Limit in simple words. Determining the limit of a function at infinity
Function y=f (x) the law (rule) is called, 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.
y element ∈ Y called function value or dependent variable.
The set X is called function scope.
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 such a number M that the following inequality holds for all:
.
The number function is called limited, if there exists a number M such that for all :
.
top face or exact upper bound real function name the smallest of the numbers that limits the range of its values \u200b\u200bfrom above. That is, this is a number s for which for all and for any , there is such an argument, the value of the function of which exceeds s′ : .
The upper bound of the function can be denoted as follows:
.
Respectively bottom face or precise lower bound real function is called the largest of the numbers that limits the range of its values from below. That is, this is such a number i , for which for all and for any , there is such an argument , the function value of which is less than i′ : .
The lower bound of a function can be denoted as follows:
.
Determining the limit of a function
Definition of the Cauchy limit of a function
Finite function limits at endpoints
Let the function be defined in some neighborhood end point except perhaps for the dot itself. at the point , if for any there exists such , depending on , that for all x , for which , the inequality
.
The limit of a function is denoted as follows:
.
Or at .
Using the logical symbols of existence and universality, the definition of the limit of a function can be written as follows:
.
Unilateral limits.
Left limit at point (left-side limit):
.
Right limit at a point (right-hand limit):
.
The limits on the left and right are often denoted as follows:
;
.
Finite limits of a function at points at infinity
Limits at infinitely distant points are defined in a similar way.
.
.
.
They are often referred to as:
;
;
.
Using the concept of a neighborhood of a point
If we introduce the concept of a punctured neighborhood of a point , then we can give a unified definition of the finite limit of a function at finite and at infinity points:
.
Here for endpoints
;
;
.
Any neighborhoods of points at infinity are punctured:
;
;
.
Infinite function limits
Definition
Let the function be defined in some punctured neighborhood of a point (finite or at infinity). Limit of function f (x) as x → x 0
equals infinity, if for any, arbitrarily a large number M > 0
, there exists a number δ M > 0
, depending on M , that for all x belonging to a punctured δ M - neighborhood of the point : , the following inequality holds:
.
The infinite limit is defined as follows:
.
Or at .
Using the logical symbols of existence and universality, the definition of the infinite limit of a function can be written as follows:
.
It is also possible to introduce definitions of infinite limits of certain signs equal to and :
.
.
Universal definition of the limit of a function
Using the concept of a neighborhood of a point, one can give a universal definition of the finite and infinite limit of a function, applicable both to finite (two-sided and one-sided) and to infinitely distant points:
.
Definition of the limit of a function according to Heine
Let the function be defined on some set X : .
The number a is called the limit of the function at point :
,
if for any sequence converging to x 0
:
,
whose elements belong to the set X : ,
.
We write this definition using the logical symbols of existence and universality:
.
If we take as the set X the left-hand neighborhood of the point x 0 , then we get the definition of the left limit. If it is right-handed, then we get the definition of the right limit. If we take the neighborhood of a point at infinity as the set X, then we obtain the definition of the limit of a function at infinity.
Theorem
The Cauchy and Heine definitions of the limit of a function are equivalent.
Proof
Properties and theorems of the limit of a function
Further, we assume that the functions under consideration are defined in the corresponding 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.
Basic properties
If the values of the function f (x) change (or make undefined) at 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 such a punctured neighborhood of the point x 0
, on which the function f (x) limited:
.
Let the function have at the point x 0
end limit other than zero:
.
Then, for any number c from the interval , there exists 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
,
then .
If , and on some neighborhood of the point
,
then .
In particular, if on some neighborhood of a point
,
then if , then and ;
if , then and .
If on some punctured neighborhood of the point x 0
:
,
and there are finite (or infinite of a certain sign) equal limits:
, then
.
Proofs of the main properties are given on the page
"Basic Properties of the Limits of a Function".
Arithmetic 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 limits 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:
.
Complex function limit
Limit theorem complex function
Let the function have a limit and map the punctured neighborhood of the point onto the punctured neighborhood of the point . Let the function be defined on this neighborhood and have a limit on it.
