How to find lim examples. How to solve limits for dummies

Limit theory - one of the sections of mathematical analysis, which one can master, others hardly calculate the limits. The question of finding the limits is quite general, since there are dozens of techniques solutions limits of various types. The same limits can be found both according to L'Hôpital's rule and without it. It happens that a schedule in a series of infinitely small functions allows you to quickly get the desired result. There are a number of techniques and tricks to find the limit of a function of any complexity. In this article, we will try to understand the main types of limits that are most often encountered in practice. We will not give the theory and definition of the limit here, there are many resources on the Internet where this is chewed. Therefore, let's get down to practical calculations, it is here that "I don’t know! I don’t know how! We were not taught!"

Computing Limits Using Substitution

Example 1. Find the limit of a function
Lim ((x ^ 2-3 * x) / (2 * x + 5), x \u003d 3).
Solution: Examples of this kind in theory are calculated by ordinary substitution

The limit is 18/11.
There is nothing complicated and wise within such limits - they substituted the value, calculated, wrote down the limit in response. However, on the basis of such limits, everyone is taught that the first thing to do is to substitute a value into a function. Further, the limits are complicated, the concept of infinity, uncertainty and the like is introduced.

Divide the limit with indefiniteness like infinity by infinity. Uncertainty Disclosure Techniques

Example 2. Find the limit of a function
Lim ((x ^ 2 + 2x) / (4x ^ 2 + 3x-4), x \u003d infinity).
Solution: A limit of the form of a polynomial is set, divided by a polynomial, and the variable tends to infinity

A simple substitution of the value to which the variable should be found to find the limits will not help, we get an uncertainty of the form infinity divided by infinity.
Sweat Limit Theory The algorithm for calculating the limit is to find the largest degree "x" in the numerator or denominator. Further, the numerator and denominator are simplified by it and the limit of the function is found

Since the value tends to zero with a variable to infinity, they are neglected, or written in the final expression in the form of zeros

Immediately from practice, you can get two conclusions that are a hint in the calculations. If the variable tends to infinity and the degree of the numerator is greater than the degree of the denominator, then the limit is equal to infinity. Otherwise, if the polynomial in the denominator is of higher order than in the numerator the limit is zero.
The limit can be written by formulas as follows

If we have a function of the form of an ordinary log without fractions, then its limit is equal to infinity

The next type of limit concerns the behavior of functions near zero.

Example 3. Find the limit of a function
Lim ((x ^ 2 + 3x-5) / (x ^ 2 + x + 2), x \u003d 0).
Solution: It is not required to take out the highest factor of the polynomial here. Quite the opposite, you need to find the smallest degree of the numerator and denominator and calculate the limit

X value ^ 2; x tend to zero when the variable tends to zero Therefore, they are neglected, thus we get

that the limit is 2.5.

Now you know how to find the limit of a function of the form polynomial divided by a polynomial if the variable tends to infinity or 0. But this is only a small and easy part of the examples. From the following material you will learn how to disclose uncertainties of the limits of a function.

Limit with uncertainty of type 0/0 and methods of its calculation

Everyone immediately remembers the rule according to which you cannot divide by zero. However, the theory of limits in this context means infinitesimal functions.
Let's look at a few examples for clarity.

Example 4. Find the limit of a function
Lim ((3x ^ 2 + 10x + 7) / (x + 1), x \u003d -1).
Solution: When substituting the value of the variable x \u003d -1 into the denominator, we get zero, the same thing we get in the numerator. So we have uncertainty of the form 0/0.
Dealing with such uncertainty is simple: you need to factor out the polynomial, or rather, select the factor that turns the function to zero.

After decomposition, the limit of the function can be written as

That's the whole technique for calculating the limit of a function. We do the same if there is a limit of the form of a polynomial divided by a polynomial.

Example 5. Find the limit of a function
Lim ((2x ^ 2-7x + 6) / (3x ^ 2-x-10), x \u003d 2).
Solution: Forward substitution shows
2*4-7*2+6=0;
3*4-2-10=0
what we have uncertainty type 0/0.
Divide the polynomials by a factor which introduces the singularity


There are teachers who teach that polynomials of the 2nd order, that is, of the form "quadratic equations", should be solved through the discriminant. But real practice shows that it is longer and more confusing, so get rid of the features within the specified algorithm. Thus, we write the function in the form of prime factors and enumerate in the limit

As you can see, there is nothing difficult in calculating such limits. At the time of studying the limits, you know how to divide polynomials, at least according to the program you should have already passed.
Among the tasks for uncertainty type 0/0there are those in which you need to apply the abbreviated multiplication formulas. But if you do not know them, then by dividing a polynomial by a monomial you can get the required formula.

