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 a point x = a if for each sequence of values ​​of x that converges to point a, the sequence of values ​​of y approaches:

• L lim f(x) = L

Concept and properties of limits

What a limit is can be understood from an example. Suppose we have the function y=1/x. If we consistently increase the value of x and look at what y is equal to, we will get increasingly 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=1/x as x tends to ∞ is equal to 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 solving problems on finding limits:

• Amount limit equal to the sum limits: lim(x+y)=lim x+lim y
• Product limit equal to the product 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 →∞, has a limit equal to zero; for x→0, the limit is equal to ∞.

• lim (sin x)/x=1 x→0

We figured out the basic elementary functions.

When moving to functions more complex type we will certainly encounter the appearance of expressions whose meaning 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 multiplied by 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 OF UNCERTAINTY ARE NOT AND TAKE A COMPLETELY SPECIFIC FINITE OR INFINITE VALUE.

Uncover uncertainty allows:

• simplifying the form of a function (transforming an expression using abbreviated multiplication formulas, trigonometric formulas, multiplication by conjugate expressions followed by reduction, etc.);
• use of remarkable limits;
• application of L'Hopital's rule;
• using the replacement of an infinitesimal expression with its equivalent (using a table of equivalent infinitesimals).

Let's group the uncertainties into uncertainty table. For each type of uncertainty we associate a method for its disclosure (method of finding the limit).

This table, together with the table of limits of basic elementary functions, will be your main tools in finding any limits.

Let's give a couple of examples when everything works out immediately after substituting the value and uncertainty does not arise.

Example.

Calculate limit

Solution.

Substitute the value:

And we immediately received an answer.

Answer:

Example.

Calculate limit

Solution.

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 indicator. This is a power function. Let's refer to the table of limits for power functions with a negative indicator. From there we have And , therefore, we can write .

Based on this, our limit will be written as:

We turn again to the table of limits, but for exponential functions with a base greater than one, whence we have:

Answer:

Let's look at examples with detailed solutions Uncovering uncertainties by transforming expressions.

Very often the expression under the limit sign needs to be slightly transformed to get rid of uncertainties.

Example.

Calculate limit

Solution.

Substitute the value:

We have arrived at uncertainty. We look at the uncertainty table to select a solution method. Let's try to simplify the expression.

Answer:

Example.

Calculate limit

Solution.

Substitute the value:

We came to uncertainty (0 to 0). We look at the uncertainty table to choose a solution method and try to simplify the expression. Let's multiply both the numerator and the denominator by the expression conjugate to the denominator.

For the denominator the conjugate expression will be

We multiplied the denominator so that we could apply the abbreviated multiplication formula - difference of squares and then reduce the resulting expression.

After a series of transformations, the uncertainty disappeared.

Answer:

COMMENT: For limits of this type, the method of multiplying by conjugate expressions is typical, so feel free to use it.

Example.

Calculate limit

Solution.

Substitute the value:

We have arrived at uncertainty. We look at the uncertainty table to choose a solution method and try to simplify the expression. Since both the numerator and the denominator vanish at x = 1, then if these expressions are used, it will be possible to reduce (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:

Let's consider limits at infinity from power expressions. If the exponents of the power expression are positive, then the limit at infinity is infinite. Moreover, the greatest degree is of primary importance; the rest can be discarded.

Example.

Example.

If the expression under the limit sign is a fraction, and both the numerator and the denominator are power expressions (m is the power of the numerator, and n is the power of the denominator), then when an uncertainty of the form infinity to infinity arises, in this case uncertainty is revealed dividing both the numerator and denominator by

Example.

Calculate limit

This online math calculator will help you if you need it calculate the limit of a function. Program solution limits not only gives the answer to the problem, it leads detailed solution with explanations, i.e. displays the limit calculation process.

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Enter a function expression

Calculate limit
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A little theory.

Limit of the function at x->x 0

Let the function f(x) be defined on some set X and let the point \(x_0 \in X\) or \(x_0 \notin X\)
Let us take from X a sequence of points different from x 0:
x 1 , x 2 , x 3 , ..., x n , ... (1)
converging to x*. The function values ​​at the points of this sequence also form a numerical sequence
f(x 1), f(x 2), f(x 3), ..., f(x n), ... (2)

and one can raise the question of the existence of its limit.. The number A is called the limit of the function f(x) at the point x = x 0 (or at x -> x 0), if for any sequence (1) of values ​​of the argument x different from x 0 converging to x 0, the corresponding sequence (2) of values function converges to number A.

$$ \lim_(x\to x_0)( f(x)) = A $$

The function f(x) can have only one limit at the point x 0. This follows from the fact that the sequence
(f(x n)) has only one limit.

There is another definition of the limit of a function.

and one can raise the question of the existence of its limit. The number A is called the limit of the function f(x) at the point x = x 0 if for any number \(\varepsilon > 0\) there is a number \(\delta > 0\) such that for all \(x \in X, \; x \neq x_0 \), satisfying the inequality \(|x-x_0| Using logical symbols, this definition can be written as
\((\forall \varepsilon > 0) (\exists \delta > 0) (\forall x \in X, \; x \neq x_0, \; |x-x_0| Note that the inequalities \(x \neq x_0 , \; |x-x_0| The first definition is based on the concept of the limit of a number sequence, so it is often called the “in the language of sequences” definition. The second definition is called the “in the language of \(\varepsilon - \delta \)”.
These two definitions of the limit of a function are equivalent and you can use either of them depending on which is more convenient for solving a particular problem.

