How to calculate the specified limit explanation. Wonderful Limits
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Topic 4.6. Calculation of limits
The limit of a function does not depend on whether it is defined at the limit point or not. But in the practice of calculating the limits of elementary functions, this circumstance is of significant importance.
1. If the function is elementary and if the limiting value of the argument belongs to its domain of definition, then calculating the limit of the function is reduced to a simple substitution of the limiting value of the argument, because limit of the elementary function f (x) at x striving forA , which is included in the domain of definition, is equal to the partial value of the function at x = A, i.e. lim f(x)=f( a) .
2. If x tends to infinity or the argument tends to a number that does not belong to the domain of definition of the function, then in each such case, finding the limit of the function requires special research.
Below are the simplest limits based on the properties of limits that can be used as formulas:
More complex cases finding the limit of a function:
each is considered separately.
This section will outline the main ways to disclose uncertainties.
1. The case when x striving forA the function f(x) represents the ratio of two infinitesimal quantities
a) First you need to make sure that the limit of the function cannot be found by direct substitution and, with the indicated change in the argument, it represents the ratio of two infinitesimal quantities. Transformations are made to reduce the fraction by a factor tending to 0. According to the definition of the limit of a function, the argument x tends to its limit value, never coinciding with him.
In general, if we are looking for the limit of a function at x striving forA , then you must remember that x does not take on a value A, i.e. x is not equal to a.
b) Bezout's theorem is applied. If you are looking for the limit of a fraction whose numerator and denominator are polynomials that vanish at the limit point x = A, then according to the above theorem both polynomials are divisible by x- A.
c) Irrationality in the numerator or denominator is destroyed by multiplying the numerator or denominator by the conjugate to the irrational expression, then after simplifying the fraction is reduced.
d) The 1st remarkable limit (4.1) is used.
e) The theorem on the equivalence of infinitesimals and the following principles are used:
2. The case when x striving forA the function f(x) represents the ratio of two infinitely large quantities
a) Dividing the numerator and denominator of a fraction by highest degree unknown.
b) B general case you can use the rule
3. The case when x striving forA the function f (x) represents the product of an infinitesimal quantity and an infinitely large one
The fraction is converted to a form whose numerator and denominator simultaneously tend to 0 or to infinity, i.e. case 3 reduces to case 1 or case 2.
4. The case when x striving forA the function f (x) represents the difference of two positive infinitely large quantities
This case is reduced to type 1 or 2 in one of the following ways:
a) bringing fractions to a common denominator;
b) converting a function to a fraction;
c) getting rid of irrationality.
5. The case when x striving forA the function f(x) represents a power whose base tends to 1 and exponent to infinity.
The function is transformed in such a way as to use the 2nd remarkable limit (4.2).
Example. Find 
.
Because x tends to 3, then the numerator of the fraction tends to the number 3 2 +3 *3+4=22, and the denominator tends to the number 3+8=11. Hence,
Example
Here the numerator and denominator of the fraction are x tending to 2 tend to 0 (uncertainty of type), we factorize the numerator and denominator, we get lim(x-2)(x+2)/(x-2)(x-5)
Example
Multiplying the numerator and denominator by the expression conjugate to the numerator, we have
Opening the parentheses in the numerator, we get
Example
Level 2. Example. Let us give an example of the application of the concept of the limit of a function in economic calculations. Let's consider an ordinary financial transaction: lending an amount S 0 with the condition that after a period of time T the amount will be refunded S T. Let's determine the value r relative growth formula
r=(S T -S 0)/S 0 (1)
Relative growth can be expressed as a percentage by multiplying the resulting value r by 100.
From formula (1) it is easy to determine the value S T:
S T= S 0 (1 + r)
When calculating long-term loans covering several full years, use the compound interest scheme. It consists in the fact that if for the 1st year the amount S 0 increases to (1 + r) times, then for the second year in (1 + r) times the sum increases S 1 = S 0 (1 + r), that is S 2 = S 0 (1 + r) 2 . It turns out similarly S 3 = S 0 (1 + r) 3 . From the above examples one can deduce general formula to calculate the increase in the amount for n years when calculated using the compound interest scheme:
S n= S 0 (1 + r) n.