Example 6. Find the limit of a function
Lim ((x ^ 2-9) / (x-3), x \u003d 3).
Solution: We have an uncertainty of type 0/0. In the numerator, we apply the formula for reduced multiplication

and calculate the required limit

The method of disclosing uncertainty by multiplying by the conjugate

The method is applied to the limits in which the uncertainties generate irrational functions. The numerator or denominator becomes zero at the point of calculation and it is not known how to find the border.

Example 7. Find the limit of a function
Lim ((sqrt (x + 2) -sqrt (7x-10)) / (3x-6), x \u003d 2).
Decision:We represent the variable in the limit formula

Substitution gives an uncertainty of type 0/0.
According to the theory of limits, the scheme for circumventing this feature is to multiply the irrational expression by the conjugate. In order for the expression not to change, the denominator must be divided by the same value

By the rule of difference of squares, we simplify the numerator and calculate the limit of the function

We simplify the terms that create a singularity in the limit and perform the substitution

Example 8. Find the limit of a function
Lim ((sqrt (x-2) -sqrt (2x-5)) / (3-x), x \u003d 3).
Solution: Forward substitution shows that the limit has a feature of the form 0/0.

To expand, we multiply and divide by the conjugate to the numerator

Writing the difference of squares

We simplify the terms that introduce the singularity and find the limit of the function

Example 9. Find the limit of a function
Lim ((x ^ 2 + x-6) / (sqrt (3x-2) -2), x \u003d 2).
Solution: Substitute 2 in the formula

We get uncertainty 0/0.
The denominator needs to be multiplied by the conjugate expression, and in the numerator you need to solve the quadratic equation or factor it into factors, taking into account the singularity. Since it is known that 2 is a root, we find the second root by Vieta's theorem

Thus, we write the numerator in the form

and substitute in the limit

By reducing the difference in squares, we get rid of the singularities in the numerator and denominator

In this way, you can get rid of the singularity in many examples, and the application should be noted wherever the given root difference turns into zero upon substitution. Other types of limits concern exponential functions, infinitesimal functions, logarithms, special limits, and other techniques. But you can read about this in the articles listed below on the limits.

application

Online limits to the site for the full consolidation of the passed material by students and schoolchildren. How to find the limit online using our resource? It is very easy to do this, you just need to write the original function with the x variable correctly, select the desired infinity from the selector and press the "Solution" button. In the case when the limit of a function must be calculated at some point x, then you need to specify the numerical value of this very point. You will receive the answer to solving the limit in a matter of seconds, in other words, instantly. However, if you provide incorrect data, the service will automatically inform you about an error. Correct the previously entered function and get the correct solution to the limit. To solve the limits, all possible tricks are used, especially the L'Hôpital's method is used, since it is universal and leads to an answer faster than other methods of calculating the limit of a function. It is interesting to consider examples in which the module is present. By the way, according to the rules of our resource, the module is denoted by the vertical line "|" or Abs (f (x)) from Latin absolute. Often a solution to the limit is required to calculate the sum of a number sequence. As everyone knows, you just need to correctly express the partial sum of the sequence under study, and then everything is much simpler, thanks to our free website service, since the calculation of the limit from the partial sum is the final sum of the numerical sequence. Generally speaking, the theory of passage to the limit is the basic concept of all mathematical analysis. Everything is based precisely on the passage to the limit, that is, the solution of the limits is laid in the foundation of the science of mathematical analysis. In integration, the passage to the limit is also used, when the integral according to the theory is represented as the sum of an unlimited number of areas. Where there is an unlimited number of something, that is, the tendency of the number of objects to infinity, then the theory of limit transitions always comes into force, and in the generally accepted form this is a solution to the familiar limits. Online solution of limits on the site The site is a unique service for receiving an accurate and instant response in real time. The limit of a function (limiting value of a function) at a given point, limiting for the domain of the function, is such a value to which the value of the function under consideration tends when its argument tends to a given point. It is not uncommon, and we would even say very often, that students have the question of solving online limits when studying mathematical analysis. When wondering about solving a limit online with a detailed solution only in special cases, it becomes clear that it is impossible to cope with a complex task without using a calculating limit calculator. The solution of the limits by our service is a guarantee of accuracy and simplicity. The limit of a function is a generalization of the concept of the limit of a sequence: initially, the limit of a function at a point was understood as the limit of a sequence of elements of the range of values \u200b\u200bof a function, composed of the images of points of a sequence of elements of the domain of definition of a function converging to a given point (limit at which is being considered); if such a limit exists, then the function is said to converge to the specified value; if such a limit does not exist, then the function is said to diverge. Solving the limits online for users becomes an easy answer provided they know how to solve the limit online using a website. Let's stay focused and don't let mistakes get us in trouble in the form of unsatisfactory ratings. 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We continue to analyze ready-made answers on the theory of limits and today we will dwell only on the case when a variable in a function or a number in a sequence tends to infinity. The instruction for calculating the limit with a variable tending to infinity was given earlier, here we will only dwell on individual cases that are not obvious and simple to everyone.

