The number of discontinuity points of a function is equal to the online calculator. Calculate function limits online

On this page we have tried to collect for you the most complete information about the study of the function. No more Googling! Just read, study, download, follow selected links.

General design of the study

What is it for? this research, you ask, if there are many services that will be built for the most sophisticated functions? In order to find out the properties and features of a given function: how it behaves at infinity, how quickly it changes sign, how smoothly or sharply it increases or decreases, where the “humps” of the convexity are directed, where the values are not defined, etc.

And on the basis of these “features” the layout of the graph is built - a picture, which is actually secondary (although for educational purposes it is important and confirms the correctness of your decision).

Let's start, of course, with plan. Function study - volumetric task(perhaps the most voluminous of the traditional higher mathematics courses, usually from 2 to 4 pages, including the drawing), therefore, in order not to forget what to do in what order, we follow the points described below.

Algorithm

Find the domain of definition. Select special points (break points).
Check for the presence of vertical asymptotes at discontinuity points and at the boundaries of the definition area.
Find the points of intersection with the coordinate axes.
Determine whether a function is even or odd.
Determine whether a function is periodic or not (trigonometric functions only).
Find extremum points and monotonicity intervals.
Find inflection points and convex-concave intervals.
Find oblique asymptotes. Investigate behavior at infinity.
Select additional points and calculate their coordinates.
Construct a graph and asymptotes.

In different sources (textbooks, manuals, lectures by your teacher), the list may have a different form from this one: some items are swapped, combined with others, shortened or removed. Please consider your teacher's requirements/preferences when making your decision.

Study diagram in pdf format: download.

Full example solution online

Conduct a complete study and plot the function $$ y(x)=\frac(x^2+8)(1-x). $$

1) The domain of the function. Since the function is a fraction, we need to find the zeros of the denominator. $$1-x=0, \quad \Rightarrow \quad x=1.$$ We exclude the only point $x=1$ from the domain of definition of the function and get: $$ D(y)=(-\infty; 1) \cup (1;+\infty). $$

2) Let us study the behavior of the function in the vicinity of the discontinuity point. Let's find one-sided limits:

Since the limits are equal to infinity, the point $x=1$ is a discontinuity of the second kind, the straight line $x=1$ is a vertical asymptote.

3) Determine the points of intersection of the function graph with the coordinate axes.

Let's find the points of intersection with the ordinate axis $Oy$, for which we equate $x=0$:

Thus, the point of intersection with the $Oy$ axis has coordinates $(0;8)$.

Let's find the points of intersection with the abscissa axis $Ox$, for which we set $y=0$:

The equation has no roots, so there are no points of intersection with the $Ox$ axis.

Note that $x^2+8>0$ for any $x$. Therefore, for $x \in (-\infty; 1)$ the function $y>0$ (takes positive values, the graph is above the x-axis), for $x \in (1; +\infty)$ the function $y\lt 0$ (takes negative values, the graph is below the x-axis).

4) The function is neither even nor odd, since:

5) We examine the function for periodicity. The function is not periodic, since it is a fractional rational function.

6) We examine the function for extrema and monotonicity. To do this, we find the first derivative of the function:

Let's equate the first derivative to zero and find stationary points (at which $y"=0$):

We got three critical points: $x=-2, x=1, x=4$. Let us divide the entire domain of definition of the function into intervals with these points and determine the signs of the derivative in each interval:

For $x \in (-\infty; -2), (4;+\infty)$ the derivative $y" \lt 0$, so the function decreases on these intervals.

When $x \in (-2; 1), (1;4)$ the derivative $y" >0$, the function increases on these intervals.

In this case, $x=-2$ is a local minimum point (the function decreases and then increases), $x=4$ is a local maximum point (the function increases and then decreases).

Let's find the values of the function at these points:

Thus, the minimum point is $(-2;4)$, the maximum point is $(4;-8)$.

7) We examine the function for kinks and convexity. Let's find the second derivative of the function:

Let us equate the second derivative to zero:

The resulting equation has no roots, so there are no inflection points. Moreover, when $x \in (-\infty; 1)$ is satisfied $y"" \gt 0$, that is, the function is concave, when $x \in (1;+\infty)$ is satisfied $y"" \ lt 0$, that is, the function is convex.