In financial calculations, schemes are used where compound interest is calculated several times a year. In this case it is stipulated annual rate r And number of accruals per year k. As a rule, accruals are made at equal intervals, that is, the length of each interval Tk forms part of the year. Then for the period in T years (here T not necessarily an integer) amount S T calculated by the formula
(2)
Where - whole part number, which coincides with the number itself, if, for example, T? integer.
Let the annual rate be r and is produced n accruals per year at regular intervals. Then for the year the amount S 0 is increased to a value determined by the formula
(3)
In theoretical analysis and practice financial activities The concept of “continuously accrued interest” is often used. To move to continuously accrued interest, you need to increase indefinitely in formulas (2) and (3), respectively, the numbers k And n(that is, to direct k And n to infinity) and calculate to what limit the functions will tend S T And S 1 . Let's apply this procedure to formula (3):
Note that the limit in curly brackets coincides with the second remarkable limit. It follows that at an annual rate r with continuously accrued interest, the amount S 0 in 1 year increases to the value S 1 *, which is determined from the formula
S 1 * = S 0 e r (4)
Let now the sum S 0 is provided as a loan with interest accrued n once a year at regular intervals. Let's denote r e annual rate at which at the end of the year the amount S 0 is increased to the value S 1 * from formula (4). In this case we will say that r e- This annual interest rate n once a year, equivalent annual interest r with continuous accrual. From formula (3) we obtain
S* 1 =S 0 (1+r e /n) n
Equating the right-hand sides of the last formula and formula (4), assuming in the latter T= 1, we can derive relationships between the quantities r And r e:
These formulas are widely used in financial calculations.
Limits give all mathematics students a lot of trouble. To solve a limit, sometimes you have to use a lot of tricks and choose from a variety of solution methods exactly the one that is suitable for a particular example.
In this article we will not help you understand the limits of your capabilities or comprehend the limits of control, but we will try to answer the question: how to understand the limits in higher mathematics? Understanding comes with experience, so at the same time we will give several detailed examples of solving limits with explanations.
The concept of limit in mathematics
The first question is: what is this limit and the limit of what? We can talk about the limits of numerical sequences and functions. We are interested in the concept of the limit of a function, since this is what students most often encounter. But first - the most general definition limit:
Let's say there is some variable value. If this value in the process of change unlimitedly approaches a certain number a , That a – the limit of this value.
For a function defined in a certain interval f(x)=y such a number is called a limit A , which the function tends to when X , tending to a certain point A . Dot A belongs to the interval on which the function is defined.
It sounds cumbersome, but it is written very simply:
Lim- from English limit- limit.
There is also a geometric explanation for determining the limit, but here we will not delve into the theory, since we are more interested in the practical rather than the theoretical side of the issue. When we say that X tends to some value, this means that the variable does not take on the value of a number, but approaches it infinitely close.
Let's give specific example. The task is to find the limit.
To solve this example, we substitute the value x=3 into a function. We get:
By the way, if you are interested, read a separate article on this topic.
In examples X can tend to any value. It can be any number or infinity. Here's an example when X tends to infinity:
It’s intuitively clear what’s what larger number in the denominator, the smaller the value the function will take. So, with unlimited growth X meaning 1/x will decrease and approach zero.
As you can see, to solve the limit, you just need to substitute the value to strive for into the function X . However, this is the simplest case. Often finding the limit is not so obvious. Within the limits there are uncertainties of the type 0/0 or infinity/infinity . What to do in such cases? Resort to tricks!
Uncertainties within
Uncertainty of the form infinity/infinity
Let there be a limit:
If we try to substitute infinity into the function, we will get infinity in both the numerator and the denominator. In general, it is worth saying that there is a certain element of art in resolving such uncertainties: you need to notice how you can transform the function in such a way that the uncertainty goes away. In our case, we divide the numerator and denominator by X in the senior degree. What will happen?
From the example already discussed above, we know that terms containing x in the denominator will tend to zero. Then the solution to the limit is:
To resolve type uncertainties infinity/infinity divide the numerator and denominator by X to the highest degree.
By the way! For our readers there is now a 10% discount on
Another type of uncertainty: 0/0
As always, substituting values into the function x=-1 gives 0 in the numerator and denominator. Look a little more closely and you will notice that in our numerator quadratic equation. Let's find the roots and write:
Let's reduce and get:
So, if you are faced with type uncertainty 0/0 – factor the numerator and denominator.
To make it easier for you to solve examples, we present a table with the limits of some functions:
L'Hopital's rule within
Another powerful way to eliminate both types of uncertainty. What is the essence of the method?
If there is uncertainty in the limit, take the derivative of the numerator and denominator until the uncertainty disappears.
L'Hopital's rule looks like this:
Important point : the limit in which the derivatives of the numerator and denominator stand instead of the numerator and denominator must exist.
And now - a real example:
There is typical uncertainty 0/0 . Let's take the derivatives of the numerator and denominator:
Voila, uncertainty is resolved quickly and elegantly.
We hope that you will be able to usefully apply this information in practice and find the answer to the question “how to solve limits in higher mathematics.” If you need to calculate the limit of a sequence or the limit of a function at a point, and there is absolutely no time for this work, contact a professional student service for a quick and detailed solution.
From the above article you can find out what the limit is and what it is eaten with - this is VERY important. Why? You may not understand what determinants are and successfully solve them; you may not understand at all what a derivative is and find them with an “A”. But if you don’t understand what a limit is, then solving practical tasks will be difficult. It would also be a good idea to familiarize yourself with the sample solutions and my design recommendations. All information is presented in a simple and accessible form.
And for the purposes of this lesson we will need the following teaching materials: Wonderful Limits And Trigonometric formulas. They can be found on the page. It is best to print out the manuals - it is much more convenient, and besides, you will often have to refer to them offline.
What is so special about remarkable limits? The remarkable thing about these limits is that they were proven by the greatest minds of famous mathematicians, and grateful descendants do not have to suffer from terrible limits with a piling up trigonometric functions, logarithms, powers. That is, when finding the limits, we will use ready-made results that have been proven theoretically.
There are several wonderful limits, but in practice, in 95% of cases, part-time students have two wonderful limits: The first wonderful limit, Second wonderful limit. It should be noted that these are historically established names, and when, for example, they talk about “the first remarkable limit,” they mean by this a very specific thing, and not some random limit taken from the ceiling.
The first wonderful limit
Consider the following limit: (instead of the native letter “he” I will use the Greek letter “alpha”, this is more convenient from the point of view of presenting the material).
According to our rule for finding limits (see article Limits. Examples of solutions ) we try to substitute zero into the function: in the numerator we get zero (the sine of zero is zero), and in the denominator, obviously, there is also zero. Thus, we are faced with an uncertainty of the form, which, fortunately, does not need to be disclosed. In the course of mathematical analysis, it is proven that:
This mathematical fact is called The first wonderful limit. I won’t give an analytical proof of the limit, but here it is: geometric meaning we'll look at it in class about infinitesimal functions .
Often in practical tasks functions can be arranged differently, this does not change anything:
- the same first wonderful limit.
But you cannot rearrange the numerator and denominator yourself! If a limit is given in the form , then it must be solved in the same form, without rearranging anything.
In practice, not only a variable can act as a parameter, but also elementary function, complex function. The only important thing is that it tends to zero.
Examples:
, , 
, 
Here , , , 
, and everything is good - the first wonderful limit is applicable.
But the following entry is heresy:
Why? Because the polynomial does not tend to zero, it tends to five.
By the way, a quick question: what is the limit? 
? The answer can be found at the end of the lesson.
In practice, not everything is so smooth; almost never a student is offered to solve a free limit and get an easy pass. Hmmm... I’m writing these lines, and a very important thought came to mind - after all, it’s better to remember “free” mathematical definitions and formulas by heart, this can provide invaluable help in the test, when the question will be decided between “two” and “three”, and the teacher decides to ask the student some simple question or offer to solve simplest example(“maybe he (s) still knows what?!”).
Let's move on to consider practical examples:
Example 1
Find the limit
If we notice a sine in the limit, then this should immediately lead us to think about the possibility of applying the first remarkable limit.
First, we try to substitute 0 into the expression under the limit sign (we do this mentally or in a draft):
So we have an uncertainty of the form be sure to indicate in making a decision. The expression under the limit sign is similar to the first wonderful limit, but this is not exactly it, it is under the sine, but in the denominator.
In such cases, we need to organize the first remarkable limit ourselves, using an artificial technique. The line of reasoning could be as follows: “under the sine we have , which means that we also need to get in the denominator.”
And this is done very simply:
That is, the denominator is artificially multiplied by in this case by 7 and is divisible by the same seven. Now our recording has taken on a familiar shape.
When the task is drawn up by hand, it is advisable to mark the first remarkable limit with a simple pencil:
What happened? In fact, our circled expression turned into a unit and disappeared in the work: 
Now all that remains is to get rid of the three-story fraction: 
Who has forgotten the simplification of multi-level fractions, please refresh the material in the reference book Hot formulas for school mathematics course.
Ready. Final answer:
If you don’t want to use pencil marks, then the solution can be written like this:
“
Let's use the first wonderful limit 
“
Example 2
Find the limit
Again we see a fraction and a sine in the limit. Let’s try to substitute zero into the numerator and denominator:
Indeed, we have uncertainty and, therefore, we need to try to organize the first wonderful limit. At the lesson Limits. Examples of solutions we considered the rule that when we have uncertainty, we need to factorize the numerator and denominator. Here it’s the same thing, we’ll represent the degrees as a product (multipliers):
Similar to the previous example, we draw a pencil around the remarkable limits (here there are two of them), and indicate that they tend to unity:
Actually, the answer is ready:
IN following examples, I will not study art in Paint, I think how to correctly format the solution in a notebook - you already understand.
Example 3
Find the limit
We substitute zero into the expression under the limit sign:
An uncertainty has been obtained that needs to be disclosed. If there is a tangent in the limit, then it is almost always converted into sine and cosine using the well-known trigonometric formula (by the way, they do approximately the same thing with cotangent, see Fig. methodological material Hot trigonometric formulas On the page Mathematical formulas, tables and reference materials ).
In this case:
The cosine of zero is equal to one, and it’s easy to get rid of it (don’t forget to mark that it tends to one):
Thus, if in the limit the cosine is a MULTIPLIER, then, roughly speaking, it needs to be turned into a unit, which disappears in the product.
Here everything turned out simpler, without any multiplications and divisions. The first remarkable limit also turns into one and disappears in the product:
As a result, infinity is obtained, and this happens.
Example 4
Find the limit
Let's try to substitute zero into the numerator and denominator:
The uncertainty is obtained (the cosine of zero, as we remember, is equal to one)
We use trigonometric formula. Take note! For some reason, limits using this formula are very common.
Let us move the constant factors beyond the limit icon:
Let's organize the first wonderful limit:
Here we have only one remarkable limit, which turns into one and disappears in the product:
Let's get rid of the three-story structure:
The limit is actually solved, we indicate that the remaining sine tends to zero:
Example 5
Find the limit 
This example is more complicated, try to figure it out yourself:
Some limits can be reduced to the 1st remarkable limit by changing a variable, you can read about this a little later in the article Methods for solving limits .
Second wonderful limit
In the theory of mathematical analysis it has been proven that:
This fact is called second wonderful limit.
Reference: 
is an irrational number.
The parameter can be not only a variable, but also a complex function. The only important thing is that it strives for infinity.
Example 6
Find the limit
When the expression under the limit sign is in a degree, this is the first sign that you need to try to apply the second wonderful limit.
But first, as always, we try to substitute an infinitely large number into the expression, the principle by which this is done is discussed in the lesson Limits. Examples of solutions .
It is easy to notice that when the base of the degree is , and the exponent is , that is, there is uncertainty of the form:
This uncertainty is precisely revealed with the help of the second remarkable limit. But, as often happens, the second wonderful limit does not lie on a silver platter, and it needs to be artificially organized. One can reason as follows: in in this example parameter, which means that in the indicator we also need to organize . To do this, we raise the base to the power, and so that the expression does not change, we raise it to the power:
When the task is completed by hand, we mark with a pencil:
Almost everything is ready, the terrible degree has turned into a nice letter:
In this case, we move the limit icon itself to the indicator:
Example 7
Find the limit
Attention! This type of limit occurs very often, please study this example very carefully.
Let's try to substitute an infinitely large number into the expression under the limit sign:
The result is uncertainty. But the second remarkable limit applies to the uncertainty of the form. What to do? We need to convert the base of the degree. We reason like this: in the denominator we have , which means that in the numerator we also need to organize .
