What is Robida s Law and what is Lobida s Law?

Lopida's Law (l'holpital's rule) is a method for determining the value of the infinitive formula by finding the derivative and then the limit of the numerator and denominator respectively under certain conditions. Let (1) when x a, the functions f(x) and f(x) both tend to zero; (2) In the decentering neighborhood of point a, f'(x) and f'(x) both exist and f'(x)≠0；(3) When x a lim f'(x)/f'(x) exists (or is infinity), then when x a lim f(x) f(x) = lim f'(x)/f'(x)。Let (1) when x, the functions f(x) and f(x) both tend to zero; (2) When|x|> n f'(x) and f'(x) both exist and f'(x)≠0；(3) When x lim f'(x)/f'(x) exists (or is infinity), then at x lim f(x) f(x)=lim f'(x)/f'(x)。

Finding the limit of the infinitive by using Lopida's rule is one of the key points in differential calculus, and it should be noted that before starting to find the limit, we should first check whether it satisfies the 0 0 or type infinitive, otherwise it will be wrong to abuse the Lopida's rule. When it does not exist (excluding cases), the law of Lobida cannot be used, and then the law of Lobida is not applicable, and the limit should be found in another way.

For example, use Taylor's formula to solve the problem. If the conditions are met, the Lopida rule can be used several times in a row until the limit is reached. Lopida's rule is an effective tool for finding the limit of the infinitive, but if you only use Lopida's rule, the calculation will often be very cumbersome, so it must be combined with other methods, such as separating the product factor of the non-zero limit in time to simplify the calculation, replacing the product factor with an equivalent quantity, etc.

Lobida's law is often used to find the limits of infinitives. Basic infinitive limit: 0 0 type; The limits of the forms (x or x a), and other forms such as 0*, 0, and 0 0 can be solved by the corresponding transformations into the above two basic infinitive forms.

Quite troublesome, high number content.

Robida's Law:

Application: Generally, when the numerator and denominator are both close to zero or infinity, the limit of the formula cannot be obtained by substituting the value, so Robida's rule is used.

Method: Find the derivative of the numerator and denominator at the same time until the numerator or denominator is not zero or infinite, and then the result of substituting the independent variable to find the formula. Premise: Both numerator and denominator can be derived.

To take the simplest example, for (x 2) (x 4), find the limit when x tends to zero, find the first derivative of the numerator to become 2x, and find the first derivative of the denominator to become 4x 3, because it is still impossible to find the limit by substituting x=0, so we use Robida's rule: the numerator becomes 2 at this time, and the denominator becomes 12x 2, and the limit of the formula can be found to be infinity.

When calculating the limit, if there is a 0 0 or type of situation, you can use the Robida rule.

lim(x a) f(x) g(x) = lim(x a) f '(x) g ' x), firstly, it must satisfy the 0 0 or type, and secondly, the limit of the latter must also exist.

Lobida's LawIt is determined by finding the derivation and then limit of the numerator and denominator respectively under certain conditionsInfinitivevalue. As we all know, the ratio of two infinitesimal or twoInfinityThe limit of the ratio may or may not exist.

Therefore, it is often necessary to find such limits in a form that can be calculated using limit algorithms or important limits. Lopida's rule is a general method applied to this type of limit calculation.

Finding the limit is higher mathematics.

It is also one of the most important parts of Advanced Mathematics, so it is of great significance to master the method of finding the limit to learn Advanced Mathematics well. Lobida's Law.

It is used to find the fractional limit where the numerator and denominator tend to zero.

Conditions of application:

Before applying Lopida's rule, two tasks must be completed: first, whether the limits of the numerator and denominator are equal to zero (or infinity); The second is whether the numerator and denominator are respectively derivable in the defined region.

If both of these conditions are satisfied, then the derivative is sought and the limit after the derivative is determined: if so, the answer is obtained directly; If it does not exist, it means that this kind of indefinite formula cannot be solved by the law of Lobida deficiency; If you are unsure, i.e. the result is still infinitive, then continue to use the Lopida's rule on the basis of validation.

The above content refers to the Encyclopedia-Lobida Rule.

Specifically, there are quite a lot, just to give an example.

lim(x→0) (x^2 / cos x) =lim(x→0) (2x/sin x) =2

The first step uses Lopida, (x 2).'=2x, (cos x)'=sin x

In the second step, an equivalent infinitesimal quantity is used. Hereinafter**.

Lopida's Law (l'hospital) rule is a method for determining the value of the infinitive formula by finding the derivative and then the limit of the numerator and denominator respectively under certain conditions.