Note that the definition of the limit of a function “in the language of sequences” is also called the definition of the limit of a function according to Heine, and the definition of the limit of a function “in the language \(\varepsilon - \delta \)” is also called the definition of the limit of a function according to Cauchy.

Limit of the function at x->x 0 - and at x->x 0 +

In what follows, we will use the concepts of one-sided limits of a function, which are defined as follows.

and one can raise the question of the existence of its limit. The number A is called the right (left) limit of the function f(x) at the point x 0 if for any sequence (1) converging to x 0, the elements x n of which are greater (less than) x 0, the corresponding sequence (2) converges to A.

Symbolically it is written like this:
$$ \lim_(x \to x_0+) f(x) = A \; \left(\lim_(x \to x_0-) f(x) = A \right) $$

We can give an equivalent definition of one-sided limits of a function “in the language \(\varepsilon - \delta \)”:

and one can raise the question of the existence of its limit. a number A is called the right (left) limit of the function f(x) at the point x 0 if for any \(\varepsilon > 0\) there exists \(\delta > 0\) such that for all x satisfying the inequalities \(x_0 Symbolic entries:

\((\forall \varepsilon > 0) (\exists \delta > 0) (\forall x, \; x_0

Limit of a function at infinity:
|f(x) - a|< ε при |x| >N

Determination of the Cauchy limit
Let the function f (x) is defined in a certain neighborhood of the point at infinity, with |x| > The number a is called the limit of the function f (x) with x tending to infinity (), if for any, however small positive number ε > 0 , there is a number N ε >K, depending on ε, which for all x, |x| > N ε, the function values ​​belong to the ε-neighborhood of point a:
|f (x) - a|< ε .
The limit of a function at infinity is denoted as follows:
.
Or at .

The following notation is also often used:
.

Let's write this definition using the logical symbols of existence and universality:
.
This assumes that the values ​​belong to the domain of the function.

One-sided limits

Left limit of a function at infinity:
|f(x) - a|< ε при x < -N

There are often cases where a function is defined only for positive or negative values variable x (more precisely in the vicinity of the point or ). Also, the limits at infinity for positive and negative values ​​of x can have different meanings. Then one-sided limits are used.

Left limit at infinity or the limit as x tends to minus infinity () is defined as follows:
.
Right limit at infinity or the limit as x tends to plus infinity ():
.
One-sided limits at infinity are often denoted as follows:
; .

Infinite limit of a function at infinity

Infinite limit of a function at infinity:
|f(x)| > M for |x| >N

Definition of the infinite limit according to Cauchy
Let the function f (x) is defined in a certain neighborhood of the point at infinity, with |x| > K, where K is a positive number. Limit of function f (x) as x tends to infinity (), is equal to infinity, if for anyone, arbitrarily large number M > 0 , there is such a number N M >K, depending on M, which for all x, |x| > N M , the function values ​​belong to the neighborhood of the point at infinity:
|f (x) | > M.
The infinite limit as x tends to infinity is denoted as follows:
.
Or at .

Using the logical symbols of existence and universality, the definition of the infinite limit of a function can be written as follows:
.

Similarly, definitions of infinite limits of certain signs equal to and are introduced:
.
.

Definitions of one-sided limits at infinity.
Left limits.
.
.
.
Right limits.
.
.
.

Determination of the limit of a function according to Heine

Let the function f (x) defined at some neighborhood of a 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 (xn), converging to x 0 : ,
whose elements belong to the neighborhood, sequence (f(xn)) converges to a:
.

If we take as a neighborhood the neighborhood of an unsigned point at infinity: , then we obtain the definition of the limit of a function as x tends to infinity, . 0 If we take a left-sided or right-sided neighborhood of the point x at infinity

: or , then we obtain the definition of the limit as x tends to minus infinity and plus infinity, respectively. Definitions of the limit according to Heine and Cauchy.

equivalent

Examples

Example 1
.

Using Cauchy's definition to show that
.
Let us introduce the following notation: Let's find the domain of definition of the function.. ;
.
Since the numerator and denominator of the fraction are polynomials, the function is defined for all x except the points at which the denominator vanishes. Let's find these points. Let's decide
; .
quadratic equation
Roots of the equation:

Since , then and .
.
Therefore the function is defined at .
.
We will use this later. -1 :
.

Let us write down the definition of the finite limit of a function at infinity according to Cauchy:
Let's transform the difference:
;
;
;
.

Divide the numerator and denominator by and multiply by
.
.
Let .
Then

So, we found that when ,
It follows that
at , and .

Since you can always increase it, let's take .

Let us write down the definition of the finite limit of a function at infinity according to Cauchy:
Then for anyone,
1) ;
2) .

at .

It means that .
Example 2
.

Using the Cauchy definition of a limit, show that:
;
.

Divide the numerator and denominator by and multiply by
.
1) Solution as x tends to minus infinity
.
Since , the function is defined for all x.
.

Let us write down the definition of the limit of a function at equal to minus infinity:

Let . Then

Enter positive numbers and :
.
It follows that for any positive number M, there is a number, so that for ,

.
It means that .
.

2) Solution as x tends to plus infinity
Therefore the function is defined at .
.
Let's transform the original function. Multiply the numerator and denominator of the fraction by and apply the difference of squares formula:
.

We have:
.
Let's transform the difference:
;
.

Divide the numerator and denominator by and multiply by
.
1) Solution as x tends to minus infinity
.
Let .
Let us write down the definition of the right limit of the function at:

Let us introduce the notation: .
.

Multiply the numerator and denominator by:
Let at and . Since this holds for any positive number, then