Example 35. We have a sequence in the form of a fraction, where the numerator and denominator are root functions.
It is necessary to find a limit when the number tends to infinity.
It is not necessary to reveal irrationality in the numerator here, but only to carefully analyze the roots and find where the higher degree of the number is.
In the first, the roots of the numerator have a factor of n ^ 4, that is, n ^ 2 can be taken out of the brackets.
Let's do the same with the denominator.
Next, we estimate the value of the radical expressions in the passage to the limit.

Got division by zero, which is wrong in school course, but this is permissible in the passage to the limit.
Only with the amendment "to assess where the function is aiming."
Therefore, not all teachers can interpret the given record as correct, although they understand that the resulting record will not change from this.
Let's look at the answer, compiled according to the requirements of teachers according to theory.
For simplicity, we will evaluate only the main dodankas under the root

Further, in the numerator the degree is 2, in the denominator 2/3, therefore the numerator grows faster, which means the limit tends to infinity.
Its sign depends on factors for n ^ 2, n ^ (2/3), so it is positive.

Example 36. Consider an example of the division limit of exponential functions. Few of such practical examples are considered, so not all students can easily see how to reveal the uncertainties that arise.
The maximum factor for the numerator and denominator is 8 ^ n, and we simplify by it

Next, we estimate the contribution of each term
The terms 3/8 tend to zero with the variable going to infinity, since 3/8<1 (свойство степенно-показательной функции).

Example 37. The limit of a sequence with factorials is expanded by assigning the factorial to the largest common factor for the numerator and denominator.
Then we reduce it and evaluate the limit by the value of the number indicators in the numerator and denominator.
In our example, the denominator grows faster, so the limit is zero.


The following is used here

factorial property.

Example 38. Without applying L'Hôpital's rules, we compare the maximum indicators of a variable in the numerator and denominator of the fraction.
Since the denominator contains the highest indicator of the variable 4\u003e 2, it grows faster.
Hence, we conclude that the limit of the function tends to zero.

Example 39. We reveal a singularity of the form infinity divided by infinity by the method of transferring x ^ 4 from the numerator and denominator of the fraction.
As a result of the passage to the limit, we get infinity.

Example 40. We have a division of polynomials, it is necessary to determine the limit as the variable tends to infinity.
The highest power of the variable in the numerator and denominator is 3, which means that the border exists and is equal to the steel.
Take out x ^ 3 and perform the passage to the limit

Example 41. We have a singularity of type one to the degree of infinity.
And this means that the expression in brackets and the indicator itself must be reduced under the second important border.
Let's write down the numerator to select an expression identical to the denominator in it.
Next, we turn to an expression containing one plus a term.
The degree must be distinguished by a factor of 1 / (term).
Thus, we get an exponent in the power of the limit of the fractional function.

The second limit was used to open the features:

Example 42. We have a singularity of type one to the degree of infinity.
For its disclosure, the function should be reduced to the second remarkable limit.
How to do this is shown in detail in the formula below.


You can find a lot of similar tasks. Their essence is to obtain the desired degree in the exponent, and it is equal to the reciprocal of the term in parentheses at unity.
Using this method, we get an exponent. Further calculation is reduced to calculating the limit of the exponent degree.
Here the exponential function tends to infinity, since the value is greater than one e \u003d 2.72\u003e 1.

