How many asymptotes can a graph of a function have? Find the asymptotes of the graph of a function Asymptotes of the graph
- The concept of asymptotes
One of the important steps in constructing graphs of functions is the search for asymptotes. We met with asymptotes more than once: when plotting functions , y=tgx, y=ctgx. We have defined them as lines that the graph of a function “tends for” but never crosses. It's time to give a precise definition of asymptotes.
There are three types of asymptotes: vertical, horizontal, and oblique. In the drawing, the asymptotes are usually denoted by dotted lines.
Consider the following artificially plotted function graph (Fig. 16.1), on the example of which all types of asymptotes are clearly visible:
We give a definition for each type of asymptote:
1. Direct x=a called vertical asymptote functions if .
2. Direct y=s called horizontal asymptote functions if .
3. Direct y=kx+b called oblique asymptote functions if .
Geometrically, the definition of an oblique asymptote means that as →∞ the graph of a function approaches a straight line arbitrarily close y=kx+b, i.e. they are practically the same. The difference of almost identical expressions tends to zero.
Note that the horizontal and oblique asymptotes are considered only under the condition →∞. Sometimes they are distinguished into horizontal and oblique asymptotes as →+∞ and →-∞.
- Asymptote Search Algorithm
The following algorithm can be used to find asymptotes:
There may be one vertical asymptote, several or none at all.
- If c is a number, then y=s is the horizontal asymptote;
- If c is infinity, then there are no horizontal asymptotes.
If a function is a ratio of two polynomials, then if the function has horizontal asymptotes, we will not look for oblique asymptotes - they do not exist.
Consider examples of finding asymptotes of a function:
Example 16.1. Find the asymptotes of the curve.
Solution X-1≠0; X≠1.
Let's check if the line is x= 1 vertical asymptote. To do this, we calculate the limit of the function at the point x= 1: .
x= 1 - vertical asymptote.
With= .
With= = . Because With=2 (number), then y=2 is the horizontal asymptote.
Since the function is a ratio of polynomials, in the presence of horizontal asymptotes we assert that there are no oblique asymptotes.
x= 1 and the horizontal asymptote y=2. For clarity, the graph of this function is shown in Fig. 16.2.
Example 16.2. Find the asymptotes of the curve.
Solution. 1. Find the domain of the function: X-2≠0; X≠2.
Let's check if the line is x= 2 vertical asymptote. To do this, we calculate the limit of the function at the point x= 2: .
We got that, therefore, x= 2 - vertical asymptote.
2. To search for horizontal asymptotes, we find: With= .
Since there is an uncertainty in the limit, we use the L'Hopital rule: With= = . Because With is infinity, then there are no horizontal asymptotes.
3. To search for oblique asymptotes, we find:
We got an uncertainty of the form , we use the L'Hopital rule: = =1. b according to the formula: .
b= = =
Got that b= 2. Then y=kx+b – oblique asymptote. In our case, it looks like: y=x+2.
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Control questions:
Lecture 17
In this lecture, we will summarize all the previously studied material. The ultimate goal of our long journey is to be able to investigate any analytically given function and build its graph. Important parts of our study will be the study of the function for extrema, the determination of intervals of monotonicity, convexity and concavity of the graph, the search for inflection points, asymptotes of the graph of the function.
Taking into account all the above aspects, we present scheme for studying the function and plotting .
1. Find the domain of the function.
2. Investigate the function for even-odd:
if , then the function is even (the graph of an even function is symmetrical with respect to the axis OU);
if , then the function is odd (the graph of an odd function is symmetrical with respect to the origin);
Otherwise, the function is neither even nor odd.
3. Investigate the function for periodicity (among the functions we study, only trigonometric functions can be periodic).
4. Find the points of intersection of the graph of the function with the coordinate axes:
· Oh: at=0 (we solve the equation only if we can use the methods known to us);
· OU: X=0.
5. Find the first derivative of the function and critical points of the first kind.
6. Find monotonicity intervals and extrema of the function.
7. Find the second derivative of the function and critical points of the second kind.
8. Find the intervals of convexity-concavity of the function graph and inflection points.
9. Find the asymptotes of the graph of the function.
10. Graph the function. When building, consider cases of possible location of the graph near the asymptotes :
11. If necessary, select control points for more accurate construction.
Consider a scheme for studying a function and plotting its graph using specific examples:
Example 17.1. Plot the function .
Solution. 1. This function is defined on the entire number line except for X=3, because at this point the denominator goes to zero.
