How many asymptotes can the graph of a function have?

Not one, one, two, three,... or infinitely many. We won’t go far for examples; let’s remember the elementary functions. A parabola, a cubic parabola, and a sine wave do not have asymptotes at all. The graph of an exponential, logarithmic function has a single asymptote. The arctangent and arccotangent have two of them, and the tangent and cotangent have infinitely many. It is not uncommon for a graph to have both horizontal and vertical asymptotes. Hyperbole, will always love you.

What does it mean to find the asymptotes of the graph of a function?

This means figuring out their equations, and drawing straight lines if the problem requires it. The process involves finding the limits of a function.

Vertical asymptotes of the graph of a function

The vertical asymptote of the graph, as a rule, is located at the point of infinite discontinuity of the function. It's simple: if at a point the function suffers an infinite discontinuity, then the straight line specified by the equation is the vertical asymptote of the graph.

Note: Please note that the entry is used to refer to two completely different concepts. Whether a point is implied or an equation of a line depends on the context.

Thus, to establish the presence of a vertical asymptote at a point, it is enough to show that at least one of the one-sided limits is infinite. Most often this is the point where the denominator of the function is zero. Essentially, we have already found vertical asymptotes in the last examples of the lesson on continuity of a function. But in some cases there is only one one-sided limit, and if it is infinite, then again - love and favor the vertical asymptote. The simplest illustration: and the ordinate axis.

From the above, an obvious fact also follows: if the function is continuous on, then there are no vertical asymptotes. For some reason a parabola came to mind. Really, where can you “stick” a straight line here? ...yes... I understand... Uncle Freud's followers became hysterical =)

The converse statement is generally false: for example, the function is not defined on the entire number line, but is completely deprived of asymptotes.

Sloping asymptotes of the graph of a function

Oblique (as a special case - horizontal) asymptotes can be drawn if the argument of the function tends to “plus infinity” or to “minus infinity”. Therefore, the graph of a function cannot have more than 2 inclined asymptotes. For example, the graph of an exponential function has a single horizontal asymptote at, and the graph of the arctangent at has two such asymptotes, and different ones at that.

When the graph in both places approaches a single oblique asymptote, then it is customary to combine the “infinities” under a single entry. For example, ...you guessed correctly: .

Asymptote of the graph of a function y = f(x) is a straight line that has the property that the distance from the point (x, f(x)) to this straight line tends to zero as the graph point moves indefinitely from the origin.

In Figure 3.10. graphic examples are given vertical, horizontal And inclined asymptote.

Finding the asymptotes of the graph is based on the following three theorems.

Vertical asymptote theorem. Let the function y = f(x) be defined in a certain neighborhood of the point x 0 (possibly excluding this point itself) and at least one of the one-sided limits of the function is equal to infinity, i.e. Then the straight line x = x 0 is the vertical asymptote of the graph of the function y = f(x).

Obviously, the straight line x = x 0 cannot be a vertical asymptote if the function is continuous at the point x 0, since in this case . Consequently, vertical asymptotes should be sought at the discontinuity points of the function or at the ends of its domain of definition.

Horizontal asymptote theorem. Let the function y = f(x) be defined for sufficiently large x and there is a finite limit of the function. Then the line y = b is the horizontal asymptote of the graph of the function.

Comment. If only one of the limits is finite, then the function has, accordingly, left-handed or right-sided horizontal asymptote.

In the event that , the function may have an oblique asymptote.

Oblique asymptote theorem. Let the function y = f(x) be defined for sufficiently large x and there be finite limits . Then the straight line y = kx + b is the slanted asymptote of the graph of the function.

No proof.

An oblique asymptote, just like a horizontal one, can be right- or left-handed if the basis of the corresponding limits is infinity of a certain sign.

Studying functions and constructing their graphs usually includes the following steps:

1. Find the domain of definition of the function.

2. Examine the function for even-odd parity.

3. Find vertical asymptotes by examining discontinuity points and the behavior of the function at the boundaries of the domain of definition, if they are finite.

4. Find horizontal or oblique asymptotes by examining the behavior of the function at infinity.

5. Find extrema and intervals of monotonicity of the function.

6. Find the intervals of convexity of the function and inflection points.

7. Find the points of intersection with the coordinate axes and, possibly, some additional points that clarify the graph.