Example 43 In the denominator of the fraction, we have an uncertainty of the type infinity minus infinity, in fact equal division to zero.
To get rid of the root, we multiply by the conjugate expression, and then rewrite the denominator using the formula for the difference of squares.
We get the uncertainty infinity divided by infinity, so we take out the variable to the greatest extent and reduce it by it.
Next, we estimate the contribution of each term and find the limit of the function at infinity

Constant number and called limit sequences(x n) if for any arbitrarily small positive numberε > 0 there is a number N that all values x n , for which n\u003e N, satisfy the inequality

| x n - a |< ε. (6.1)

They write it as follows: or x n → a.

Inequality (6.1) is equivalent to the double inequality

a- ε< x n < a + ε, (6.2)

which means that the points x n, starting from some number n\u003e N, lie inside the interval (a-ε, a + ε ), i.e. fall into any smallε -the neighborhood of the point and.

A sequence that has a limit is called converging, otherwise - diverging.

The concept of a limit of a function is a generalization of the concept of a limit of a sequence, since the limit of a sequence can be considered as the limit of a function x n \u003d f (n) of an integer argument n.

Let a function f (x) be given and let a - limit point the domain of this function D (f), i.e. a point, any neighborhood of which contains points of the set D (f) different from a... Point a may or may not belong to the set D (f).

Definition 1. The constant number A is called limit function f (x) atx →a if for any sequence (x n) of argument values \u200b\u200btending to and, the corresponding sequences (f (x n)) have the same limit A.

This definition is called the definition of the Heine limit of a function, or " in sequence language”.

Definition 2... The constant number A is called limit function f (x) at x →a if, by setting an arbitrary arbitrarily small positive number ε , one can find such δ \u003e 0 (depending on ε), which for all xlying inε-neighborhoods of the number and, i.e. for xsatisfying the inequality
0 < x-a< ε , the values \u200b\u200bof the function f (x) will lie inε-neighborhood of the number A, i.e.| f (x) -A |< ε.

This definition is called the definition of the Cauchy limit of a function,or “In the language ε - δ “.

Definitions 1 and 2 are equivalent. If the function f (x) as x → a has limitequal to A, this is written as

. (6.3)

In the event that the sequence (f (x n)) increases (or decreases) indefinitely for any method of approximation x to your limit and, then we say that the function f (x) has endless limit, and write it down as:

A variable (i.e., a sequence or function) whose limit is zero is called infinitely small value.

A variable whose limit is equal to infinity is called infinitely large.

To find the limit in practice, use the following theorems.

Theorem 1 ... If there is every limit

(6.4)

(6.5)

(6.6)

Comment... Expressions like 0/0, ∞/∞, ∞-∞ , 0*∞ , - are uncertain, for example, the ratio of two infinitesimal or infinitely large quantities, and finding a limit of this kind is called "disclosure of uncertainties."

Theorem 2. (6.7)

those. you can go to the limit based on the degree with a constant exponent, in particular, ;

(6.8)

(6.9)

Theorem 3.

(6.10)

(6.11)

where e » 2.7 is the base of the natural logarithm. Formulas (6.10) and (6.11) are called the first wonderful limitand the second remarkable limit.

The consequences of formula (6.11) are also used in practice:

(6.12)

(6.13)

(6.14)

in particular the limit

If x → a and at the same time x\u003e a, then they write x → a + 0. If, in particular, a \u003d 0, then instead of the symbol 0 + 0, write +0. Similarly, if x →a and, moreover, x a-0. Numbers and are called accordingly right limit and left limit function f (x) at the point and... For the function f (x) to have a limit as x →a is necessary and sufficient to ... The function f (x) is called continuous at the pointx 0 if limit

. (6.15)

Condition (6.15) can be rewritten as:

,

that is, the passage to the limit under the sign of the function is possible if it is continuous at a given point.

If equality (6.15) is violated, then it is said that at x \u003d x o function f (x) it has break.Consider the function y \u003d 1 / x. The domain of this function is the set R, except for x \u003d 0. The point x \u003d 0 is the limit point of the set D (f), since in any of its neighborhood, that is, any open interval containing point 0 contains points from D (f), but it does not itself belong to this set. The value f (x o) \u003d f (0) is undefined, so at the point x o \u003d 0 the function has a discontinuity.

