimal sequence if its limit is zero:
.
Sum and Difference finite number of infinitesimal sequences is an infinitesimal sequence.
Product of a bounded sequence to an infinitesimal is an infinitesimal sequence.
Product of a finite number infinitesimal sequences is an infinitesimal sequence.
For a sequence to have a limit a , it is necessary and sufficient that , where is an infinitesimal sequence.
Proofs of properties of infinitesimal sequences given on the page
Infinitely small sequences - definition and properties >>> .
Infinitely large sequence
Sequence is called an infinite sequence, if for any positive number there exists such a natural number N , depending on , that for all natural numbers the inequality
.
In this case, write
.
Or at .
They say it tends to infinity.
If , starting from some number N , then
.
If , then
.
If the sequences are infinitely large, then starting from some number N , a sequence is defined that is infinitely small. If are an infinitesimal sequence with non-zero elements, then the sequence is infinitely large.
If the sequence is infinitely large and the sequence is bounded, then
.
If the absolute values of the elements of the sequence are bounded from below by a positive number (), and is infinitely small with non-zero elements, then
.
In details definition of an infinitely large sequence with examples given on the page
Definition of an infinitely large sequence >>> .
Proofs for properties of infinitely large sequences given on the page
Properties of infinitely large sequences >>> .
Sequence Convergence Criteria
Monotonic sequences
The sequence is called strictly increasing, if for all n the following inequality holds:
.
Accordingly, for strictly decreasing sequence, the following inequality holds:
.
For non-decreasing:
.
For non-increasing:
.
It follows that a strictly increasing sequence is also nondecreasing. A strictly decreasing sequence is also non-increasing.
The sequence is called monotonous if it is non-decreasing or non-increasing.
A monotonic sequence is bounded on at least one side by . A non-decreasing sequence is bounded from below: . A non-increasing sequence is bounded from above: .
Weierstrass theorem. In order for a non-decreasing (non-increasing) sequence to have a finite limit, it is necessary and sufficient that it be bounded from above (from below). Here M is some number.
Since any non-decreasing (non-increasing) sequence is bounded from below (from above), the Weierstrass theorem can be rephrased as follows:
For a monotone sequence to have a finite limit, it is necessary and sufficient that it be bounded: .
Monotonic unbounded sequence has an infinite limit, equal for non-decreasing and non-increasing sequences.
Proof of the Weierstrass theorem given on the page
Weierstrass' theorem on the limit of a monotone sequence >>> .
Cauchy criterion for sequence convergence
Cauchy condition. A sequence satisfies the Cauchy condition if for any there exists a natural number such that for all natural numbers n and m satisfying the condition , the inequality
.
Sequences satisfying the Cauchy condition are also called fundamental sequences.
Cauchy criterion for sequence convergence. For a sequence to have a finite limit, it is necessary and sufficient that it satisfies the Cauchy condition.
Proof of the Cauchy Convergence Criterion given on the page
Cauchy's convergence criterion for a sequence >>> .
Subsequences
Bolzano-Weierstrass theorem. From any bounded sequence, a convergent subsequence can be distinguished. And from any unlimited sequence - an infinitely large subsequence converging to or to .
Proof of the Bolzano-Weierstrass theorem given on the page
Bolzano–Weierstrass theorem >>> .
Definitions, theorems, and properties of subsequences and partial limits are discussed on page
Subsequences and partial limits of sequences >>>.
References:
CM. Nikolsky. Course of mathematical analysis. Volume 1. Moscow, 1983.
L.D. Kudryavtsev. Course of mathematical analysis. Volume 1. Moscow, 2003.
V.A. Zorich. Mathematical analysis. Part 1. Moscow, 1997.
V.A. Ilyin, E.G. Pozniak. Fundamentals of mathematical analysis. Part 1. Moscow, 2005.
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Theory of limits- one of the sections of mathematical analysis, which one can master, others hardly calculate the limits. The question of finding limits is quite general, since there are dozens of tricks limit solutions various types. The same limits can be found both by L'Hopital's rule and without it. It happens that the schedule in a series of infinitesimal functions allows you to quickly get the desired result. There are a set of tricks and tricks that allow you to find the limit of a function of any complexity. In this article, we will try to understand the main types of limits that are most often encountered in practice. We will not give the theory and definition of the limit here, there are many resources on the Internet where this is chewed. Therefore, let's do practical calculations, it is here that you start "I don't know! I don't know how! We weren't taught!"
Calculation of limits by the substitution method
Example 1 Find the limit of a function
Lim((x^2-3*x)/(2*x+5),x=3).
Solution: In theory, examples of this kind are calculated by the usual substitution
The limit is 18/11.
There is nothing complicated and wise within such limits - they substituted the value, calculated, wrote down the limit in response. However, on the basis of such limits, everyone is taught that, first of all, you need to substitute a value into the function. Further, the limits complicate, introduce the concept of infinity, uncertainty, and the like.
Limit with uncertainty of type infinity divided by infinity. Uncertainty disclosure methods
Example 2 Find the limit of a function
Lim((x^2+2x)/(4x^2+3x-4),x=infinity).
Solution: A limit of the form polynomial divided by a polynomial is given, and the variable tends to infinity 
A simple substitution of the value to which the variable should find the limits will not help, we get the uncertainty of the form infinity divided by infinity.