Here - final or infinitely distant points: . Neighborhoods and their corresponding limits can be either two-sided or one-sided.
Then there is a limit of the complex function and it is equal to:
.
The complex function limit theorem applies when the function is not defined at a point or has a value other than the limit value. To apply this theorem, there must be a punctured neighborhood of the point on which 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 of the continuous function:
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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 (t) as t → t 0
, and it is equal to x 0
:
.
Here point t 0
can be finite or at infinity: .
And let the function f (x) continuous at x 0
.
Then there is a limit of the composite function f (g(t)), and it is equal to f (x0):
.
The proofs of the theorems are given on the page
"The Limit and Continuity of a Complex Function".
Infinitesimal and infinitely large functions
Infinitely small functions
Definition
A function is called infinitesimal for if
.
Sum, difference and product of a finite number of infinitely small functions for is an infinitesimal function for .
The product of a function bounded on some punctured neighborhood of the point , to an infinitesimal for is an infinitesimal function of for .
For a function to have a finite limit, it is necessary and sufficient that
,
where is an infinitesimal function for .
"Properties of infinitesimal functions".
Infinitely large functions
Definition
The function is called infinitely large for 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 at , 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 infinitely small for:
, and (on some punctured neighborhood of the point ), then
.
Proofs of properties are set out in the section
"Properties of infinitely large functions".
Relationship between infinitely large and infinitely small functions
The connection between infinitely large and infinitely small functions follows from the two previous properties.
If the function is infinitely large at , then the function is infinitely small at .
If the function is infinitely small 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 definite sign at , that is, it is positive (or negative) on some punctured neighborhood of the point , then this fact can be expressed as follows:
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Similarly, if an infinitely large function has a certain sign at , then they write:
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Then the symbolic connection between infinitely small and infinitely large functions can be supplemented by the following relations:
,
,
,
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Additional formulas relating infinity symbols can be found on the page
"Points at infinity and their properties".
Limits of monotonic functions
Definition
A function defined on some set of real numbers X is called strictly increasing, if for all such that the following inequality holds:
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Accordingly, for strictly decreasing function, the following inequality holds:
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For non-decreasing:
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For non-increasing:
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This implies that a strictly increasing function is also nondecreasing. A strictly decreasing function is also nonincreasing.
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 from above by the number M : , then there is a finite limit . If not bounded above, then .
If it is bounded from below by the number m : , then there is a finite limit . If not bounded below, then .
If the 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:
;
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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 stated on the page
"Limits of Monotonic Functions".
References:
L.D. Kudryavtsev. Well mathematical analysis. Volume 1. Moscow, 2003.
CM. Nikolsky. Course of mathematical analysis. Volume 1. Moscow, 1983.
The main elementary functions have been sorted out.
When moving to functions more complex type we will definitely encounter expressions whose value is not defined. Such expressions are called uncertainties.
Let's list everything main types of uncertainties: zero divided by zero (0 by 0), infinity divided by infinity, zero times infinity, infinity minus infinity, one to the power of infinity, zero to the power of zero, infinity to the power of zero.
ALL OTHER EXPRESSIONS ARE NOT UNCERTAINTY AND TAKE A COMPLETELY SPECIFIC FINITE OR INFINITE VALUE.
Reveal Uncertainties allows:
- simplification of the form of a function (transformation of an expression using abbreviated multiplication formulas, trigonometric formulas, multiplication by conjugate expressions followed by reduction, etc.);
- usage wonderful limits;
- application of L'Hospital's rule;
- the use of replacing an infinitesimal expression with its equivalent (using a table of equivalent infinitesimals).
We group the uncertainties into uncertainty table. For each type of uncertainty, we put in correspondence the method of its disclosure (the method of finding the limit).
This table, together with the table of limits of the basic elementary functions, will be your main tools in finding any limits.
Let's give a couple of examples when everything is immediately obtained after substituting the value and uncertainty does not arise.
Example.
Calculate Limit 
Decision.
We substitute the value: 
And we got an answer right away.
Answer:
Example.
Calculate Limit 
Decision.
We substitute the value x=0 into the base of our exponential power function: 
That is, the limit can be rewritten as 
Now let's take a look at the index. This is a power function. Let us turn to the table of limits for power functions with a negative exponent. From there we have 
and 
, therefore, we can write 
.
Based on this, our limit can be written as: 
Again we turn to the table of limits, but for exponential functions with a base greater than one, from which we have:
Answer:
Let's look at examples with detailed solutions disclosure of ambiguities by transforming expressions.
Very often, the expression under the limit sign needs to be slightly transformed to get rid of ambiguities.
Example.
Calculate Limit
Decision.
We substitute the value: 
Came to uncertainty. We look at the table of uncertainties to choose a solution method. Let's try to simplify the expression. 
Answer:
Example.
Calculate Limit 
Decision.
We substitute the value: 
Came to uncertainty (0 by 0). We look at the table of uncertainties to select a solution method and try to simplify the expression. We multiply both the numerator and the denominator by the expression conjugate to the denominator.
For the denominator, the adjoint expression is 
We multiplied the denominator so that we could apply the abbreviated multiplication formula - the difference of squares and then reduce the resulting expression. 
After a series of transformations, the uncertainty disappeared.
Answer:
COMMENT: for limits of this kind, the method of multiplication by conjugate expressions is typical, so feel free to use it.
Example.
Calculate Limit 
Decision.
We substitute the value: 
Came to uncertainty. We look at the table of uncertainties to select a solution method and try to simplify the expression. Since both the numerator and denominator vanish at x=1, if these expressions can be reduced (x-1) and the uncertainty will disappear.
Let's factorize the numerator: 
Let's factorize the denominator: 
Our limit will take the form: 
After the transformation, the uncertainty was revealed.
Answer:
Consider the limits at infinity of power expressions. If the exponents of the exponential expression are positive, then the limit at infinity is infinite. Moreover, the main value has the greatest degree, the rest can be discarded.
Example.
Example.
If the expression under the limit sign is a fraction, and both the numerator and denominator are power expressions(m is the degree of the numerator, and n is the degree of the denominator), then when there is an uncertainty of the form infinity to infinity, in this case uncertainty is revealed division and numerator and denominator by
Example.
Calculate Limit 
Function limit at infinity:
|f(x) - a|< ε
при |x| >N
Definition of the Cauchy limit
Let the function f (x) is defined in some neighborhood of a point at infinity, for |x| > The number a is called the limit of the function f (x) for x tending to infinity (), if for any, arbitrarily small positive number ε > 0
, there exists a number N ε > K, depending on ε , such that for all x, |x| > N ε , the values of the function belong to the ε neighborhood of the point a :
|f (x) - a|< ε
.
The limit of a function at infinity is denoted as follows:
.
Or at .
The following notation is also often used:
.
We write this definition using the logical symbols of existence and universality:
.
Here it is assumed that the values belong to the scope of the function.
One-sided limits
Left limit of the function at infinity:
|f(x) - a|< ε
при x < -N
Often there are cases when a function is defined only for positive or negative values variable x (more precisely, in the vicinity of the point or ). Also limits at infinity for positive and negative values of x can have various meanings. Then one-sided limits are used.
Left limit at infinity or the limit as x tends to minus infinity () is defined as follows:
.
Right limit at infinity or limit as x tends to plus infinity () :
.
One-sided limits at infinity are often written like this:
;
.
Infinite function limit at infinity
Infinite function limit at infinity:
|f(x)| > M for |x| > N
Definition of the infinite limit according to Cauchy
Let the function f (x) is defined in some neighborhood of a point at infinity, for |x| > K , where K is a positive number. Limit of function f (x) when x tends to infinity (), is equal to infinity, if for any arbitrarily large number M > 0
, there exists a number N M > K, depending on M , such that for all x, |x| > N M , the values of the function belong to the neighborhood of the point at infinity:
|f (x) | > M.
The infinite limit as x tends to infinity is denoted as follows:
.
Or at .
Using the logical symbols of existence and universality, the definition of the infinite limit of a function can be written as follows:
.
The definitions of the infinite limits of certain signs equal to and are introduced similarly:
.
.
Definitions of one-sided limits at infinity.
Left limits.
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.
.
Right limits.
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.
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Definition of the limit of a function according to Heine
Let the function f (x) is defined on some neighborhood of the point at infinity x 0
, where or or .
The number a (finite or at infinity) is called the limit of the function f (x) at point x 0
:
,
if for any sequence ( x n ), converging to x 0
:
,
whose elements belong to the neighborhood , the sequence (f(xn)) converges to a :
.
If we take the neighborhood of an unsigned point at infinity as a neighborhood: , then we obtain the definition of the limit of the function as x tends to infinity, . If we take the left-hand or right-hand neighborhood of the point at infinity x 0 : or , then we get the definition of the limit as x tends to minus infinity and plus infinity, respectively.
The Heine and Cauchy definitions of the limit are equivalent.
Examples
Example 1
Using the Cauchy definition, show that
.
Let us introduce the notation:
.
Find the domain of the function . Since the numerator and denominator of a fraction are polynomials, the function is defined for all x except for the points where the denominator vanishes. Let's find these points. We solve a quadratic equation. ;
.
Equation roots:
;
.
Since , then and .
Therefore, the function is defined for . This we will use in the future.
We write out the definition of the finite limit of a function at infinity according to Cauchy:
.
Let's transform the difference:
.
Divide the numerator and denominator by and multiply by -1
:
.
Let be .
Then
;
;
;
.
So, we have found that at ,
.
.
Hence it follows that
at , and .
Since it is always possible to increase, we take . Then for any ,
at .
It means that .
Example 2
Let be .
Using the definition of the Cauchy limit, show that:
1)
;
2)
.
1) Solution for x tending to minus infinity
Since , then the function is defined for all x .
Let us write out the definition of the limit of the function at equal to minus infinity:
.
Let be . Then
;
.
So, we have found that at ,
.
We enter positive numbers and:
.
It follows that for any positive number M , there is a number , so that for ,
.
It means that .
2) Solution for x tending to plus infinity
Let's transform the original function. Multiply the numerator and denominator of the fraction and apply the difference of squares formula:
.
We have:
.
Let us write the definition of the right limit of the function for :
.
Let's introduce the notation: .
Let's transform the difference:
.
Multiply the numerator and denominator by:
.
Let be
.
Then
;
.
So, we have found that at ,
.
We enter positive numbers and:
.
Hence it follows that
at and .
Since this holds for any positive number, then
.
References:
CM. Nikolsky. Course of mathematical analysis. Volume 1. Moscow, 1983.
There is such a thing in mathematics as the limit of a function. To understand how to find limits, you need to remember the definition of the limit of a function: a function f (x) has a limit L at the point x = a, if for each sequence of x values \u200b\u200bconverging to a point a, the sequence of y values approaches:
- L lim f(x) = L
The concept and properties of limits
What is a limit can be understood from an example. Suppose we have a function y=1/x. If we consistently increase the value of x and look at what y is equal to, we will get ever-decreasing values: at x=10000 y=1/10000; at x=1000000 y=1/1000000. Those. the more x, the less y. If x = ∞, y will be so small that it can be considered equal to 0. Thus, the limit of the function y \u003d 1 / x as x tends to ∞ is 0. It is written like this:
- lim1/х=0
The limit of a function has several properties that you need to remember: this will greatly facilitate the solution of problems to find limits:
- The sum limit is equal to the sum of the limits: lim(x+y)=lim x+lim y
- The limit of the product is equal to the product of the limits: lim(xy)=lim x*lim y
- The limit of the quotient is equal to the quotient of the limits: lim(x/y)=lim x/lim y
- The constant factor is taken out of the limit sign: lim(Cx)=C lim x
The function y=1 /x, in which x →∞, the limit is zero, when x→0, the limit is ∞.
- lim (sin x)/x=1 x→0