Can the equivalent infinitesimal be replaced locally?

It cannot be replaced. Equivalent infinitesimal

The premise of substitution is the numerator (denominator.

The factor in is fine.

For example, let a b (ab be equivalent infinitesimal ) a*a, a (c + d) where a can be replaced by b; But (a+c) and so on a is not a factor. When finding the limit, infinitesimal amounts.

Multiply and divide, the infinitesimal quantity in the period can be replaced by an infinitesimal quantity of the same order. Two infinitesimal quantities of the same order mean: infinitesimal a infinitesimal b=n (constant).

When finding the limit. Use the condition of equivalent infinitesimal:

The limit value of the amount to be substituted is 0 when taking the limit;

The quantity to be substituted can be substituted as an element to be multiplied or divided, but not as an element of addition or subtraction.

The equivalent infinitesimal is locally replaceable. The condition is that the neglected part is infinitesimal than the other terms that are "higher".

Equivalent infinitesimal use condition: the amount to be substituted, the limit value is 0 when the limit is removed. The quantity to be substituted can be substituted as an element to be multiplied or divided, but not as an element of addition or subtraction.

The equivalent infinitesimal cannot be replaced in the local part, because the operation process may be cumbersome to achieve in the replacement process. So it's better not to replace.

The product factor can be replaced, and a very small number of additions and subtractions can also be replaced, and you obviously can't replace sincosx, which tends to sin1, and your substitution results tend to 0, which is obviously not equivalent.

Teacher Wu's handout: I think of sinxcosx as 1 2sin2x and (x+1 2sin2x) 2x

The equivalent infinitesimal substitution formula is as follows:

There are two main principles for using equivalent infinitesimal beings:

1. The multiplication and division limit is used directly.

2. Look at the numerator denominator order when adding or subtracting the limit. If the order of the numerator and denominator is the same after the equivalent infinitesimal is used; Not available if the order is different.

When finding the limit, use the condition of the equivalent infinitesimal :

1. The limit value of the amount to be substituted is 0 when taking the limit;

2. The amount to be substituted can be substituted as an equivalent infinitesimal when it is an element to be multiplied or divided, but it cannot be substituted as an element of addition or subtraction, and it can be substituted as a whole when adding or subtracting, and it may not be substituted separately or separately at will.

The substitution formula for the equivalent infinitesimal is as follows: when x approaches 0: e x-1 x; ln(x+1) ~x；sinx ~ x；arcsinx ~ x；tanx ~ x；arctanx ~ x；1-cosx ~ x^2)/2；tanx-sinx ~ x^3)/2；(1+bx)^a-1 ~ abx；It is an equivalent infinitesimal substitution, which is generally used in multiplication and division, and is generally not used in addition and subtraction.

An infinitesimal is a variable with a limit of zero. However, constants are a special class of variables, just as a straight line is a type of curve. Therefore, constants can also be studied as coarse bonds as variables.

is a constant that can be used as an infinitesimal number. On the other hand, the equivalent infinitesimal can also be seen as Taylor's formula from zero to the first order.

2. When x tends to 0, find the limit, which can be solved by using the equivalent infinitesimal to solve it. When x tends to 0, finding f(x sin x) can also be solved using the equivalent infinitesimal solution. x and sin x are equal infinitesimals, so the limit of the function can be found.

3. Equivalent infinitesimal: high numbers are often used to find the limit when x tends to 0, of course, x tends to be hidden in infinity, and it can also be found when it is converted into a reciprocal to become an equivalent infinitesimal.

Replaceable.

If each of the addition and subtraction terms is infinitesimal, each is equivalent infinitesimal.

If the result is not 0, it can be replaced. Use Taylor's Hill formula with caution.

Finding the limit is based on this idea. For example, find the limit of (tanx-sinx) (x 3) when x >0. Use the Lopita Rule.

It is easy to find this limit as 1 2.

Equivalent infinitesimal

If lim b a n = constant, then b is the infinitesimal of the n order of a, and b and a n are infinitesimal of the same order.

In particular, if this constant is 1 and n = 1, i.e. lim b a=1, then a and b are equal infinitesimal relations and denoted as a b.

The equivalent infinitesimal has an important application in finding the limit of amusement, and we have the following theorem: assuming lim a a'、b～b'Then: lim a b=lim a'/b'。

Now we require this limit lim(x 0) sin(x) (x+3).

According to the above fixed fixation and dismantling principle, when x 0 sin(x) x (important limit 1) x+3 x+3 , then lim(x 0) sin(x) (x+3)=lim(x 0) x (x+3)=1.

The equivalent infinitesimal in ln parentheses can be replaced, and it can be used on the applicable occasion, the equivalent infinitesimal is the infinitesimal of the same order When the infinitesimal is infinitesimal, the problem of the ordinal number of the same order should be taken into account, and the infinitesimal substitution itself is similar in characterization, but at this time, the direct substitution may cause too large an error, so Robbie is generally used to tease the higher-order situation.

When the independent variable x is infinitely close to a value x0 (x0 can be 0, , or some other number), and the function value f(x) is infinitely close to zero, i.e., f(x)=0 (or f(x0)=0), then f(x) is said to be an infinitesimal quantity when x0.

It is worth noting that the equivalent infinitesimal can generally only be replaced in multiplication and division, and substitution in addition and subtraction sometimes causes errors in the source of sensitivity, and can be substituted as a whole when adding and subtracting, and cannot be substituted separately or separately. Hail finger branches.

The equivalent infinitesimal in ln parentheses.

It can be replaced, it can be used on the applicable occasions, the equivalent infinitesimal is the infinitesimal of the same order when the infinitesimal is infinitesimal to the infinitesimite, it is necessary to take into account the lead problem of the ordinal number of the same order, and the infinitesimal confession of the small substitution itself is similar in characterization, but at this time, the direct substitution may cause too large an error, so Robida is generally used to seek the high-order situation.

When the independent variable. When x is infinitely close to a value x0 (x0 can be 0, or some other number), and the function value f(x) is infinitely close to zero, i.e., f(x) = 0 (or f(x0)=0), then f(x) is said to be an infinitesimal quantity when x x0.

It is worth noting that the equivalent infinitesimal can generally only be replaced in multiplication and division, and sometimes the prejudgment will be wrong in addition and subtraction, and it can be substituted as a whole when adding and subtracting, and cannot be substituted separately or separately.

It cannot be replaced. The premise of an equivalent infinitesimal substitution is the factor in the numerator (denominator).

For example, let a b (ab be equivalent infinitesimal ) a*a, a (c + d) where a can be replaced by b; But (a+c) and so on a is not a factor. When finding the limit, the infinitesimal quantity is multiplied and dust-based operation, and the infinitesimal quantity in the period can be replaced by the infinitesimal quantity of the same order.

Two infinitesimal quantities of the same order, i.e., the Infinitesimal Sect Hall is careful to say: infinitesimal A infinitesimal b = n (constant).

When finding the limit. Use the condition of equivalent infinitesimal:

The limit value of the amount to be substituted is 0 when taking the limit;

The amount to be substituted can be substituted as an element to be multiplied or subtracted, but not as an element of addition or subtraction.

If each of the addition and subtraction terms is infinitesimal and the result obtained by substituting each of them with the equivalent infinitesimal is not 0, it can be replaced. Finding the limit with Taylor's formula is based on this idea. For example:

Find the limit of (tanx-sinx) (x 3) when x 0. It is easy to find this limit of 1 2 using the Lopita rule.

Limits

Basic concepts of mathematical analysis. It refers to the value (limit value) of a variable in a certain process of change, from such a branch that gradually stabilizes in general to a certain change trend and trend.

The limit method is the basic method used by mathematical analysis to study functions, and the basic concepts of analysis (continuous, differential, integral, and series) are based on the concept of limits, and then all the theories, calculations, and applications of analysis are established. Therefore, a precise definition of the limit is necessary, and it is the fundamental question of the reliability of the theories and calculations involved in the analysis.

Historically, it was Cauchy, A-l.First, a general definition of the limit is given more clearly.

He said, "When all the values of the same variable are infinitely close to a fixed value, and the final difference from it is as small as it is" (Tutorial of Analysis, 1821), this line of modulus fixed values is called the limit of the variable.

Subsequently, Weierstrass (Weierstrass, K. Waierstrass).(Giving a strictly quantitative definition of the limit along this line of thought, this is what is used in mathematical analysis as the -δ definition or the - definition, etc.) Since then, there have been practical criteria for judging various limit problems.

The concept of limits is equally important in other disciplines of analytics, and there are some generalizations in disciplines such as functional analysis and point-set topology.