The function f (x) is called continuous on the right at the point x o, if the limit

,

and left continuous at the point x o, if the limit

.

Continuity of a function at a point x o is equivalent to its continuity at this point both on the right and on the left.

For the function to be continuous at the point x o, for example, on the right, it is necessary, firstly, that there is a finite limit, and secondly, that this limit be equal to f (x o). Therefore, if at least one of these two conditions is not met, then the function will have a gap.

1. If the limit exists and is not equal to f (x o), then they say that function f (x) at the point x o has break of the first kind, or leap.

2. If the limit is + ∞ or -∞ or does not exist, then they say that in point x o function has a gap second kind.

For example, the function y \u003d ctg x for x→ +0 has a limit equal to + ∞therefore, at the point x \u003d 0 it has a discontinuity of the second kind. Function y \u003d E (x) (integer part of x) at points with integer abscissas has discontinuities of the first kind, or jumps.

A function that is continuous at each point of the interval is called continuous in . A continuous function is shown as a solid curve.

Many problems associated with the continuous growth of any quantity lead to the second remarkable limit. Such tasks, for example, include: the growth of the contribution according to the law of compound interest, the growth of the country's population, the decay of radioactive substances, the reproduction of bacteria, etc.

Consider example of Ya.I. Perelmangiving an interpretation of the number e in the problem of compound interest. Number ethere is a limit ... In savings banks, interest money is added to the fixed capital annually. If the connection is made more often, then the capital grows faster, since a large amount is involved in the formation of interest. Let's take a purely theoretical, highly simplified example. Let the bank put 100 den. units at the rate of 100% per annum. If interest money will be added to the fixed capital only after a year, then by this date 100 den. units will turn into 200 monetary units. Now let's see what will turn into 100 den. units, if interest money is added to the fixed capital every six months. After half a year 100 den. units grow to 100× 1.5 \u003d 150, and six months later - 150× 1.5 \u003d 225 (monetary units). If the connection is done every 1/3 of the year, then after the year 100 den. units turn into 100× (1 +1/3) 3 " 237 (monetary units). We will speed up the terms for joining interest-bearing money to 0.1 years, to 0.01 years, to 0.001 years, etc. Then from 100 den. units after a year it will turn out:

100 × (1 +1/10) 10 "259 (monetary units),

100 × (1 + 1/100) 100 * 270 (monetary units),

100 × (1 + 1/1000) 1000 * 271 (monetary units).

With an unlimited reduction in the terms of interest attachment, the accrued capital does not grow infinitely, but approaches a certain limit equal to approximately 271. The capital allocated at 100% per annum cannot increase by more than 2.71 times, even if the accrued interest was added to the capital each second because the limit

Example 3.1. Using the definition of the limit of a number sequence, prove that the sequence x n \u003d (n-1) / n has a limit equal to 1.

Decision.We need to prove that whateverε We did not take\u003e 0, for it there is a natural number N such that for all n N the inequality | x n -1 |< ε.

Take any e\u003e 0. Since; x n -1 \u003d (n + 1) / n - 1 \u003d 1 / n, then to find N it is enough to solve the inequality 1 / n< e. Hence n\u003e 1 / e and, therefore, N can be taken as the integer part of 1 /e, N \u003d E (1 / e ). We have thus proved that the limit.

Example 3.2 ... Find the limit of a sequence given by a common term .

Decision.We apply the sum limit theorem and find the limit of each term. For n→ ∞, the numerator and denominator of each term tends to infinity, and we cannot directly apply the quotient limit theorem. Therefore, we first transform x nby dividing the numerator and denominator of the first term by n 2, and the second on n... Then, applying the quotient limit and sum limit theorem, we find:

.

Example 3.3. ... To find .

Decision. .

Here we used the degree limit theorem: the degree limit is equal to the degree of the base limit.

Example 3.4 ... To find ( ).

Decision.The limit difference theorem cannot be applied, since we have an uncertainty of the form ∞-∞ ... We transform the formula for the common member:

.

Example 3.5 ... A function f (x) \u003d 2 1 / x is given. Prove that there is no limit.

Decision.Let us use the definition 1 of the limit of a function through a sequence. Take a sequence (x n) converging to 0, i.e. Let us show that the value f (x n) \u003d behaves differently for different sequences. Let x n \u003d 1 / n. Obviously, then the limit We now choose as x n a sequence with a common term x n \u003d -1 / n, also tending to zero. Therefore, there is no limit.

Example 3.6 ... Prove that there is no limit.

Decision.Let x 1, x 2, ..., x n, ... be a sequence for which
... How the sequence (f (x n)) \u003d (sin x n) behaves for different x n → ∞

If x n \u003d p n, then sin x n \u003d sin p n \u003d 0 for all n and the limit If
x n \u003d 2p n + p / 2, then sin x n \u003d sin (2 p n + p / 2) \u003d sin p / 2 \u003d 1 for all n and hence the limit. So it doesn't exist.

Widget for calculating limits on-line

In the upper window, instead of sin (x) / x, enter the function whose limit you want to find. In the lower window, enter the number to which x tends and click the Calcular button, get the desired limit. And if in the result window you click on Show steps in the upper right corner, you will get a detailed solution.

Function entry rules: sqrt (x) - square root, cbrt (x) - cube root, exp (x) - exponent, ln (x) - natural logarithm, sin (x) - sine, cos (x) - cosine, tan (x) - tangent, cot (x) - cotangent, arcsin (x) - arcsine, arccos (x) - arccosine, arctan (x) - arctangent. Signs: * multiplication, / division, ^ exponentiation, instead of infinity Infinity. Example: the function is entered like this sqrt (tan (x / 2)).

Limits give all math students a lot of trouble. To solve the 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 specific 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 the limits with explanations.

Limit concept in mathematics

The first question: what is this limit and what is the limit? We can talk about the limits of numerical sequences and functions. We are interested in the concept of the limit of a function, since it is with them that students most often encounter. But first, the most general definition of a limit:

Let's say there is some variable. If this value in the process of change is unlimitedly approaching a certain number a then a Is the limit of this value.

For a function defined in a certain interval f (x) \u003d y the limit is such a number A , to which the function tends at x tending to a certain point and ... Point and belongs to the interval on which the function is defined.

It sounds cumbersome, but it's very simple to write:

Lim - from English limit is the limit.

There is also a geometric explanation for the definition of the limit, but here we will not go into theory, since we are more interested in the practical than the theoretical side of the issue. When we say that x tends to some value, this means that the variable does not take the value of the number, but is infinitely close to it.

Let's give a concrete example. The challenge is to find the limit.

To solve this example, substitute the value x \u003d 3 into a function. We get:

By the way, if you are interested, read a separate article on this topic.

In examples x can strive for any value. It can be any number or infinity. Here is an example when x tends to infinity:

It is intuitively clear that the larger the number in the denominator, the lower the value the function will take. So, with unlimited growth x value 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. Finding the limit is often not so obvious. Uncertainties such as 0/0 or infinity / infinity ... What to do in such cases? To 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 get infinity in both the numerator and denominator. In general, it is worth saying that there is a certain element of art in resolving such uncertainties: it must be noted how a function can be transformed in such a way that the uncertainty disappears. In our case, we divide the numerator and denominator by x in the senior degree. What happens?

From the example already considered above, we know that the terms containing x in the denominator will tend to zero. Then the solution to the limit is:

To disclose uncertainties like infinity / infinity divide the numerator and denominator by x to the highest degree.

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Another type of uncertainty: 0/0

As always, substitution in the value function x \u003d -1 gives 0 in the numerator and denominator. Look a little more closely and you will notice that we have a quadratic equation in the numerator. Find the roots and write:

Let's shorten and get:

So, if you are faced with an uncertainty like 0/0 - factor out the numerator and denominator.

To make it easier for you to solve examples, we give a table with the limits of some functions:

L'Hôpital's rule within

Another powerful way to remove both types of uncertainty. What is the essence of the method?

If there is uncertainty in the limit, we take the derivative of the numerator and denominator until the uncertainty disappears.

Lopital's rule looks like this:

An important point : the limit in which instead of the numerator and denominator are derivatives of the numerator and denominator, must exist.

And now for a real example:

Typical uncertainty 0/0 ... Let's take the derivatives of the numerator and denominator:

Voila, ambiguity is resolved quickly and elegantly.

We hope that you can usefully apply this information in practice and find an 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 no time for this work from the word "at all", contact a professional student service for a quick and detailed solution.