2. To determine the evenness and oddness of the function, we find:
We see that and , therefore, the function is neither even nor odd.
3. The function is non-periodic.
4. Find the points of intersection with the coordinate axes. To find the point of intersection with the axis Oh accept at=0. We get the equation: . So, the point (0; 0) is the point of intersection with the coordinate axes.
5. Find the derivative of the function according to the rule of differentiation of a fraction: = = = = .
To find the critical points, we find the points at which the derivative of the function is equal to 0 or does not exist.
If =0, therefore, . The product is then 0 when at least one of the factors is 0: or .
X-3) 2 is equal to 0, i.e. does not exist at X=3.
So, the function has three critical points of the first kind: ; ; .
6. On the real axis, we mark the critical points of the first kind, and mark the point with a punctured dot, because it does not define a function.
Arrange the signs of the derivative = on each interval:
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On intervals where , the original function increases (at (-∞;0] ), where - decreases (at ).
Dot X=0 is the maximum point of the function. To find the maximum of the function, let's find the value of the function at the point 0: .
Dot X=6 is the minimum point of the function. To find the minimum of the function, let's find the value of the function at point 6: .
The results of the research can be entered in the table. The number of rows in the table is fixed and equal to four, and the number of columns depends on the function under study. In the cells of the first row, the intervals into which the critical points divide the domain of the function definition are sequentially entered, including the critical points themselves. In order to avoid errors when constructing points that do not belong to the definition area, it is possible not to include them in the table.
The second row of the table contains the signs of the derivative on each of the considered intervals and the value of the derivative at the critical points. In accordance with the signs of the derivative of the function, the intervals of increase, decrease, and extrema of the function are marked in the third line.
The last line is used to denote the maximum and minimum of the function.
| X | (-∞;0) | (0;3) | (3;6) | (6;+ ∞) | |||
| + | - | - | + | ||||
| f(x) | |||||||
| conclusions | max | min |
7. Find the second derivative of the function as a derivative of the first derivative: = =
Take out in the numerator X-3 outside the brackets and do the reduction:
We present in the numerator like terms: .
Let us find critical points of the second kind: points at which the second derivative of the function is equal to zero or does not exist.
0 if =0. This fraction cannot be equal to zero, therefore, there are no points at which the second derivative of the function is equal to zero.
Does not exist if the denominator ( X-3) 3 is 0, i.e. does not exist at X=3. :Oh , OU, origin, units of measure for each axis.
Before plotting a function, you need to:
draw asymptotes with dotted lines;
mark the points of intersection with the coordinate axes;
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· Using the obtained data on the intervals of increase, decrease, convexity and concavity, construct a graph of the function. The branches of the graph should "tend" to the asymptotes, but not cross them.
Check whether the graph of the function corresponds to the study: if the function is even or odd, then whether the symmetry is observed; whether the theoretically found intervals of increase and decrease, convexity and concavity, inflection points.
11. For more precise construction, you can select multiple control points. For example, let's find the function values at points -2 and 7:
We adjust the graph taking into account the control points.
Control questions:
- What is the algorithm for plotting a function graph?
- Can a function have an extremum at points that do not belong to the domain of definition?
CHAPTER 3. 3. INTEGRAL CALCULUS OF THE FUNCTION
- (from Greek a negative part, and symptotos coinciding together). A straight line constantly approaching a curve and meeting it only at infinity. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910. ASYMPTOE from ... ... Dictionary of foreign words of the Russian language
ASYMPTOTE- (from the Greek asymptotos non-coincident), a straight line to which the infinite branch of the curve approaches indefinitely, for example, the asymptote of a hyperbola ... Modern Encyclopedia
ASYMPTOTE- (from Greek asymptotos mismatched) a curve with an infinite branch is a straight line to which this branch approaches indefinitely, for example, an asymptote of a hyperbola ... Big Encyclopedic Dictionary
asymptote- A straight line that is gradually approached by a curve. asymptote A straight line approached (never reaching it) by a curve having an infinite branch of some function when its argument increases indefinitely or ... Technical Translator's Handbook
Asymptote- (from Greek asymptotos mismatched), a straight line to which an infinite branch of a curve approaches indefinitely, such as the asymptote of a hyperbola. … Illustrated Encyclopedic Dictionary
ASYMPTOTE- female, geom. a straight line, always approaching a curve (hyperbola), but never converging with it. An example to explain this: if any number is all divided in half, then it will decrease to infinity, but will never become zero. ... ... Dahl's Explanatory Dictionary
asymptote- noun, number of synonyms: 1 line (182) ASIS synonym dictionary. V.N. Trishin. 2013 ... Synonym dictionary
Asymptote- (from the Greek words: a, sun, piptw) mismatched. By asymptote is meant such a line, which, being indefinitely continued, approaches a given curved line or some part of it, so that the distance between the common lines becomes less ... ...
