Calculate limits lim examples. Limit of a function – definitions, theorems and properties

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 pile of 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 we’ll look at its geometric meaning in the lesson 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, but also an elementary function or a complex function can act as a parameter. It is only important 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 a simple example (“maybe he/she 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 in this case by 7 and divided 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. In class 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 the following examples, I will not do art in Paint, I think how to correctly draw up a 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 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 the 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. You can reason as follows: in this example the parameter is , which means that we also need to organize in the indicator. 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 .

Solving problems on finding limits When solving problems on finding limits, you should remember some limits so as not to calculate them again each time. Combining these known limits, we will find new limits using the properties indicated in § 4. For convenience, we present the most frequently encountered limits: Limits 1 lim x - a x a 2 lim 1 = 0 3 lim x- ± co X ± 00 4 lim -L, = oo X->o\X\ 5 lim sin*- l X -о X 6 lim f(x) = f(a), if f (x) is continuous x a If it is known that the function is continuous, then instead of finding the limit, we calculate the value of the function. Example 1. Find lim (x*-6l:+ 8). Since the multi-term X->2 term function is continuous, then lim (x*-6x4- 8) = 2*-6-2 + 8 = 4. x-+2 x*_2x 4-1 Example 2. Find lim -G. . First, we find the limit of the denominator: lim [xr-\-bx)= 12 + 5-1 =6; it is not equal to X-Y1 zero, which means we can apply property 4 § 4, then x™i *" + &* ~~ lim (x2 bx) - 12 + 5-1 ""6 1. The limit of the denominator X X is equal to zero, therefore, property 4 of § 4 cannot be applied. Since the numerator is a constant number, and the denominator is [x2x) -> -0 for x - 1, then the entire fraction increases indefinitely in absolute value, i.e. lim " 1 X - * - - 1 x* + x Example 4. Find lim\-ll*"!"" "The limit of the denominator is zero: lim (xr-6lg+ 8) = 2*-6-2 + 8 = 0, so X property 4 § 4 not applicable. But the limit of the numerator is also zero: lim (x2 - 5d; + 6) = 22 - 5-2-f 6 = 0. So, the limits of the numerator and denominator are simultaneously equal to zero. However, the number 2 is the root of both the numerator and the denominator, so the fraction can be reduced by the difference x-2 (according to Bezout’s theorem). In fact, x*-5x + 6 (x-2) (x-3) x-3 x"-6x + 8~ (x-2) (x-4) ~~ x-4" therefore, xr- -f- 6 g x-3 -1 1 Example 5. Find lim xn (n integer, positive). X with We have xn = X* X . . X, n times Since each factor grows without limit, the product also grows without limit, i.e. lim xn=oo. x oo Example 6. Find lim xn(n integer, positive). X -> - CO We have xn = x x... x. Since each factor grows in absolute value while remaining negative, then in the case of an even degree the product will grow unlimitedly while remaining positive, i.e. lim *n = + oo (for even n). *-* -о In the case of an odd degree, the absolute value of the product increases, but it remains negative, i.e. lim xn = - oo (for n odd). p -- 00 Example 7. Find lim . x x-*- co * If m>pu then we can write: m = n + kt where k>0. Therefore xm b lim -=- = lim -=-= lim x. UP Yn x - x> A x yu We came to example 6. If ti uTL xm I lim lim lim t. X - O x-* yu A X ->co Here the numerator remains constant, and the denominator grows in absolute value, so lim -ь = 0. X-*oo X* It is recommended to remember the result of this example in the following form: The power function grows as faster, the larger the exponent. $хв_Зхг + 7 Example 8. Find lim g L -г-=. In this example x-*® «J* "Г bХ -ох-о and the numerator and denominator increase without limit. Let us divide both the numerator and the denominator by the highest power of x, i.e. on xb, then 3 7_ Example 9. Find lira. Performing transformations, we obtain lira ^ = lim X CO + 3 7 3 Since lim -5 = 0, lim -, = 0. , then the limit of the denominator is equal to 1. Therefore, the whole fraction increases without limit, i.e. t. lim Let's calculate the limit S of the denominator, remembering that the cos*-function is continuous: lira (2 + cos x) = 2 + cozy =2. Then x->- S lim (l-fsin*) Example 15. Find lim *<*-e>2 and lim e "(X"a)\ Polo X-+ ± co X ± CO press (l: - a)2 = z; since (l;-a)2 always grows non-negatively and without limit with x, then for x - ±oo the new variable z-*oc. Therefore we obtain qt £<*-«)* = X ->± 00 s=lim ег = oo (see note to §5). g -*■ co Similarly lim e~(X-a)2 = lim e~z=Q, since x ± oo g m - (x- a)z decreases without limit as x ->±oo (see note to §

Let's look at some illustrative examples.

Let x be a numerical variable, X the area of its change. If each number x belonging to X is associated with a certain number y, then they say that a function is defined on the set X, and write y = f(x).
The X set in this case is a plane consisting of two coordinate axes – 0X and 0Y. For example, let's depict the function y = x 2. The 0X and 0Y axes form X - the area of its change. The figure clearly shows how the function behaves. In this case, they say that the function y = x 2 is defined on the set X.

The set Y of all partial values of a function is called the set of values f(x). In other words, the set of values is the interval along the 0Y axis where the function is defined. The depicted parabola clearly shows that f(x) > 0, because x2 > 0. Therefore, the range of values will be . We look at many values by 0Y.

The set of all x is called the domain of f(x). We look at many definitions by 0X and in our case the range of acceptable values is [-; +].