8) Let us examine the behavior of the function at infinity, that is, at .

Since the limits are infinite, there are no horizontal asymptotes.

Let's try to determine oblique asymptotes of the form $y=kx+b$. We calculate the values of $k, b$ using known formulas:

We found that the function has one oblique asymptote $y=-x-1$.

9) Additional points. Let's calculate the value of the function at some other points in order to more accurately construct the graph.

$$ y(-5)=5.5; \quad y(2)=-12; \quad y(7)=-9.5. $$

10) Based on the data obtained, we will construct a graph, supplement it with asymptotes $x=1$ (blue), $y=-x-1$ (green) and mark characteristic points (purple intersection with the ordinate axis, orange extrema, black additional points):

Examples of function exploration solutions

Various functions (polynomials, logarithms, fractions) have its own characteristics during research(discontinuities, asymptotes, number of extrema, limited domain of definition), so here we tried to collect examples from control ones for studying functions of the most common types. Have fun learning!

Task 1. Investigate a function using differential calculus methods and construct a graph.

$$y=\frac(e^x)(x).$$

Task 2. Explore the function and build its graph.

$$y=-\frac(1)(4)(x^3-3x^2+4).$$

Task 3. Explore a function using its derivative and plot a graph.

$$y=\ln \frac(x+1)(x+2).$$

Task 4. Conduct a complete study of the function and draw a graph.

$$y=\frac(x)(\sqrt(x^2+x)).$$

Task 5. Investigate the function using differential calculus and construct a graph.

$$y=\frac(x^3-1)(4x^2).$$

Task 6. Examine the function for extrema, monotonicity, convexity and construct a graph.

$$y=\frac(x^3)(x^2-1).$$

Task 7. Conduct a study of the function by plotting a graph.

$$y=\frac(x^3)(2(x+5)^2).$$

How to build a chart online?

Even if the teacher requires you to turn in an assignment, handwritten, with a drawing on a piece of paper in a box, it will be extremely useful for you, during the decision, to build a graph in a special program (or service) in order to check the progress of the solution, compare its appearance with what is obtained manually, and perhaps find errors in your calculations (when the graphs clearly behave differently).

Below you will find several links to sites that allow you to build convenient, fast, beautiful and, of course, free graphics for almost any function. In fact, there are many more such services, but is it worth looking if the best ones are chosen?

Desmos graphing calculator

The second link is practical, for those who want to learn how to build beautiful charts in Desmos.com (see description above): Complete instructions for working with Desmos. This instruction is quite old, since then the site interface has changed for the better, but the basics have remained unchanged and will help you quickly understand the important functions of the service.

Official instructions, examples and video instructions in English can be found here: Learn Desmos.

Reshebnik

Need a completed task urgently? More than a hundred different functions with full research are already waiting for you. Detailed solution, fast payment via SMS and low price - approx. 50 rubles. Maybe your task is already ready? Check it out!

Useful videos

Webinar on working with Desmos.com. This is already a full review of the site’s functions, for as much as 36 minutes. Unfortunately, it is in English, but basic knowledge of the language and attentiveness is enough to understand most of it.

Cool old popular science film "Mathematics. Functions and Graphs." Explanations at your fingertips in the literal sense of the word, the very basics.

Practical work No. 3

Investigation of a function for continuity

Purpose of the work: Develop and improve the ability to determine the continuity of a function, find break points of a function, consolidate the skill of calculating limits

Learning Tools: textbook Mathematics pp. 62-71, handouts, workbook on mathematics.

Form: frontal.

Reference material

Definition : The function f (x) is called continuous at x0 if:

1) there is a function value at point f (x 0)

2) there is a finite limit at the point x0

3) the limit is equal to the value of the function at point x0

Definition : Function is continuous on the interval, if it is continuous at all points of this interval.

Definition : If at any point x0 function at = f (x) is not continuous , then point x0 called break point this function, and function y = f (x) called explosive at this point.

Discontinuity points of the 1st kind

Point x=1 removable break point

=-1

Discontinuity points of the 2nd kind

Operating procedure:

Task 1.

a) y=x2+3 at point x=-2

Solution:

y (-2)=(-2)2+3=7

, the function is continuous at the point x=-2

b) y=at point x=2

Solution:

, the function is continuous at the point x=2

Task 2.

solution

The function is indefinite at the point x=2, therefore the function at this point is not continuous and suffers a discontinuity.Let's plot the function:

Let's find one-sided limits at the point x=2:

https://pandia.ru/text/79/377/images/image027_20.gif" width="93" height="29 src=">, since one-sided limits are finite and equal, then point x = 2 point 1st kind rupture (point of reparable rupture)

solution

Let's plot the function:

https://pandia.ru/text/79/377/images/image030_17.gif" width="89" height="29 src=">.gif" width="36" height="41">

solution

The function is indefinite at the point x = -1, therefore the function at this point is not continuous and suffers a discontinuity.Let's plot the function:

Let's find one-sided limits at point x=-1:

https://pandia.ru/text/79/377/images/image035_13.gif" width="111" height="41 src="> since there is no final limit, then point x = -1 point rupture of the 2nd kind.

Self-administered task

Task 3. Based on the definition of a continuous function, prove the continuity of these functions at the indicated points

A) y=2x2+1 at point x=1

b) y=at point x=-1

Task 4. Examine functions for continuity. Find break points and determine their type.

Security questions:

The concept of continuity of a function at a point. Continuity of a function on an interval. Types of function break points. Examples.

Summing up the work: Analysis of completed tasks.

Evaluation criteria:

"5"-correct completion of tasks 3 (a, b), 4 (a, b, c)

"4"-correct execution of any 4 examples parts yourself.

"3"-completing tasks 1(a, b), 2(a, b, c)

Main sources :

Grigoriev. M., Academy, 2013.

Bogomolov: textbook. For Suz. -M.: Bustard, 2009. -395 p.

Additional sources

Bugrov S. M. Differential and integral calculus. High School 1990

Mathematical analysis in questions and problems. High School 1987

Govorov P. T. Collection of competition problems in mathematics. Academy 2000

Higher mathematics in exercises and problems. Academy 2001

Pekhletsky I. D..Mathematics. Academy 2001

Collection of problems in mathematics: Textbook for secondary specialized educational institutions. Academy 2004

Application

Limits online on the site for students and schoolchildren to fully consolidate the material they have covered. How to find the limit online using our resource? This is very easy to do, you just need to correctly write the original function with the variable x, select the desired infinity from the selector and click the “Solve” button. In the case where the limit of a function must be calculated at some point x, then you need to indicate the numerical value of this very point. You will receive an answer to the solution of the limit in a matter of seconds, in other words - instantly. However, if you provide incorrect data, the service will automatically notify you of the error. Correct the previously introduced function and obtain the correct solution to the limit. To solve limits, all possible techniques are used, L'Hopital's method is especially often used, since it is universal and leads to an answer faster than other methods of calculating the limit of a function. 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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.

This program can be useful for high school students in general education schools when preparing for tests and exams, when testing knowledge before the Unified State Exam, and for parents to control the solution of many problems in mathematics and algebra. Or maybe it’s too expensive for you to hire a tutor or buy new textbooks? Or do you just want to get your math or algebra homework done as quickly as possible? In this case, you can also use our programs with detailed solutions.

In this way, you can conduct your own training and/or training of your younger brothers or sisters, while the level of education in the field of solving problems increases.

Enter a function expression
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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.

Definition. 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.

Definition 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.

Definition 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 $”:

Definition 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

Educational institution "Belarusian State

Agricultural Academy"

Department of Higher Mathematics

Guidelines

to study the topic “Continuity of functions of one variable”

students of the accounting faculty in correspondence form

education (NISPO)

Gorki, 2013

Continuity of functions of one variable

One-sided limits

Let the function
defined on the set
. Let us introduce the concept of one-sided limits of a function at
. We will consider the following values X, What
. This means that
, remaining all the time to the left of
at
then it's called left limit this function at the point (or when
) and is denoted

Let it now
, remaining all the time to the right of , i.e. staying longer . If there is a limit of the function
, then it is called right limit this function at the point and is designated

The left and right limits are called one-way limits functions at a point.

If there are one-sided limits of a function at a point and they are equal to each other, then the function has the same limit at this point:

If one-sided limits of a function at a point exist, but are not equal to each other, then the limit of the function at this point does not exist .

Continuity of a function at a point

Let the function
defined on some set D. Let the independent variable X goes from one of its (initial) values
to another (final) value . The difference between the final and initial values is called increment quantities X and is designated
. The increment can be either positive or negative. In the first case, the value X when moving from To X increases, and in the second case - decreases.

If the independent variable X gets some increment
, then the function
gets increment
. Because
, That.

Function increment
at the point is called the difference, where
– increment of the independent variable.

Several definitions of the continuity of a function at a point can be given.

The function is called continuous in the interval , if it is continuous at every point of this interval. Geometrically continuity of a function
in a closed interval means that the graph of the function is a continuous line without breaks.

Functions that are continuous on an interval have important properties that are expressed by the following statements.

If the function
is continuous on the interval [ a, b], then it is limited on this segment.

If the function
is continuous on the interval [ a, b], then it reaches its minimum and maximum values on this segment.

If the function
is continuous on the interval [ a, b] And
, then whatever the number is WITH, enclosed between numbers A And IN, there is a point
, What
.

From this statement it follows that if the function
is continuous on [ a, b] and at the ends of this segment takes values of different signs, then there is at least one point on this segment c, in which the function vanishes.

The following statement is true: if arithmetic operations are performed on continuous functions, the result is a continuous functionI.

Example 1 .

at the point
.

Solution . Function value at
There is
. Let's calculate the one-sided limits of the function at the point
:

Since one-sided limits at
are equal to each other and equal to the value of the function at this point, then this function is continuous at the point
.

3. Continuity of elementary functions

Consider the function
. This constant function is continuous at any point , because
.

Function
is also continuous at every point
, because
. Because
, then based on the above statement about arithmetic operations on continuous functions
will be continuous. The functions will also be continuous
.

Similarly, we can show the continuity of the remaining elementary functions.

Thus, any elementary function is continuous in its domain of definition, i.e. The domain of definition of an elementary function coincides with the domain of its continuity.

Continuity of complex and inverse functions

Let the function
continuous at a point , and the function
continuous at a point
. Then the complex function
continuous at a point . This means that if a complex function is made up of continuous functions, then it will also be continuous, i.e. a continuous function from a continuous function is a continuous function . This definition extends to a finite number of continuous functions.

From this definition it follows that under the sign of a continuous function we can go to the limit:

This means that if the function is continuous, then the sign of the limit and the sign of the function can be swapped.

Let the function
defined, strictly monotone and continuous on the interval [ a, b]. Then its inverse function
defined, strictly monotone and continuous on the interval [ A, B], Where
.

Breakpoints and their classification I

As is already known, if the function
defined on the set D and at the point
condition is met
, then the function is continuous at this point. If this continuity condition is not satisfied, then at the point X 0 function has a gap.

Dot called discontinuity point of the first kind functions
, if at this point the function has finite one-sided limits that are not equal to each other, i.e. . In this case, the value

called abruptly functions
at the point .

Dot called removable break point functions
, if the one-sided limits of the function at this point are equal to each other and not equal to the value of the function at this point, i.e. In this case, to eliminate the gap at the point need to put

Dot X 0 is called point of discontinuity of the second kind functions
if at least one of the one-sided limits
or
at this point either does not exist or is equal to infinity.

Example 2 . Examine the continuity of a function

Solution . The function is defined and continuous on the entire number line, except for the point
. At this point the function has a discontinuity. Let's find one-sided limits of the function at the point
:

Since at the point
one-sided limits are equal to each other, and the function at this point is not defined, then the point
is a removable break point. To eliminate the gap at this point, it is necessary to further define the function by putting
.

Example 3 . Examine the continuity of a function

Solution . The function is defined and continuous on the entire set of real numbers except
. At this point the function has a discontinuity. Let's find one-sided limits of the function at
:

Since this function at the point
has finite one-sided limits that are not equal to each other, then this point is a discontinuity point of the first kind. Jump of a function at a point
equal.