Asymptote A surface is a straight line that intersects the surface at least at two points at infinity... Encyclopedia of Brockhaus and Efron
ASYMPTOTE- (asymptote) The value that this function tends to when the argument (argument) changes, but does not reach it with any final value of the argument. For example, if the total cost of output x is given by the function TC=a+bx, where a and b are constants... Economic dictionary
Asymptote- a straight line, which tends (never reaching it), having an infinite branch of a curve of some function, when its argument increases or decreases indefinitely. For example, in the function: y = c + 1/x, the value of y approaches with ... ... Economic and Mathematical Dictionary
How many asymptotes can a graph of a function have?
None, one, two, three... or an infinite number. We will not go far for examples, we will recall elementary functions. Parabola, cubic parabola, sinusoid have no asymptotes at all. The graph of an exponential, logarithmic function has a single asymptote. The arctangent, arccotangent has two of them, and the tangent, cotangent has an infinite number. 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 a graph of a function?
This means finding out their equations, and drawing straight lines if the condition of the problem requires it. The process involves finding the limits of the function.
Vertical asymptotes of a graph of a function
The vertical asymptote of the graph, as a rule, is at the point of infinite discontinuity of the function. It's simple: if at a point the function suffers an infinite break, then the straight line given by the equation is the vertical asymptote of the graph.
Note: Please note that the notation is used to refer to two completely different concepts. The point is implied or the equation of a straight line - depends on the context.
Thus, to establish the presence of a vertical asymptote at a point, it suffices 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 equal to zero. In fact, we have already found vertical asymptotes in the last examples of the lesson on the continuity of a function. But in a number of 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 y-axis.
The obvious fact also follows from the above: if the function is continuous on, then there are no vertical asymptotes. For some reason, a parabola came to mind. Indeed, where can you “stick” a straight line here? ... yes ... I understand ... the followers of Uncle Freud huddled in hysterics =)
The converse statement is generally not true: for example, the function is not defined on the entire real line, but it is completely deprived of asymptotes.
Oblique asymptotes of a graph of a function
Inclined (as a special case - horizontal) asymptotes can be drawn if the function argument tends to "plus infinity" or "minus infinity". Therefore, the graph of a function cannot have more than 2 oblique 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.
When the graph here and there approaches the only oblique asymptote, then it is customary to combine “infinities” under a single entry. For example, ... you guessed it right: .
Asymptotes of the graph of a function
The ghost of the asymptote has been wandering around the site for a long time in order to finally materialize in a single article and bring special delight to readers puzzled full function study. Finding the asymptotes of the graph is one of the few parts of the specified task, which is covered in the school course only in an overview order, since events revolve around the calculation function limits, but they still belong to higher mathematics. Visitors who are poorly versed in mathematical analysis, I think the hint is understandable ;-) ... stop-stop, where are you going? limits- it's easy!
Examples of asymptotes met immediately in the first lesson about graphs of elementary functions, and now the topic is receiving detailed consideration.
So what is an asymptote?
Imagine variable point, which "travels" along the graph of the function. The asymptote is straight, to whcih unlimited close the graph of the function approaches as its variable point goes to infinity.
Note : the definition is meaningful, if you need a formulation in the notation of mathematical analysis, please refer to the textbook.
On a plane, asymptotes are classified according to their natural arrangement:
1) Vertical asymptotes, which are given by an equation of the form , where "alpha" is a real number. The popular representative defines the y-axis itself,
with an attack of mild nausea, we recall hyperbole.
2) Oblique asymptotes traditionally written straight line equation with a slope factor. Sometimes a special case is singled out as a separate group - horizontal asymptotes. For example, the same hyperbola with asymptote .
Quickly off we go, let's hit the topic with a short automatic burst:
How many asymptotes can a graph of a function have?
None, one, two, three... or an infinite number. We will not go far for examples, we will remember elementary functions. Parabola, cubic parabola, sinusoid have no asymptotes at all. The graph of an exponential, logarithmic function has a single asymptote. The arctangent, arccotangent has two of them, and the tangent, cotangent has an infinite number. It is not uncommon for a graph to have both horizontal and vertical asymptotes. Hyperbole, will always love you.
What means ?