Function differential

It can be proven that if a function has a limit equal to a finite number for a certain base, then it can be represented as the sum of this number and an infinitesimal value for the same base (and vice versa): .

Let's apply this theorem to a differentiable function: .

Thus, the increment of the function Dу consists of two terms: 1) linear with respect to Dx, i.e. f `(x)Dх; 2) nonlinear with respect to Dx, i.e. a(Dx)Dx. At the same time, since , this second term represents an infinitesimal more high order than Dx (as Dx tends to zero, it tends to zero even faster).

Differential function is the main, linear relative to Dx part of the increment of the function, equal to the product of the derivative and the increment of the independent variable dy = f `(x)Dx.

Let's find the differential of the function y = x.

Since dy = f `(x)Dх = x`Dх = Dх, then dx = Dх, i.e. the differential of an independent variable is equal to the increment of this variable.

Therefore, the formula for the differential of a function can be written as dy = f `(x)dх. That is why one of the notations for the derivative is the fraction dy/dх.

The geometric meaning of the differential is illustrated
Figure 3.11. Let us take an arbitrary point M(x, y) on the graph of the function y = f(x). Let's give the argument x the increment Dx. Then the function y = f(x) will receive the increment Dy = f(x + Dх) - f(x). Let us draw a tangent to the graph of the function at point M, which forms an angle a with the positive direction of the abscissa axis, i.e. f `(x) = tan a. From right triangle MKN
KN = MN*tg a = Dх*tg a = f `(x)Dх = dy.

Thus, the differential of a function is the increment in the ordinate of the tangent drawn to the graph of the function at a given point when x receives the increment Dx.

The properties of a differential are basically the same as those of a derivative:

3. d(u ± v) = du ± dv.

4. d(uv) = v du + u dv.

5. d(u/v) = (v du - u dv)/v 2.

However, there is important property differential of a function that its derivative does not have is invariance of differential form.

From the definition of the differential for the function y = f(x), the differential dy = f `(x)dх. If this function y is complex, i.e. y = f(u), where u = j(x), then y = f and f `(x) = f `(u)*u`. Then dy = f `(u)*u`dх. But for the function
u = j(x) differential du = u`dх. Hence dy = f `(u)*du.

Comparing the equalities dy = f `(x)dх and dy = f `(u)*du, we make sure that the differential formula does not change if instead of a function of the independent variable x we ​​consider a function of the dependent variable u. This property of a differential is called invariance (i.e., immutability) of the form (or formula) of the differential.

However, there is still a difference in these two formulas: in the first of them, the differential of the independent variable is equal to the increment of this variable, i.e. dx = Dx, and secondly, the differential of the function du is only the linear part of the increment of this function Du and only for small Dx du » Du.

There will also be tasks for independent decision, to which you can see the answers.

The concept of asymptote

If you first construct the asymptotes of the curve, then in many cases the construction of a graph of the function becomes easier.

The fate of the asymptote is full of tragedy. Imagine what it’s like: all your life moving in a straight line towards your cherished goal, getting as close as possible to it, but never achieving it. For example, striving to connect your life path with the path of the desired person, at some point approaching him almost closely, but not even touching him. Or strive to earn a billion, but before achieving this goal and entering the Guinness Book of Records for your case, hundredths of a cent are missing. Etc. So it is with an asymptote: it constantly strives to reach the curve of the function graph, approaches it to the minimum possible distance, but never touches it.

Definition 1. Asymptotes are those straight lines to which the graph of a function approaches arbitrarily closely when the variable tends to plus infinity or minus infinity.

Definition 2. A straight line is called an asymptote of the graph of a function if the distance from the variable point M the graph of the function up to this line tends to zero as the point moves away indefinitely M from the origin along any branch of the function graph.

There are three types of asymptotes: vertical, horizontal and oblique.

Vertical asymptotes

The first thing you need to know about vertical asymptotes is that they are parallel to the axis Oy .

Definition. Straight x = a is vertical asymptote of the graph of the function , if point x = a is point of discontinuity of the second kind for this function.

From the definition it follows that the straight line x = a is the vertical asymptote of the graph of the function f(x) if at least one of the conditions is met:

In this case, the function f(x) may not be defined at all, respectively, when x ≥ a And x ≤ a .

Comment:

Example 1. Graph of a function y=ln x has a vertical asymptote x= 0 (i.e. coinciding with the axis Oy) on the boundary of the domain of definition, since the limit of the function as x tends to zero from the right is equal to minus infinity:

(picture above).

yourself and then see the solutions

Example 2. Find the asymptotes of the graph of the function.

