Consider the function %%f(x)%% defined at least in some punctured neighborhood %%\stackrel(\circ)(\text(U))(a)%% of the point %%a \in \overline( \mathbb(R))%% extended number line.
The concept of a Cauchy limit
The number %%A \in \mathbb(R)%% is called limit of the function%%f(x)%% at the point %%a \in \mathbb(R)%% (or at %%x%% tending to %%a \in \mathbb(R)%%), if, what Whatever the positive number %%\varepsilon%%, there is a positive number %%\delta%% such that for all points in the punctured %%\delta%% neighborhood of the point %%a%% the function values belong to %%\varepsilon %%-neighborhood of point %%A%%, or
$$ A = \lim\limits_(x \to a)(f(x)) \Leftrightarrow \forall\varepsilon > 0 ~\exists \delta > 0 \big(x \in \stackrel(\circ)(\text (U))_\delta(a) \Rightarrow f(x) \in \text(U)_\varepsilon (A) \big) $$
This definition is called the %%\varepsilon%% and %%\delta%% definition, proposed by the French mathematician Augustin Cauchy and used with early XIX century to the present, since it has the necessary mathematical rigor and accuracy.
Combining various neighborhoods of the point %%a%% of the form %%\stackrel(\circ)(\text(U))_\delta(a), \text(U)_\delta (\infty), \text(U) _\delta (-\infty), \text(U)_\delta (+\infty), \text(U)_\delta^+ (a), \text(U)_\delta^- (a) %% with surroundings %%\text(U)_\varepsilon (A), \text(U)_\varepsilon (\infty), \text(U)_\varepsilon (+\infty), \text(U) _\varepsilon (-\infty)%%, we get 24 definitions of the Cauchy limit.
Geometric meaning
Geometric meaning of the limit of a function
Let us find out what the geometric meaning of the limit of a function at a point is. Let's build a graph of the function %%y = f(x)%% and mark the points %%x = a%% and %%y = A%% on it.
The limit of the function %%y = f(x)%% at the point %%x \to a%% exists and is equal to A if for any %%\varepsilon%% neighborhood of the point %%A%% one can specify such a %%\ delta%%-neighborhood of the point %%a%%, such that for any %%x%% from this %%\delta%%-neighborhood the value %%f(x)%% will be in the %%\varepsilon%%-neighborhood points %%A%%.
Note that by the definition of the limit of a function according to Cauchy, for the existence of a limit at %%x \to a%%, it does not matter what value the function takes at the point %%a%%. Examples can be given where the function is not defined when %%x = a%% or takes a value other than %%A%%. However, the limit may be %%A%%.
Determination of the Heine limit
The element %%A \in \overline(\mathbb(R))%% is called the limit of the function %%f(x)%% at %% x \to a, a \in \overline(\mathbb(R))%% , if for any sequence %%\(x_n\) \to a%% from the domain of definition, the sequence of corresponding values %%\big\(f(x_n)\big\)%% tends to %%A%%.
The definition of a limit according to Heine is convenient to use when doubts arise about the existence of a limit of a function at a given point. If it is possible to construct at least one sequence %%\(x_n\)%% with a limit at the point %%a%% such that the sequence %%\big\(f(x_n)\big\)%% has no limit, then we can conclude that the function %%f(x)%% has no limit at this point. If for two various sequences %%\(x"_n\)%% and %%\(x""_n\)%% having same limit %%a%%, sequences %%\big\(f(x"_n)\big\)%% and %%\big\(f(x""_n)\big\)%% have various limits, then in this case there is also no limit of the function %%f(x)%%.
Example
Let %%f(x) = \sin(1/x)%%. Let's check whether the limit of this function exists at the point %%a = 0%%.
Let us first choose a sequence $$ \(x_n\) = \left\(\frac((-1)^n)(n\pi)\right\) converging to this point. $$
It is clear that %%x_n \ne 0~\forall~n \in \mathbb(N)%% and %%\lim (x_n) = 0%%. Then %%f(x_n) = \sin(\left((-1)^n n\pi\right)) \equiv 0%% and %%\lim\big\(f(x_n)\big\) = 0 %%.
Then take a sequence converging to the same point $$ x"_n = \left\( \frac(2)((4n + 1)\pi) \right\), $$
for which %%\lim(x"_n) = +0%%, %%f(x"_n) = \sin(\big((4n + 1)\pi/2\big)) \equiv 1%% and %%\lim\big\(f(x"_n)\big\) = 1%%. Similarly for the sequence $$ x""_n = \left\(-\frac(2)((4n + 1) \pi) \right\), $$
also converging to the point %%x = 0%%, %%\lim\big\(f(x""_n)\big\) = -1%%.
All three sequences gave different results, which contradicts the condition of the Heine definition, i.e. this function has no limit at the point %%x = 0%%.
Theorem
The Cauchy and Heine definitions of the limit are equivalent.
The formulation of the main theorems and properties of the limit of a function is given. Definitions of finite and infinite limits at finite points and at infinity (two-sided and one-sided) according to Cauchy and Heine are given. Arithmetic properties are considered; theorems related to inequalities; Cauchy convergence criterion; limit of a complex function; properties of infinitesimal, infinitely large and monotonic functions. The definition of a function is given.
ContentSecond definition according to Cauchy
The limit of a function (according to Cauchy) as its argument x tends to x 0 is a finite number or point at infinity a for which the following conditions are met:1) there is such a punctured neighborhood of the point x 0 , on which the function f (x) determined;
2) for any neighborhood of the point a belonging to , there is such a punctured neighborhood of the point x 0 , on which the function values belong to the selected neighborhood of point a:
at .
Here a and x 0
can also be either finite numbers or points at infinity. Using the logical symbols of existence and universality, this definition can be written as follows:
.
If we take the left or right neighborhood of an end point as a set, we obtain the definition of a Cauchy limit on the left or right.
Theorem
The Cauchy and Heine definitions of the limit of a function are equivalent.
Proof
Applicable neighborhoods of points
Then, in fact, the Cauchy definition means the following.
For any positive numbers , there are numbers , so that for all x belonging to the punctured neighborhood of the point : , the values of the function belong to the neighborhood of the point a: ,
Where , .
This definition is not entirely convenient to work with, since neighborhoods are defined using four numbers. But it can be simplified by introducing neighborhoods with equidistant ends. That is, you can put , . Then we will get a definition that is easier to use when proving theorems. Moreover, it is equivalent to the definition in which arbitrary neighborhoods are used. The proof of this fact is given in the section “Equivalence of Cauchy definitions of the limit of a function”.
Then you can give single definition limit of a function at finite and infinite points:
.
Here for endpoints
;
;
.
Any neighborhood of points at infinity is punctured:
;
;
.
Finite limits of function at end points
The number a is called the limit of the function f (x) at point x 0 , If1) the function is defined on some punctured neighborhood of the end point;
2) for any there is such that depends on , such that for all x for which , the inequality holds
.
Using the logical symbols of existence and universality, the definition of the limit of a function can be written as follows:
.
One-sided limits.
Left limit at a point (left-sided limit):
.
Right limit at a point (right-hand limit):
.
The left and right limits are often denoted as follows:
;
.
Finite limits of a function at points at infinity
Limits at points at infinity are determined in a similar way.
.
.
.
Infinite Function Limits
You can also introduce definitions of infinite limits of certain signs equal to and :
.
.
Properties and theorems of the limit of a function
We further assume that the functions under consideration are defined in the corresponding punctured neighborhood of the point , which is a finite number or one of the symbols: . It can also be a one-sided limit point, that is, have the form or . The neighborhood is two-sided for a two-sided limit and one-sided for a one-sided limit.
Basic properties
If the values of the function f (x) change (or make undefined) a finite number of points x 1, x 2, x 3, ... x n, then this change will not affect the existence and value of the limit of the function at an arbitrary point x 0 .
If there is a finite limit, then there is a punctured neighborhood of the point x 0
, on which the function f (x) limited:
.
Let the function have at point x 0
finite non-zero limit:
.
Then, for any number c from the interval , there is such a punctured neighborhood of the point x 0
, what for ,
, If ;
, If .
If, on some punctured neighborhood of the point, , is a constant, then .
If there are finite limits and and on some punctured neighborhood of the point x 0
,
That .
If , and on some neighborhood of the point
,
That .
In particular, if in some neighborhood of a point
,
then if , then and ;
if , then and .
If on some punctured neighborhood of a point x 0
:
,
and there are finite (or infinite of a certain sign) equal limits:
, That
.
Proofs of the main properties are given on the page
"Basic properties of the limit of a function."
Let the functions and be defined in some punctured neighborhood of the point . And let there be finite limits:
And .
And let C be a constant, that is, a given number. Then
;
;
;
, If .
If, then.
Proofs of arithmetic properties are given on the page
"Arithmetic properties of the limit of a function".
Cauchy criterion for the existence of a limit of a function
Theorem
In order for a function defined on some punctured neighborhood of a finite or at infinity point x 0
, had a finite limit at this point, it is necessary and sufficient that for any ε > 0
there was such a punctured neighborhood of the point x 0
, that for any points and from this neighborhood, the following inequality holds:
.
Limit of a complex function
Theorem on the limit of a complex function
Let the function have a limit and map a punctured neighborhood of a point onto a punctured neighborhood of a point. Let the function be defined on this neighborhood and have a limit on it.
Here are the final or infinitely distant points: . Neighborhoods and their corresponding limits can be either two-sided or one-sided.
Then there is a limit of a complex function and it is equal to:
.
The limit theorem of a complex function is applied when the function is not defined at a point or has a value different from the limit. To apply this theorem, there must be a punctured neighborhood of the point where the set of values of the function does not contain the point:
.
If the function is continuous at the point , then the limit sign can be applied to the argument continuous function:
.
The following is a theorem corresponding to this case.
Theorem on the limit of a continuous function of a function
Let there be a limit of the function g (x) as x → x 0
, and it is equal to t 0
:
.
Here is point x 0
can be finite or infinitely distant: .
And let the function f (t) continuous at point t 0
.
Then there is a limit of the complex function f (g(x)), and it is equal to f (t 0):
.
Proofs of the theorems are given on the page
"Limit and continuity of a complex function".
Infinitesimal and infinitely large functions
Infinitesimal functions
Definition
A function is said to be infinitesimal if
.
Sum, difference and product of a finite number of infinitesimal functions at is an infinitesimal function at .
Product of a function bounded on some punctured neighborhood of the point , to an infinitesimal at is an infinitesimal function at .
In order for a function to have a finite limit, it is necessary and sufficient that
,
where is an infinitesimal function at .
"Properties of infinitesimal functions".
Infinitely large functions
Definition
A function is said to be infinitely large if
.
The sum or difference of a bounded function, on some punctured neighborhood of the point , and an infinitely large function at is an infinitely large function at .
If the function is infinitely large for , and the function is bounded on some punctured neighborhood of the point , then
.
If the function , on some punctured neighborhood of the point , satisfies the inequality:
,
and the function is infinitesimal at:
, and (on some punctured neighborhood of the point), then
.
Proofs of the properties are presented in section
"Properties of infinitely large functions".
Relationship between infinitely large and infinitesimal functions
From the two previous properties follows the connection between infinitely large and infinitesimal functions.
If a function is infinitely large at , then the function is infinitesimal at .
If a function is infinitesimal for , and , then the function is infinitely large for .
The relationship between an infinitesimal and an infinitely large function can be expressed symbolically:
,
.
If an infinitesimal function has a certain sign at , that is, it is positive (or negative) on some punctured neighborhood of the point , then this fact can be expressed as follows:
.
In the same way, if an infinitely large function has a certain sign at , then they write:
.
Then the symbolic connection between infinitely small and infinitely large functions can be supplemented with the following relations:
,
,
,
.
Additional formulas relating infinity symbols can be found on the page
"Points at infinity and their properties."
Limits of monotonic functions
Definition
Function defined on some set real numbers X is called strictly increasing, if for all such that the following inequality holds:
.
Accordingly, for strictly decreasing function the following inequality holds:
.
For non-decreasing:
.
For non-increasing:
.
It follows that a strictly increasing function is also non-decreasing. A strictly decreasing function is also non-increasing.
The function is called monotonous, if it is non-decreasing or non-increasing.
Theorem
Let the function not decrease on the interval where .
If it is bounded above by the number M: then there is a finite limit. If not limited from above, then .
If it is limited from below by the number m: then there is a finite limit. If not limited from below, then .
If points a and b are at infinity, then in the expressions the limit signs mean that .
This theorem can be formulated more compactly.
Let the function not decrease on the interval where . Then there are one-sided limits at points a and b:
;
.
A similar theorem for a non-increasing function.
Let the function not increase on the interval where . Then there are one-sided limits:
;
.
The proof of the theorem is presented on the page
"Limits of monotonic functions".
Function Definition
Function y = f (x) is a law (rule) according to which each element x of the set X is associated with one and only one element y of the set Y.
Element x ∈ X called function argument or independent variable.
Element y ∈ Y called function value or dependent variable.
The set X is called domain of the function.
Set of elements y ∈ Y, which have preimages in the set X, is called area or set of function values.
The actual function is called limited from above (from below), if there is a number M such that the inequality holds for all:
.
The number function is called limited, if there is a number M such that for all:
.
Top edge or accurate upper limit
A real function is called the smallest number that limits its range of values from above. That is, this is a number s for which, for everyone and for any, there is an argument whose function value exceeds s′: .
The upper bound of a function can be denoted as follows:
.
Respectively bottom edge or accurate lower limit
A real function is called the largest number that limits its range of values from below. That is, this is a number i for which, for everyone and for any, there is an argument whose function value is less than i′: .
The infimum of a function can be denoted as follows:
.
Used literature:
L.D. Kudryavtsev. Course of mathematical analysis. Volume 1. Moscow, 2003.
CM. Nikolsky. Course of mathematical analysis. Volume 1. Moscow, 1983.
For those who want to learn how to find limits, in this article we will talk about this. We won’t delve into the theory; teachers usually give it at lectures. So the “boring theory” should be jotted down in your notebooks. If this is not the case, then you can read textbooks borrowed from the library. educational institution or on other Internet resources.
So, the concept of a limit is quite important in studying a higher mathematics course, especially when you come across integral calculus and understand the connection between the limit and the integral. This material will look at simple examples, as well as ways to solve them.
Examples of solutions
| Example 1 |
| Calculate a) $ \lim_(x \to 0) \frac(1)(x) $; b)$ \lim_(x \to \infty) \frac(1)(x) $ |
| Solution |
|
a) $$ \lim \limits_(x \to 0) \frac(1)(x) = \infty $$ b)$$ \lim_(x \to \infty) \frac(1)(x) = 0 $$ People often send us these limits with a request to help solve them. We decided to highlight them as a separate example and explain that these limits just need to be remembered, as a rule. If you cannot solve your problem, then send it to us. We will provide detailed solution. You will be able to view the progress of the calculation and gain information. This will help you get your grade from your teacher in a timely manner! |
| Answer |
| $$ \text(a)) \lim \limits_(x \to 0) \frac(1)(x) = \infty \text( b))\lim \limits_(x \to \infty) \frac(1 )(x) = 0 $$ |
What to do with uncertainty of the form: $ \bigg [\frac(0)(0) \bigg ] $
| Example 3 |
| Solve $ \lim \limits_(x \to -1) \frac(x^2-1)(x+1) $ |
| Solution |
|
As always, we start by substituting the value $ x $ into the expression under the limit sign. $$ \lim \limits_(x \to -1) \frac(x^2-1)(x+1) = \frac((-1)^2-1)(-1+1)=\frac( 0)(0) $$ What's next now? What should happen in the end? Since this is uncertainty, this is not an answer yet and we continue the calculation. Since we have a polynomial in the numerators, we will factorize it using the formula familiar to everyone from school $$ a^2-b^2=(a-b)(a+b) $$. Do you remember? Great! Now go ahead and use it with the song :) We find that the numerator $ x^2-1=(x-1)(x+1) $ We continue to solve taking into account the above transformation: $$ \lim \limits_(x \to -1)\frac(x^2-1)(x+1) = \lim \limits_(x \to -1)\frac((x-1)(x+ 1))(x+1) = $$ $$ = \lim \limits_(x \to -1)(x-1)=-1-1=-2 $$ |
| Answer |
| $$ \lim \limits_(x \to -1) \frac(x^2-1)(x+1) = -2 $$ |
Let's push the limit in the last two examples to infinity and consider the uncertainty: $ \bigg [\frac(\infty)(\infty) \bigg ] $
| Example 5 |
| Calculate $ \lim \limits_(x \to \infty) \frac(x^2-1)(x+1) $ |
| Solution |
|
$ \lim \limits_(x \to \infty) \frac(x^2-1)(x+1) = \frac(\infty)(\infty) $ What to do? What should I do? Don't panic, because the impossible is possible. It is necessary to take out the x in both the numerator and the denominator, and then reduce it. After this, try to calculate the limit. Let's try... $$ \lim \limits_(x \to \infty) \frac(x^2-1)(x+1) =\lim \limits_(x \to \infty) \frac(x^2(1-\frac (1)(x^2)))(x(1+\frac(1)(x))) = $$ $$ = \lim \limits_(x \to \infty) \frac(x(1-\frac(1)(x^2)))((1+\frac(1)(x))) = $$ Using the definition from Example 2 and substituting infinity for x, we get: $$ = \frac(\infty(1-\frac(1)(\infty)))((1+\frac(1)(\infty))) = \frac(\infty \cdot 1)(1+ 0) = \frac(\infty)(1) = \infty $$ |
| Answer |
| $$ \lim \limits_(x \to \infty) \frac(x^2-1)(x+1) = \infty $$ |
Algorithm for calculating limits
So, let's briefly summarize the examples and create an algorithm for solving the limits:
- Substitute point x into the expression following the limit sign. If a certain number or infinity is obtained, then the limit is completely solved. IN otherwise we have uncertainty: “zero divided by zero” or “infinity divided by infinity” and move on to the next points of the instructions.
- To eliminate the uncertainty of “zero divided by zero,” you need to factor the numerator and denominator. Reduce similar ones. Substitute point x into the expression under the limit sign.
- If the uncertainty is “infinity divided by infinity,” then we take out both the numerator and the denominator x to the greatest degree. We shorten the X's. We substitute the values of x from under the limit into the remaining expression.
In this article, you learned the basics of solving limits, often used in the Calculus course. Of course, these are not all types of problems offered by examiners, but only the simplest limits. We'll talk about other types of assignments in future articles, but first you need to learn this lesson in order to move forward. Let's discuss what to do if there are roots, degrees, study infinitesimal equivalent functions, remarkable limits, L'Hopital's rule.
If you can't figure out the limits yourself, don't panic. We are always happy to help!
Limit of a function at a point and at
The limit of a function is the main apparatus of mathematical analysis. With its help, the continuity of a function, derivative, integral, and sum of a series are subsequently determined.
Let the function y=f(x)defined in some neighborhood of the point, except perhaps the point 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). Number b called limit of the function y=f(x) at the point (or at ), if for any sequence acceptable values argument converging to (i.e. ), the sequence of corresponding function values converges to the number b(ie).
In this case they write or at. The geometric meaning of the limit of a function: means that for all points X, sufficiently close to the point , the corresponding values of the function differ as little as desired from the number b.
Definition 2 (in "language e-d ", or according to Cauchy). Number b called limit of the function y=f(x) at the point (or for ), if for any positive number e there is a positive number d such that for all satisfying the inequality , the inequality .
Recorded.
This definition can be briefly written as follows:
Note that you can write it like this.
The geometric meaning of the limit of the function: if for any e-neighborhood of the point b there is such a d-neighborhood of the point , that for all from this d-neighborhood the corresponding values of the function f(x) lie in the e-neighborhood of the point b. In other words, the points on the graph of the function y=f(x) lie inside a strip of width 2e bounded by straight lines at = b+e, at = b- e (Figure 17). Obviously, the value of d depends on the choice of e, so we write d = d(e).
In determining the limit of a function it is assumed that X strives for in any way: remaining less than (to the left of ), greater than (to the right of ), or fluctuating around a point .
There are cases when the method of approximating an argument X To significantly affects the value of the function limit. Therefore, the concepts of one-sided limits are introduced.
Definition. The number is called limit of the function y=f(x) left at the point , if for any number e > 0 there is a number d = d(e) > 0 such that for , the inequality .
The limit on the left is written this way or briefly (Dirichlet notation) (Figure 18).
Defined similarly limit of the function on the right , let's write it using symbols:
Briefly, the limit on the right is denoted by .
The left and right limits of a function are called one-way limits . Obviously, if , then both one-sided limits exist, and .
The converse is also true: if both limits exist and and they are equal, then there exists a limit and .
If, then it does not exist.
Definition. Let the function y=f(x) is defined in the interval . Number b called limit of the function y=f(x) at X® ¥, if for any number e > 0 there is such a number M = M(e) > 0, which for all X, satisfying the inequality the inequality is satisfied. Briefly this definition can be written as follows:
If X® +¥, then write if X® -¥, then write , if = , then their general meaning usually denoted by .
The geometric meaning of this definition is as follows: for , that for and the corresponding values of the function y=f(x) fall into the e-neighborhood of the point b, i.e. the points of the graph lie in a strip of width 2e, bounded by straight lines and (Figure 19).
Infinitely large functions (b.b.f)
Infinitesimal functions (infinitesimal functions)
Definition. Function y=f(x) is called infinitely large at , if for any number M> 0 there is a number d = d( M) > 0, which is for everyone X, satisfying the inequality, the inequality is satisfied. Write or at .
For example, the function is b.b.f. at .
If f(x) tends to infinity at and takes only positive values, then write ; unless negative values, That .
Definition. Function y=f(x), defined on the entire numerical axis, is called infinitely large at , if for any number M> 0 there is such a number N = N(M) > 0, which is for everyone X satisfying the inequality, the inequality is written. Short:
For example, there is b.b.f. at .
Note that if the argument X, tending to infinity, takes only natural values, i.e. , then the corresponding b.b.f. becomes an infinitely large sequence. For example, the sequence is an infinitely large sequence. Obviously, any b.b.f. in the vicinity points is unlimited in this vicinity. The converse is not true: an unbounded function may not be b.b.f. (For example, )
However, if where b - final number, then the function f(x limited in the vicinity of the point.
Indeed, from the definition of the limit of a function it follows that when the condition is satisfied. Therefore, for , and this means that the function f(x) is limited.
Definition. Function y=f(x) is called infinitesimal at , If
By definition of the limit of a function, this equality means: for any number there is a number such that for all X satisfying the inequality, the inequality is satisfied.
The b.m.f. is determined similarly. at
: In all these cases.
Infinitesimal functions are often called infinitesimal quantities or infinitesimal ; usually denoted by the Greek letters a, b, etc.
Examples of b.m.f. serve functions when
Another example: - an infinitesimal sequence.
Example Prove that .
Solution . Feature 5+ X can be represented as the sum of the number 7 and b.m.f. X- 2 (at ), i.e. equality is satisfied. Therefore, by Theorem 3.4.6 we obtain .
Basic theorems about limits
Let's consider theorems (without proof) that make it easier to find the limits of a function. The formulation of the theorems for the cases when and is similar. In the theorems presented, we will assume that the limits exist.
Theorem 5.8 The limit of the sum (difference) of two functions is equal to the sum (difference) of their limits: .
Theorem 5.9 The limit of the product of two functions is equal to the product of their limits:
Note that the theorem is valid for the product of any finite number of functions.
Corollary 3 The constant factor can be taken beyond the limit sign: .
Corollary 4 The limit of a degree with a natural exponent is equal to the same degree of the limit: . In particular,
Theorem 5.10 The limit of a fraction is equal to the limit of the numerator divided by the limit of the denominator, unless the limit of the denominator is zero:
Example Calculate
Solution .
Example Calculate
Solution . Here the theorem on the limit of a fraction cannot be applied, because the limit of the denominator, at is equal to 0. In addition, the limit of the numerator is equal to 0. In such cases we say that we have type uncertainty. To expand it, we factorize the numerator and denominator of the fraction, then reduce it by:
Example Calculate
Solution . Here we are dealing with type uncertainty. To find the limit of a given fraction, divide the numerator and denominator by:
The function is the sum of the number 2 and b.m.f., therefore
Signs of Limits
Not every function, even a limited one, has a limit. For example, the function at has no limit. In many questions of analysis, it is enough just to verify the existence of a limit of a function. In such cases, signs of the existence of a limit are used.
The first and second remarkable limits
Definition. When calculating the limits of expressions containing trigonometric functions, the limit is often used
called the first remarkable limit .
It reads: the limit of the ratio of a sine to its argument is equal to one when the argument tends to zero.
Example Find
Solution . We have uncertainty of the form . The fraction limit theorem does not apply. Let us then denote at and
Example 3 Find
Solution.
Definition. Equalities are called second remarkable limit .
Comment. It is known that the limit of a number sequence
Has a limit equal to e: . The number e is called the Neper number. The number e is irrational, its approximate value is 2.72 (e = 2, 718281828459045...). Some properties of the number e make it especially convenient to choose this number as the base of logarithms. Logarithms to base e are called natural logarithms and are denoted by Note that
Let us accept without proof the statement that the function also tends to the number e
If you put it then follows. These equalities are widely used in calculating limits. In applications of analysis, an important role is played by the exponential function with base e. The function is called exponential, and the notation is also used
Example Find
Solution . We denote obviously, with We have
Calculation of limits
To reveal uncertainties of the form, it is often useful to apply the principle of replacing infinitesimals with equivalent ones and other properties of equivalent infinitesimal functions. As is known ~ x when ~ x at , because
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 of a limit:
Let's say there is some variable quantity. 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 concrete 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 in basic operations on matrices, 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 a 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.
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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 we have a quadratic equation in the numerator. 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, allowing to eliminate uncertainties of both types. 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.