Pot theory of limits The algorithm for calculating the limit is to find the largest degree of "x" in the numerator or denominator. Next, the numerator and denominator are simplified on it and the limit of the function is found 
Since the value tends to zero when the variable goes to infinity, they are neglected, or written in the final expression as zeros 
Immediately from practice, you can get two conclusions that are a hint in the calculations. If the variable tends to infinity and the degree of the numerator is greater than the degree of the denominator, then the limit is equal to infinity. Otherwise, if the polynomial in the denominator is higher order than in the numerator, the limit is zero.
The limit formula can be written as 
If we have a function of the form of an ordinary log without fractions, then its limit is equal to infinity 
The next type of limits concerns the behavior of functions near zero.
Example 3 Find the limit of a function
Lim((x^2+3x-5)/(x^2+x+2), x=0).
Solution: Here it is not required to take out the leading multiplier of the polynomial. Exactly the opposite, it is necessary to find the smallest power of the numerator and denominator and calculate the limit 
x^2 value; x tend to zero when the variable tends to zero Therefore, they are neglected, thus we get 
that the limit is 2.5.
Now you know how to find the limit of a function kind of a polynomial divided by a polynomial if the variable tends to infinity or 0. But this is only a small and easy part of the examples. From the following material you will learn how to uncover the uncertainties of the limits of a function.
Limit with uncertainty of type 0/0 and methods for its calculation
Immediately everyone remembers the rule according to which you cannot divide by zero. However, the theory of limits in this context means infinitesimal functions.
Let's look at a few examples to illustrate.
Example 4 Find the limit of a function
Lim((3x^2+10x+7)/(x+1), x=-1).
Solution: When substituting the value of the variable x = -1 into the denominator, we get zero, we get the same in the numerator. So we have uncertainty of the form 0/0.
It is easy to deal with such uncertainty: you need to factorize the polynomial, or rather, select a factor that turns the function into zero. 
After decomposition, the limit of the function can be written as 
That's the whole technique for calculating the limit of a function. We do the same if there is a limit of the form of a polynomial divided by a polynomial.
Example 5 Find the limit of a function
Lim((2x^2-7x+6)/(3x^2-x-10), x=2).
Solution: Direct substitution shows
2*4-7*2+6=0;
3*4-2-10=0
what do we have type uncertainty 0/0.
Divide the polynomials by the factor that introduces the singularity 
There are teachers who teach that polynomials of the 2nd order, that is, the type of "quadratic equations" should be solved through the discriminant. But real practice shows that this is longer and more complicated, so get rid of features within the limits according to the specified algorithm. Thus, we write the function in the form of simple factors and calculate in the limit 
As you can see, there is nothing complicated in calculating such limits. You know how to divide polynomials at the time of studying the limits, at least according to the program, you should already pass.
Among the tasks for type uncertainty 0/0 there are those in which it is necessary to apply the formulas of abbreviated multiplication. But if you do not know them, then by dividing the polynomial by the monomial, you can get the desired formula.
Example 6 Find the limit of a function
Lim((x^2-9)/(x-3), x=3).
Solution: We have an uncertainty of type 0/0 . In the numerator, we use the formula for abbreviated multiplication 
and calculate the desired limit 
Uncertainty disclosure method by multiplication by the conjugate
The method is applied to the limits in which irrational functions generate uncertainty. The numerator or denominator turns to zero at the calculation point and it is not known how to find the boundary.
Example 7 Find the limit of a function
Lim((sqrt(x+2)-sqrt(7x-10))/(3x-6), x=2).
Solution: Let's represent the variable in the limit formula
When substituting, we get an uncertainty of type 0/0.
According to the theory of limits, the scheme for bypassing this singularity consists in multiplying an irrational expression by its conjugate. To keep the expression unchanged, the denominator must be divided by the same value 
By the difference of squares rule, we simplify the numerator and calculate the limit of the function
We simplify the terms that create a singularity in the limit and perform the substitution
Example 8 Find the limit of a function
Lim((sqrt(x-2)-sqrt(2x-5))/(3-x), x=3).
Solution: Direct substitution shows that the limit has a singularity of the form 0/0. 
To expand, multiply and divide by the conjugate to the numerator 
Write down the difference of squares
We simplify the terms that introduce a singularity and find the limit of the function
Example 9 Find the limit of a function
Lim((x^2+x-6)/(sqrt(3x-2)-2), x=2).
Solution: Substitute the deuce in the formula 
Get uncertainty 0/0.
The denominator must be multiplied by the conjugate expression, and in the numerator, solve the quadratic equation or factorize, taking into account the singularity. Since it is known that 2 is a root, then the second root is found by the Vieta theorem
Thus, we write the numerator in the form 
and put in the limit
Having reduced the difference of squares, we get rid of the features in the numerator and denominator
In the above way, you can get rid of the singularity in many examples, and the application should be noticed everywhere where the given difference of the roots turns into zero when substituting. Other types of limits concern exponential functions, infinitesimal functions, logarithms, singular limits, and other techniques. But you can read about this in the articles below on limits.
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? Fine! Now go ahead and apply it with the song :) We get that the numerator $ x^2-1=(x-1)(x+1) $ We continue to solve given the above transformation: $$ \lim \limits_(x \to -1)\frac(x^2-1)(x+1) = \lim \limits_(x \to -1)\frac((x-1)(x+ 1))(x+1) = $$ $$ = \lim \limits_(x \to -1)(x-1)=-1-1=-2 $$ |
| Answer |
| $$ \lim \limits_(x \to -1) \frac(x^2-1)(x+1) = -2 $$ |
Let'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:
- 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.
- 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.
- 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!