Example 3. Find asymptotes of the graph of a function

Horizontal asymptotes

The first thing you need to know about horizontal asymptotes is that they are parallel to the axis Ox .

If (the limit of a function as the argument tends to plus or minus infinity is equal to a certain value b), That y = b – horizontal asymptote crooked y = f(x ) (right when X tends to plus infinity, left when X tends to minus infinity, and two-sided if the limits as X tends to plus or minus infinity are equal).

Example 5. Graph of a function

at a> 1 has left horizontal asympotote y= 0 (i.e. coinciding with the axis Ox), since the limit of the function as “x” tends to minus infinity is zero:

The curve does not have a right horizontal asymptote, since the limit of the function as “x” tends to plus infinity is equal to infinity:

Oblique asymptotes

The vertical and horizontal asymptotes that we examined above are parallel to the coordinate axes, so to construct them we only needed a certain number - the point on the abscissa or ordinate axis through which the asymptote passes. For an oblique asymptote, a larger slope is needed k, which shows the angle of inclination of the line, and the free term b, which shows how much the line is above or below the origin. Those who have not forgotten analytical geometry, and from it the equations of the straight line, will notice that for the oblique asymptote they find equation of a line with slope. The existence of an oblique asymptote is determined by the following theorem, on the basis of which the coefficients just mentioned are found.

Theorem. To make the curve y = f(x) had an asymptote y = kx + b , it is necessary and sufficient for there to be finite limits k And b of the function under consideration as the variable tends x to plus infinity and minus infinity:

(1)

(2)

The numbers found in this way k And b and are the oblique asymptote coefficients.

In the first case (as x tends to plus infinity), a right inclined asymptote is obtained, in the second (as x tends to minus infinity), a left oblique asymptote is obtained. The right oblique asymptote is shown in Fig. below.

When finding the equation for an oblique asymptote, it is necessary to take into account the tendency of X to both plus infinity and minus infinity. For some functions, for example, fractional rational ones, these limits coincide, but for many functions these limits are different and only one of them can exist.

If the limits coincide and x tends to plus infinity and minus infinity, the straight line y = kx + b is the two-sided asymptote of the curve.

If at least one of the limits defining the asymptote y = kx + b , does not exist, then the graph of the function does not have an oblique asymptote (but may have a vertical one).

It is easy to see that the horizontal asymptote y = b is a special case of oblique y = kx + b at k = 0 .

Therefore, if in any direction a curve has a horizontal asymptote, then in this direction there is no inclined one, and vice versa.

Example 6. Find asymptotes of the graph of a function

Solution. The function is defined on the entire number line except x= 0, i.e.

Therefore, at the breaking point x= 0 the curve may have a vertical asymptote. Indeed, the limit of the function as x tends to zero from the left is equal to plus infinity:

Hence, x= 0 – vertical asymptote of the graph of this function.

The graph of this function does not have a horizontal asymptote, since the limit of the function as x tends to plus infinity is equal to plus infinity:

Let us find out the presence of an oblique asymptote:

Got finite limits k= 2 and b= 0 . Straight y = 2x is the two-way slanted asymptote of the graph of this function (figure inside the example).

Example 7. Find asymptotes of the graph of a function

Solution. The function has one break point x= −1 . Let's calculate one-sided limits and determine the type of discontinuity:

Conclusion: x= −1 is a discontinuity point of the second kind, so the straight line x= −1 is the vertical asymptote of the graph of this function.

We are looking for oblique asymptotes. Since this function is fractional-rational, the limits at and at will coincide. Thus, we find the coefficients for substituting the straight line - oblique asymptote into the equation:

Substituting the found coefficients into the equation of the straight line with the slope coefficient, we obtain the equation of the oblique asymptote:

y = −3x + 5 .

In the figure, the graph of the function is indicated in burgundy, and the asymptotes are indicated in black.

Example 8. Find asymptotes of the graph of a function

Solution. Since this function is continuous, its graph has no vertical asymptotes. We are looking for oblique asymptotes:

.

Thus, the graph of this function has an asymptote y= 0 at and has no asyptote at .

Example 9. Find asymptotes of the graph of a function

Solution. First we look for vertical asymptotes. To do this, we find the domain of definition of the function. A function is defined when the inequality and . Sign of the variable x matches the sign. Therefore, consider the equivalent inequality. From this we obtain the domain of definition of the function: . A vertical asymptote can only be on the boundary of the domain of definition of the function. But x= 0 cannot be a vertical asymptote, since the function is defined at x = 0 .

Consider the right-hand limit at (there is no left-hand limit):

.

Dot x= 2 is a discontinuity point of the second kind, so the straight line x= 2 - vertical asymptote of the graph of this function.

We are looking for oblique asymptotes:

So, y = x+ 1 - oblique asymptote of the graph of this function at . We are looking for an oblique asymptote at:

So, y = −x − 1 - oblique asymptote at .

Example 10. Find asymptotes of the graph of a function

Solution. A function has a domain of definition . Since the vertical asymptote of the graph of this function can only be on the boundary of the domain of definition, we find the one-sided limits of the function at .

Asymptotes of the graph of a function

The ghost of the asymptote has been wandering around the site for a long time to finally materialize in a separate article and bring particular delight to readers who are puzzled full study of the function. Finding the asymptotes of a graph is one of the few parts of this task that is covered in the school course only in an overview manner, since the events revolve around the calculation function limits, but they still belong to higher mathematics. For visitors who have little understanding of mathematical analysis, I think the hint is clear ;-) ...stop, stop, where are you going? Limits- it's easy!

Examples of asymptotes were encountered immediately in the first lesson about graphs of elementary functions, and the topic is now receiving detailed consideration.

So what is an asymptote?

Imagine variable point, which “travels” along the graph of the function. Asymptote is straight, to whcih indefinitely close the graph of a function approaches as its variable point moves to infinity.

Note : The definition is meaningful, if you need a formulation in calculus notation, please refer to the textbook.

On the plane, asymptotes are classified according to their natural location:

1) Vertical asymptotes, which are given by an equation of the form , where “alpha” is real number. A popular representative defines the ordinate axis itself,
with a slight feeling of nausea we remember the hyperbole.

2) Oblique asymptotes traditionally written equation of a straight line with an angle coefficient. Sometimes a special case is identified as a separate group - horizontal asymptotes. For example, the same hyperbola with asymptote.

Let's go quickly, let's hit the topic with a short burst of machine gun fire:

How many asymptotes can the graph of a function have?

Not one, one, two, three,... or infinitely many. We won’t go far for examples, let’s remember elementary functions. A parabola, a cubic parabola, and a sine wave do not have asymptotes at all. The graph of an exponential, logarithmic function has a single asymptote. The arctangent and arccotangent have two of them, and the tangent and cotangent have infinitely many. It is not uncommon for a graph to have both horizontal and vertical asymptotes. Hyperbole, will always love you.

What means ?

Vertical asymptotes of the graph of a function

The vertical asymptote of the graph is usually located at the point of infinite discontinuity functions. It's simple: if at a point the function suffers an infinite discontinuity, then the straight line specified by the equation is the vertical asymptote of the graph.

Note : Note that the entry is used to refer to two completely different concepts. Whether a point is implied or an equation of a line depends on the context.

Thus, to establish the presence of a vertical asymptote at a point, it is enough to show that at least one from one-sided limits infinite. Most often this is the point where the denominator of the function is zero. Essentially, we have already found vertical asymptotes in the last examples of the lesson on the continuity of a function. But in some cases there is only one one-sided limit, and if it is infinite, then again - love and favor the vertical asymptote. The simplest illustration: and the ordinate axis (see. Graphs and properties of elementary functions).

From the above, an obvious fact also follows: if the function is continuous on, then there are no vertical asymptotes. For some reason a parabola came to mind. Really, where can you “stick” a straight line here? ...yes... I understand... Uncle Freud's followers became hysterical =)

The converse statement is generally false: for example, the function is not defined on the entire number line, but is completely deprived of asymptotes.

Sloping asymptotes of the graph of a function

Oblique (as a special case - horizontal) asymptotes can be drawn if the argument of the function tends to “plus infinity” or to “minus infinity”. That's why the graph of a function cannot have more than two slanting asymptotes. For example, the graph of an exponential function has a single horizontal asymptote at , and the graph of the arctangent at has two such asymptotes, and different ones at that.

When the graph in both places approaches a single oblique asymptote, then the “infinities” are usually combined under a single entry. For example, ...you guessed correctly: .

General rule of thumb:

If there are two final limit , then the straight line is the oblique asymptote of the graph of the function at . If at least one of the listed limits is infinite, then there is no oblique asymptote.

Note : the formulas remain valid if “x” tends only to “plus infinity” or only to “minus infinity”.

Let us show that the parabola has no oblique asymptotes:

The limit is infinite, which means there is no oblique asymptote. Note that in finding the limit the need has disappeared since the answer has already been received.

Note : If you have (or will have) difficulties understanding the plus-minus, minus-plus signs, please see the help at the beginning of the lesson
on infinitesimal functions, where I told you how to correctly interpret these signs.

Obviously, for any quadratic, cubic function, a polynomial of 4th and higher degrees also does not have oblique asymptotes.

Now let’s make sure that the graph also does not have an oblique asymptote. To reveal uncertainty we use L'Hopital's rule:
, which was what needed to be checked.

When the function grows indefinitely, but there is no straight line to which its graph would approach infinitely close.

Let's move on to the practical part of the lesson:

How to find the asymptotes of the graph of a function?

This is exactly how the typical task is formulated, and it involves finding ALL asymptotes of the graph (vertical, inclined/horizontal). Although, to be more precise in posing the question, we are talking about research for the presence of asymptotes (after all, there may not be any at all). Let's start with something simple:

Example 1

Find asymptotes of the graph of a function

Solution It’s convenient to break it down into two points:

1) First we check whether there are vertical asymptotes. The denominator goes to zero at , and it is immediately clear that at this point the function suffers endless gap, and the straight line given by the equation is the vertical asymptote of the graph of the function. But, before drawing such a conclusion, it is necessary to find one-sided limits:

I remind you of the calculation technique that I similarly focused on in the article Continuity of function. Break points. In the expression under the limit sign we substitute . There's nothing interesting in the numerator:
.

But in the denominator it turns out infinitesimal negative number:
, it determines the fate of the limit.

The left-hand limit is infinite, and, in principle, it is already possible to make a verdict about the presence of a vertical asymptote. But unilateral limits are needed not only for this - they HELP TO UNDERSTAND HOW locate the graph of the function and build it CORRECTLY. Therefore, we must also calculate the right-handed limit:

Conclusion: one-sided limits are infinite, which means that the straight line is the vertical asymptote of the graph of the function at .

First limit finite, which means it is necessary to “continue the conversation” and find the second limit:

The second limit too finite.

Thus, our asymptote is:

Conclusion: the straight line given by the equation is the horizontal asymptote of the graph of the function at .

To find the horizontal asymptote
you can use a simplified formula:

If exists finite limit, then the straight line is the horizontal asymptote of the graph of the function at .

It is easy to see that the numerator and denominator of the function same order of growth, which means the sought limit will be finite:

Answer:

According to the condition, you do not need to complete the drawing, but if in full swing function study, then on the draft we immediately make a sketch:

Based on the three found limits, try to figure out for yourself how the graph of the function might be located. Is it at all difficult? Find 5-6-7-8 points and mark them on the drawing. However, the graph of this function is constructed using transformations of the graph of an elementary function, and readers who carefully examined Example 21 of the above article can easily guess what kind of curve this is.

Example 2

Find asymptotes of the graph of a function

This is an example for you to solve on your own. Let me remind you that it is convenient to divide the process into two points – vertical asymptotes and oblique asymptotes. In the sample solution, the horizontal asymptote is found using a simplified scheme.

In practice, fractional-rational functions are most often encountered, and after training on hyperbolas, we will complicate the task:

Example 3

Find asymptotes of the graph of a function

Solution: One, two and done:

1) Vertical asymptotes are located at points of infinite discontinuity, so you need to check whether the denominator goes to zero. Let's decide quadratic equation:

The discriminant is positive, so the equation has two real roots, and the work is significantly increased =)

In order to further find one-sided limits, it is convenient to factorize the square trinomial:
(for compact notation, the “minus” was included in the first bracket). To be on the safe side, let’s check by opening the brackets mentally or on a draft.

Let's rewrite the function in the form

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

And at the point:

Thus, the straight lines are vertical asymptotes of the graph of the function in question.

2) If you look at the function , then it is quite obvious that the limit will be finite and we have a horizontal asymptote. Let's show its presence in a short way:

Thus, the straight line (abscissa axis) is the horizontal asymptote of the graph of this function.

Answer:

The found limits and asymptotes provide a lot of information about the graph of the function. Try to mentally imagine the drawing taking into account the following facts:

Sketch your version of the graph on your draft.

Of course, the limits found do not clearly determine the appearance of the graph, and you may make a mistake, but the exercise itself will provide invaluable help during full function study. The correct picture is at the end of the lesson.

Example 4

Find asymptotes of the graph of a function

Example 5

Find asymptotes of the graph of a function

These are tasks for independent solution. Both graphs again have horizontal asymptotes, which are immediately detected by the following features: in Example 4 growth order denominator more, than the order of growth of the numerator, and in Example 5 the numerator and denominator same order of growth. In the sample solution, the first function is examined for the presence of oblique asymptotes in full, and the second – through the limit.

Horizontal asymptotes, in my subjective impression, are noticeably more common than those that are “truly tilted.” The long-awaited general case:

Example 6

Find asymptotes of the graph of a function

Solution: classics of the genre:

1) Since the denominator is positive, then the function continuous along the entire number line, and there are no vertical asymptotes. …Is it good? Not the right word - excellent! Point No. 1 is closed.

2) Let's check the presence of oblique asymptotes:

First limit finite, so let's move on. During the calculation of the second limit to eliminate uncertainty "infinity minus infinity" We bring the expression to a common denominator:

The second limit too finite Therefore, the graph of the function in question has an oblique asymptote:

Conclusion:

Thus, when the graph of the function infinitely close approaches a straight line:

Note that it intersects its oblique asymptote at the origin, and such intersection points are quite acceptable - it is important that “everything is normal” at infinity (in fact, this is where we are talking about asymptotes).

Example 7

Find asymptotes of the graph of a function

Solution: There’s nothing special to comment on, so I’ll draw up an approximate example of a clean solution:

1) Vertical asymptotes. Let's explore the point.

The straight line is the vertical asymptote for the graph at .

2) Oblique asymptotes:

The straight line is the slanted asymptote for the graph at .

Answer:

The found one-sided limits and asymptotes allow us to predict with high confidence what the graph of this function looks like. Correct drawing at the end of the lesson.

Example 8

Find asymptotes of the graph of a function

This is an example for independent solution; for the convenience of calculating some limits, you can divide the numerator by the denominator term by term. Again, when analyzing your results, try to draw a graph of this function.

Obviously, the owners of the “real” oblique asymptotes are the graphs of those fractional rational functions, which have a higher degree of the numerator one more the highest degree of the denominator. If it is more, there will be no oblique asymptote (for example, ).

But other miracles happen in life:

Example 9

Example 11

Examine the graph of a function for the presence of asymptotes

Solution: it's obvious that , therefore we consider only the right half-plane, where there is a graph of the function.

Thus, the straight line (ordinate axis) is the vertical asymptote for the graph of the function at .

2) The study on oblique asymptote can be carried out according to the full scheme, but in the article L'Hopital's rules we found out that linear function higher order of growth than logarithmic, therefore: (See Example 1 of the same lesson).

Conclusion: the x-axis is the horizontal asymptote of the graph of the function at .

Answer:
, If ;
, If .

Drawing for clarity:

It is interesting that a seemingly similar function has no asymptotes at all (those who wish can check this).

Two final examples for self-study:

Example 12

Examine the graph of a function for the presence of asymptotes

The solution can be conveniently divided into two points:

1) First we check whether there are vertical asymptotes. The denominator goes to zero at, and it is immediately clear that at this point the function suffers an infinite discontinuity, and the straight line specified by the equation is the vertical asymptote of the graph of the function. But, before drawing such a conclusion, it is necessary to find one-sided limits:

I remind you of the calculation technique that I similarly focused on in the article Continuity of a function. Breaking points. We substitute “X” in the expression under the limit sign. There's nothing interesting in the numerator:

But the denominator results in an infinitesimal negative number:

It determines the fate of the limit.

The left-hand limit is infinite, and, in principle, it is already possible to make a verdict about the presence of a vertical asymptote. But one-sided limits are needed not only for this - they HELP TO UNDERSTAND HOW the graph of a function is located and to construct it CORRECTLY. Therefore, we must also calculate the right-handed limit:

Conclusion: one-sided limits are infinite, which means that the straight line is the vertical asymptote of the graph of the function at.

The first limit is finite, which means we need to “continue the conversation” and find the second limit:

The second limit is also finite.

Thus, our asymptote is:

Conclusion: the straight line specified by the equation is the horizontal asymptote of the graph of the function at.

To find the horizontal asymptote, you can use a simplified formula:

If there is a finite limit, then the straight line is the horizontal asymptote of the graph of the function at.

It is easy to notice that the numerator and denominator of the function are of the same order of growth, which means that the sought limit will be finite:

According to the condition, there is no need to make a drawing, but if we are in the midst of researching a function, then we immediately make a sketch on the draft:

Based on the three limits found, try to figure out for yourself how the graph of the function might be located. Is it at all difficult? Find 5-6-7-8 points and mark them on the drawing. However, the graph of this function is constructed using transformations of the graph of an elementary function, and readers who carefully examined Example 21 of this article can easily guess what kind of curve this is.

This is an example for you to solve on your own. Let me remind you that the process is conveniently divided into two points - vertical asymptotes and oblique asymptotes. In the sample solution, the horizontal asymptote is found using a simplified scheme.

In practice, fractional-rational functions are most often encountered, and after training on hyperbolas, we will complicate the task:

Find asymptotes of the graph of a function

Solution: One, two and done:

1) Vertical asymptotes are at points of infinite discontinuity, so you need to check whether the denominator goes to zero. Let's solve the quadratic equation:

The discriminant is positive, so the equation has two real roots, and the work is added significantly

In order to further find one-sided limits, it is convenient to factorize the square trinomial:

(for compact notation, the “minus” was included in the first bracket). To be on the safe side, let’s check by opening the brackets mentally or on a draft.

Let's rewrite the function in the form

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

asymptote graph function limit

And at the point:

Thus, the straight lines are vertical asymptotes of the graph of the function in question.

2) If you look at the function, it is quite obvious that the limit will be finite and we have a horizontal asymptote. Let's show its presence in a short way:

Thus, the straight line (abscissa axis) is the horizontal asymptote of the graph of this function.

The found limits and asymptotes provide a lot of information about the graph of the function. Try to mentally imagine the drawing taking into account the following facts:

Sketch your version of the graph on your draft.

Of course, the found limits do not clearly determine the appearance of the graph, and you may make a mistake, but the exercise itself will provide invaluable assistance in the course of a complete study of the function. The correct picture is at the end of the lesson.

Find asymptotes of the graph of a function

Find asymptotes of the graph of a function

These are tasks for independent solution. Both graphs again have horizontal asymptotes, which are immediately detected by the following features: in Example 4, the order of growth of the denominator is greater than the order of growth of the numerator, and in Example 5, the numerator and denominator are of the same order of growth. In the sample solution, the first function is examined for the presence of oblique asymptotes in full, and the second - through the limit.

Horizontal asymptotes, in my subjective impression, are noticeably more common than those that are “truly tilted.” The long-awaited general case:

Find asymptotes of the graph of a function

Solution: classic of the genre:

• 1) Since the denominator is positive, the function is continuous along the entire number line, and there are no vertical asymptotes. …Is it good? Not the right word - excellent! Point No. 1 is closed.
• 2) Let's check the presence of oblique asymptotes:

The second limit is also finite, therefore, the graph of the function in question has an oblique asymptote:

Thus, when the graph of the function approaches a straight line infinitely close.

Note that it intersects its oblique asymptote at the origin, and such intersection points are quite acceptable - it is important that “everything is normal” at infinity (in fact, this is where we are talking about asymptotes).

Find asymptotes of the graph of a function

Solution: there’s nothing special to comment on, so I’ll draw up an approximate example of a final solution:

1) Vertical asymptotes. Let's explore the point.

The straight line is the vertical asymptote for the graph at.

2) Oblique asymptotes:

The straight line is the slanted asymptote for the graph at.

The found one-sided limits and asymptotes allow us to predict with high confidence what the graph of this function looks like.

Find asymptotes of the graph of a function

This is an example for independent solution; for the convenience of calculating some limits, you can divide the numerator by the denominator term by term. Again, when analyzing your results, try to draw a graph of this function.

Obviously, the owners of “real” oblique asymptotes are the graphs of those fractional rational functions in which the leading degree of the numerator is one greater than the leading degree of the denominator. If it is more, there will no longer be an oblique asymptote (for example).

But other miracles happen in life.