Vertical asymptotes of a graph of a function
The vertical asymptote of a graph is usually at the point of infinity functions. It's simple: if at a point the function suffers an infinite break, then the straight line given by the equation is the vertical asymptote of the graph.
Note : note that the notation is used to refer to two completely different concepts. The point is implied or the equation of a straight line - depends on the context.
Thus, to establish the presence of a vertical asymptote at a point, it suffices to show that at least one from unilateral limits 
endless. Most often, this is the point where the denominator of the function is equal to zero. In fact, we have already found vertical asymptotes in the last examples of the lesson. on the continuity of the 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 y-axis (see. Graphs and properties of elementary functions).
From the above, the obvious fact also follows: if the function is continuous on, then there are no vertical asymptotes. For some reason, a parabola came to mind. Indeed, where can you “stick” a straight line here? ... yes ... I understand ... the followers of Uncle Freud huddled in hysterics =)
The converse statement is generally not true: for example, the function is not defined on the entire real line, but it is completely deprived of asymptotes.
Oblique asymptotes of a graph of a function
Oblique (as a special case - horizontal) asymptotes can be drawn if the function argument tends to "plus infinity" or "minus infinity". That's why the graph of a function cannot have more than two oblique asymptotes. For example, the graph of an exponential function has a single horizontal asymptote at , and the graph of the arc tangent at has two such asymptotes, and different ones.
When the graph here and there approaches the only oblique asymptote, then it is customary to unite “infinities” under a single entry. For example, ... you guessed it right: .
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 above 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, so there is no oblique asymptote. Note that in finding the limit 
no longer needed because the answer has already been received.
Note
: if you have (or will have) difficulty understanding the plus-minus, minus-plus signs, please see the help at the beginning of the lesson
about infinitesimal functions, where I told how to correctly interpret these signs.
It is obvious that any quadratic, cubic function, polynomial of the 4th and higher degrees also does not have oblique asymptotes.
And now let's make sure that at the graph also does not have an oblique asymptote. To uncover uncertainty, we use L'Hopital's rule:
, which was to be verified.
When the function grows indefinitely, however, there is no such 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 a graph of a function?
This is how a typical task is formulated, and it involves finding ALL asymptotes of the graph (vertical, oblique / horizontal). Although, to be more precise in the formulation of the question, we are talking about a study 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 is convenient to break it into two points:
1) First we check if there are vertical asymptotes. The denominator vanishes 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, which I likewise dwelled on in the article Function continuity. break points. In the expression under the limit sign, instead of "x" we substitute . There is 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 pass a verdict on the presence of a vertical asymptote. But one-sided limits are needed not only for this - they HELP TO UNDERSTAND, HOW the graph of the function is located and plot it CORRECTLY. Therefore, we must also calculate the right-hand limit: 
Conclusion: one-sided limits are infinite, which means that the line is a vertical asymptote of the graph of the function at .
First limit finite, which means that it is necessary to “continue the conversation” and find the second limit:
The second limit too finite.
So 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 the simplified formula:
If exists finite limit , then the 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 one order of growth, which means that the desired limit will be finite: 
Answer:
According to the condition, it is not necessary to complete the drawing, but if in full swing function research, then on the draft we immediately make a sketch: 
Based on the three limits found, try to independently figure out how the graph of the function can be located. Quite 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 elementary function graph, and readers who have carefully examined Example 21 of this article will easily guess what kind of curve it is.
Example 2
Find asymptotes of the graph of a function
This is a do-it-yourself example. The process, I remind you, 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:
Example 3
Find asymptotes of the graph of a function 
Solution: One, two and done:
1) The vertical asymptotes are found at the points of infinite discontinuity, so you need to check if the denominator goes to zero. We will decide quadratic equation:
The discriminant is positive, so the equation has two real roots, and work is added significantly =) 
In order to further find one-sided limits, it is convenient to factorize the square trinomial:
(for compact notation, "minus" was introduced in the first bracket). For safety net, we will perform a check, mentally or on a draft, opening the brackets.
Let's rewrite the function in the form 
Find one-sided limits at the point : 
And at the point: 
Thus, the straight lines are the vertical asymptotes of the graph of the function under consideration.
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 it in a short way: 
Thus, the straight line (abscissa) is the horizontal asymptote of the graph of this function.
Answer:
The found limits and asymptotes give 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 a draft.