A point a (a belongs to or X) is called a limit point of the set X if in any neighborhood of the point a there are points of the set X different from a.

The time has come to understand what is the limit of a function?

The pure b to which the function tends as x tends to the number a is called limit of the function. This is written as follows:

For example, f(x) = x 2. We need to find out what the function tends to (is not equal to) at x 2. First, we write down the limit:

Let's look at the graph.

Let's draw a line parallel to the 0Y axis through point 2 on the 0X axis. It will intersect our graph at point (2;4). Let’s drop a perpendicular from this point onto the 0Y axis and get to point 4. This is what our function strives for at x 2. If we now substitute the value 2 into the function f(x), the answer will be the same.

Now before we move on to calculation of limits, let us introduce basic definitions.

Introduced by the French mathematician Augustin Louis Cauchy in the 19th century.

Let's say the function f(x) is defined on a certain interval that contains the point x = A, but it is not at all necessary that the value of f(A) be defined.

Then, according to Cauchy's definition, limit of the function f(x) will be a certain number B with x tending to A if for every C > 0 there is a number D > 0 for which

Those. if the function f(x) at x A is limited by limit B, this is written as

Sequence limit a certain number A is called if for any arbitrarily small positive number B > 0 there is a number N for which all values in the case n > N satisfy the inequality

This limit looks like .

A sequence that has a limit will be called convergent; if not, we will call it divergent.

As you have already noticed, limits are indicated by the lim icon, under which some condition for the variable is written, and then the function itself is written. Such a set will be read as “the limit of a function subject to...”. For example:

- the limit of the function as x tends to 1.

The expression “approaching 1” means that x successively takes on values that approach 1 infinitely close.

Now it becomes clear that to calculate this limit it is enough to substitute the value 1 for x:

In addition to a specific numerical value, x can also tend to infinity. For example:

The expression x means that x is constantly increasing and approaching infinity indefinitely. Therefore, substituting infinity for x, it becomes obvious that the function 1-x will tend to , but with the opposite sign:

Thus, calculation of limits comes down to finding its specific value or a certain area in which the function limited by the limit falls.

Based on the above, it follows that when calculating limits it is important to use several rules:

Understanding essence of limit and basic rules limit calculations, you'll gain key insight into how to solve them. If any limit causes you difficulties, then write in the comments and we will definitely help you.

Note: Jurisprudence is the science of laws, which helps in conflicts and other life difficulties.

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 of 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 it.

In general, if one is 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 the highest power of the unknown.

b) In general, 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 transformed 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, a compound interest scheme is used. 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, you can derive a general formula for calculating the growth of 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 is the integer part of the 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 in the practice of financial activity, the concept of “continuously accrued interest” is often encountered. 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 to 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.

Function limit- number a will be the limit of some variable quantity if, in the process of its change, this variable quantity indefinitely approaches a.

Or in other words, the number A is the limit of the function y = f(x) at the point x 0, if for any sequence of points from the domain of definition of the function , not equal x 0, and which converges to the point x 0 (lim x n = x0), the sequence of corresponding function values converges to the number A.

The graph of a function whose limit, given an argument that tends to infinity, is equal to L:

Meaning A is limit (limit value) of the function f(x) at the point x 0 in case for any sequence of points , which converges to x 0, but which does not contain x 0 as one of its elements (i.e. in the punctured vicinity x 0), sequence of function values converges to A.

Limit of a Cauchy function.

Meaning A will be limit of the function f(x) at the point x 0 if for any non-negative number taken in advance ε the corresponding non-negative number will be found δ = δ(ε) such that for each argument x, satisfying the condition 0 < | x - x0 | < δ , the inequality will be satisfied | f(x)A |< ε .

It will be very simple if you understand the essence of the limit and the basic rules for finding it. What is the limit of the function f (x) at x striving for a equals A, is written like this:

Moreover, the value to which the variable tends x, can be not only a number, but also infinity (∞), sometimes +∞ or -∞, or there may be no limit at all.

To understand how find the limits of a function, it is best to look at examples of solutions.

It is necessary to find the limits of the function f (x) = 1/x at:

x→ 2, x→ 0, x→ ∞.

Let's find a solution to the first limit. To do this, you can simply substitute x the number it tends to, i.e. 2, we get:

Let's find the second limit of the function. Here substitute pure 0 instead x it is impossible, because You cannot divide by 0. But we can take values close to zero, for example, 0.01; 0.001; 0.0001; 0.00001 and so on, and the value of the function f (x) will increase: 100; 1000; 10000; 100,000 and so on. Thus, it can be understood that when x→ 0 the value of the function that is under the limit sign will increase without limit, i.e. strive towards infinity. Which means:

Regarding the third limit. The same situation as in the previous case, it is impossible to substitute ∞ in its purest form. We need to consider the case of unlimited increase x. We substitute 1000 one by one; 10000; 100000 and so on, we have that the value of the function f (x) = 1/x will decrease: 0.001; 0.0001; 0.00001; and so on, tending to zero. That's why:

It is necessary to calculate the limit of the function

Starting to solve the second example, we see uncertainty. From here we find the highest degree of the numerator and denominator - this is x 3, we take it out of brackets in the numerator and denominator and then reduce it by:

Answer

The first step in finding this limit, substitute the value 1 instead x, resulting in uncertainty. To solve it, let’s factorize the numerator and do this using the method of finding the roots of a quadratic equation x 2 + 2x - 3:

D = 2 2 - 4*1*(-3) = 4 +12 = 16→ √ D=√16 = 4

x 1.2 = (-2±4)/2→ x 1 = -3;x 2= 1.

So the numerator will be:

Answer