Establish. 1) When x a, the functions f(x) and f(x) tend to zero;

(2) In the decentering neighborhood of point a, f'(x) and f'(x) both exist and f'(x)≠0；

(3) When x a lim f'(x)/f'(x) existence (or infinity), then.

lim f(x) f(x)=lim f. at x a'(x)/f'(x).

Again. 1) When x, the functions f(x) and f(x) tend to zero;

(2) When|x|> n f'(x) and f'(x) both exist and f'(x) 0 ≠ bridge judgment;

(3) When x lim f'(x)/f'(x) Shedding means in (or for infinity), then.

lim f(x) f(x)=lim f'(x) Chain elimination f'(x).

Lopida's rule is a method to determine the value of the infinitive formula by finding the derivation of the numerator and denominator respectively and then finding the polar mountain elimination limit under certain conditions. This method is mainly used to determine the value of the infinitive formula by finding the derivative and then the limit of the numerator and denominator respectively under certain conditions.

Before applying Lopida's rule, two tasks must be accomplished: whether the limits of the numerator and denominator are equal to zero (or infinity); The second is whether the numerator and denominator are respectively derivable in the defined region. If both of these conditions are satisfied, then the derivative is sought and the limit after the derivative is determined: if so, the answer is obtained directly.

If it does not exist, then the infinitive cannot be solved by Lopida's rule; If you are unsure, i.e. the result is still infinitive, then continue to use the Lopida's rule on the basis of validation.

Lopida's Law Formulas and Conditions:

Let the functions f(x) and f(x) meet the following conditions:

1. When x a, lim f(x)=0, lim f(x)=0;

2. Both f(x) and f(x) are derivative in a decentered neighborhood of point a, and the derivative of f(x) is not equal to 0;

3. When x a, lim(f'(x)/f'(x)) exists or is infinity.

Robida's rule: When the numerator and denominator tend to zero or infinity, the limit of the formula cannot be obtained by substituting the value, so the application method of Robida's law is used

Finding the derivative of the numerator and denominator at the same time until the numerator or denominator is not zero or infinite is calculated to calculate the result of substituting the independent variables. Premise: Both numerator and denominator can be derived.

To take the simplest example, for (x 2) (x 4), find the limit when x tends to zero, find the first derivative of the numerator to become 2x, and find the first derivative of the denominator to become 4x 3, because it is still impossible to find the limit by substituting x=0, so we use Robida's rule: the numerator becomes 2 at this time, and the denominator becomes 12x 2, and the limit of the formula can be found to be infinity. The landlord can try to calculate the limit of x tending to 1 and tending to infinity himself, and it will be clear.

1) When x a, the functions f(x) and g(x) tend to zero; (2) In the decentering neighborhood of point a, f'(x) and g'(x) both exist and g'(x)≠0；(3) When x a lim f'(x)/g'(x) exists (or is infinity), then when x a lim f(x) f(x) = lim f'(x)/f'(x)。In addition, the limit of the sequence cannot be used using Robida's rule because it does not satisfy condition (2).

Theorem 1:

1) When x approaches a, the functions f(x) and f(x) both tend to 0;

2) In some decentric neighborhood of point a, f'(x) and f'(x) both exist and f'(x) is not equal to 0;

3）limf'(x)/f'(x) (x tends to a) exists or is infinite;

Then limf(x) f(x)=limf'(x)/f'(x), x approaches a

Theorem 2: 1) When x approaches infinity, both f(x) and f(x) tend to 0;

2) When the absolute value of x is greater than a certain value n, f'(x) and f'(x) both exist and f'(x) is not equal to 0;

3）limf'(x)/f'(x) (x tends to infinity) exists or is infinite;

Then limf(x) f(x)=limf'(x)/f'(x), x tends to infinity.

(Actually, you don't have to worry about the law of Robida, as long as you can use it.) The derivative will be used as a tool to find the limit of the general 0 0 type infinitive, which is called l'Hospital Law.

is a law for finding the limit of a function, for infinitives.

For example, when x tends to 0, the (sinx) x numerator and denominator limit are all 0, so the limit law of the quotient does not work, then Lobida's law can be applied.

For example, when x tends to a, if f(x) and g(x) both tend to 0 (or infinity), then f(x) g(x) is equal to f at the limit of x tending to a'(x)/g'(x) The limit at which x tends to a.

Other infinitives (e.g. 0 0, infinity to the power of 0, infinity minus infinity, etc.) can be reduced to the form f(x) g(x) and Lobida's rule is applied.

That is, when finding the limit, if the limits of the numerator and the denominator are both infinite or the limits are zero, the derivatives of the numerator and denominator are respectively derived.