Of course, the limits found do not unequivocally determine the type of graph, and you may make a mistake, but the exercise itself will be of 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 decision. Both graphs again have horizontal asymptotes, which are immediately detected by the following features: in Example 4 order of growth denominator more than the order of growth of the numerator, and in Example 5 the numerator and denominator one order of growth. In the sample solution, the first function is investigated for the presence of oblique asymptotes in a full way, and the second - through the limit .
Horizontal asymptotes, in my subjective impression, are noticeably more common than those that are "truly tilted". 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, the function continuous on the entire number line, and there are no vertical asymptotes. …Is it good? Not the right word - excellent! Item #1 is closed.
2) 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 under consideration has an oblique asymptote:
Conclusion:
Thus, for 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 (actually, we are talking about asymptotes exactly there).
Example 7
Find asymptotes of the graph of a function
Solution: there is nothing much to comment on, so I will draw up an approximate sample of a final solution:
1) Vertical asymptotes. Let's explore the point. 
The straight line is the vertical asymptote for the plot at .
2) Oblique asymptotes: 
The straight line is the oblique asymptote for the graph at .
Answer: 
The found one-sided limits and asymptotes allow us to assume with high certainty 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 an independent solution, for the convenience of calculating some limits, you can divide the numerator by the denominator term by term. And again, analyzing the 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 for which the highest degree of the numerator one more the highest degree of the denominator. If 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 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 (y-axis) is the vertical asymptote for the graph of the function at .
2) The study of the oblique asymptote can be carried out according to the full scheme, but in the article Rules of L'Hospital we found that a linear function of a higher order of growth than a logarithmic one, therefore: 
(see example 1 of the same lesson).
Conclusion: the abscissa axis is the horizontal asymptote of the graph of the function at .
Answer:
, If ;
, If .
Drawing for clarity: 
Interestingly, a seemingly similar function has no asymptotes at all (those who wish can check this).
Two final self-study examples:
Example 12
Examine the graph of a function for asymptotes 
This is how a typical task is formulated, and it involves finding ALL asymptotes of the graph (vertical, oblique / horizontal). Although, to be more precise in the formulation of the question, we are talking about a study for the presence of asymptotes (after all, there may not be any at all).
Let's start with something simple:
Example 1
Solution It is convenient to break it into two points:
1) First we check if there are vertical asymptotes. The denominator vanishes 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, which I likewise dwelled on in the article function continuity. break points. In the expression under the limit sign, instead of "x" we substitute . There is 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 pass a verdict on the presence of a vertical asymptote. But one-sided limits are needed not only for this - they HELP TO UNDERSTAND HOW the graph of the function is located and plot it CORRECTLY. Therefore, we must also calculate the right-hand limit: 
Conclusion: one-sided limits are infinite, which means that the line is a vertical asymptote of the graph of the function at .
First limit finite, which means that it is necessary to “continue the conversation” and find the second limit:
The second limit too finite.
So 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 the simplified formula:
If there is a finite limit, then the line is a horizontal asymptote of the graph of the function at .
It is easy to see that the numerator and denominator of the function one order of growth, which means that the desired limit will be finite: 
Answer:
According to the condition, it is not necessary to complete the drawing, but if in full swing function research, then on the draft we immediately make a sketch: 
Based on the three limits found, try to independently figure out how the graph of the function can be located. Quite 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 elementary function graph, and readers who have carefully examined Example 21 of this article will easily guess what kind of curve it is.
Example 2
Find asymptotes of the graph of a function
This is a do-it-yourself example. The process, I remind you, 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:
Example 3
Find asymptotes of the graph of a function 
Solution: One, two and done:
1) The vertical asymptotes are found at the points of infinite discontinuity, so you need to check if the denominator goes to zero. We will decide quadratic equation :
The discriminant is positive, so the equation has two real roots, and work is added significantly =) 
In order to further find one-sided limits, it is convenient to factorize the square trinomial:
(for compact notation, "minus" was introduced in the first bracket). For safety net, we will perform a check, mentally or on a draft, opening the brackets.
Let's rewrite the function in the form 
Find one-sided limits at the point : 
And at the point: 
Thus, the straight lines are the vertical asymptotes of the graph of the function under consideration.
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 it in a short way: 
Thus, the straight line (abscissa) is the horizontal asymptote of the graph of this function.
Answer:
The found limits and asymptotes give 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 a draft.
Of course, the limits found do not unequivocally determine the type of graph, and you may make a mistake, but the exercise itself will be of 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 decision. Both graphs again have horizontal asymptotes, which are immediately detected by the following features: in Example 4 order of growth the denominator is greater than the order of growth of the numerator, and in Example 5 the numerator and denominator one order of growth. In the sample solution, the first function is investigated for the presence of oblique asymptotes in a full way, and the second - through the limit .
Horizontal asymptotes, in my subjective impression, are noticeably more common than those that are "truly tilted". 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, the function continuous on the entire number line, and there are no vertical asymptotes. …Is it good? Not the right word - great! Item #1 is closed.
2) 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 under consideration has an oblique asymptote:
Conclusion:
Thus, for 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 (actually, it is there that the discussion of asymptotes comes up).
Example 7
Find asymptotes of the graph of a function
Solution: there is nothing much to comment on, so I will draw up an approximate sample of a final solution:
1) Vertical asymptotes. Let's explore the point. 
The straight line is the vertical asymptote for the plot at .
2) Oblique asymptotes: 
The straight line is the oblique asymptote for the graph at .
Answer: 
The found one-sided limits and asymptotes allow us to assume with high certainty 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 an independent solution, for the convenience of calculating some limits, you can divide the numerator by the denominator term by term. And again, analyzing the 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 for which the highest degree of the numerator one more the highest degree of the denominator. If more, there will be no oblique asymptote (for example, ).
But other miracles happen in life:
Example 9
Solution: function continuous on the entire number line, which means that there are no vertical asymptotes. But there may well be slopes. We check:
I remember how I came across a similar function at the university and simply could not believe that it had an oblique asymptote. Until I calculated the second limit:
Strictly speaking, there are two uncertainties here: and , but one way or another, you need to use the solution method, which is discussed in Examples 5-6 of the article about the limits of increased complexity. Multiply and divide by the conjugate expression to use the formula:
Answer:
Perhaps the most popular oblique asymptote.
Until now, infinity has managed to be "cut with the same brush", but it happens that the graph of the function two different oblique asymptotes for and for :
Example 10
Examine the graph of a function for asymptotes 
Solution: the root expression is positive, which means domain- any real number, and there can be no vertical sticks.
Let's check if oblique asymptotes exist.
If "x" tends to "minus infinity", then:
(when introducing "x" under the square root, you must add a "minus" sign so as not to lose the negative denominator)
It looks unusual, but here the uncertainty is "infinity minus infinity." Multiply the numerator and denominator by the adjoint expression:
Thus, the straight line is the oblique asymptote of the graph at .
With "plus infinity" everything is more trivial: 
And the straight line - at .
Answer:
If ;
, If .
I can't resist the graphic image: 
This is one of the branches hyperbole .
It is not uncommon when the potential presence of asymptotes is initially limited function scope:
Example 11
Examine the graph of a function for asymptotes
Solution: it's obvious that 
, therefore, we consider only the right half-plane, where there is a graph of the function.
1) Function continuous on the interval , which means that if the vertical asymptote exists, then it can only be the y-axis. We study the behavior of the function near the point on right:
Note, there is NO ambiguity here(on such cases, attention was focused at the beginning of the article Limit solution methods).
Thus, the straight line (y-axis) is the vertical asymptote for the graph of the function at .
2) The study of the oblique asymptote can be carried out according to the full scheme, but in the article Lopital Rules we found that a linear function of a higher order of growth than a logarithmic one, therefore: 
(see example 1 of the same lesson).
Conclusion: the abscissa axis is the horizontal asymptote of the graph of the function at .
Answer:
If ;
, If .
Drawing for clarity: 
Interestingly, a seemingly similar function has no asymptotes at all (those who wish can check this).
Two final self-study examples:
Example 12
Examine the graph of a function for asymptotes 
To test for vertical asymptotes, we first need to find function scope, and then calculate a pair of one-sided limits at "suspicious" points. Oblique asymptotes are also not excluded, since the function is defined to "plus" and "minus" infinity.
Example 13
Examine the graph of a function for asymptotes
And here there can only be oblique asymptotes, and the directions , should be considered separately.
I hope you found the right asymptote =)
I wish you success!
Solutions and answers:
Example 2:Solution
:
. Let's find one-sided limits:
Straight is the vertical asymptote of the graph of the function at .
2) Oblique asymptotes.
Straight .
Answer:
Drawing
to Example 3:
Example 4:Solution
:
1) Vertical asymptotes. The function suffers an infinite break at a point . Let's calculate one-sided limits:
Note: an infinitesimal negative number to an even power is equal to an infinitesimal positive number: .
Straight is the vertical asymptote of the graph of the function.
2) Oblique asymptotes.
Straight (abscissa) is the horizontal asymptote of the graph of the function at .
Answer:
