How do you understand higher order infinitesimal quantities? What is an infinitesimal of higher orde

Let f(x) and g(x) be infinitesimal of x x* in the process of change of a variable, and g(x) ≠ 0, then.

1) If limf(x) g(x)=0, then f(x) is said to be an infinitesimal of order higher than g(x) (or an infinitesimal of higher order, denoted as f(x)=o(g(x))(x x*); Habitually, place an infinitesimal amount.

denoted as o(1);

2) If limf(x) g(x)= then f(x) is said to be an infinitesimal of order lower than g(x);

3) If limf(x) g(x)=a≠0, then f(x) and g(x) are infinitesimal of the same order.

4) If limf(x) g(x)=1, then f(x) and g(x) are said to be equal infinitesimals.

and denoted as f(x) g(x); Equivalent infinitesimal is a special case of infinitesimal of the same order;

5) If limf(x) gk(x)=a≠0(k>0), then f(x) is said to be infinitesimal of order k with respect to g(x).

If lim( )=0, then " is said to be infinitesimal than the higher order. This means that in a certain process (x x0 or something like x), 0 is faster than 0.

When the ratio of two different infinitesimal limits results in 0, , constants (non-0 and 1), and 1 correspond to the former being the latter's higher-order infinitesimal, lower-order infinitesimal, congantianeous infinitesimal, and equivalent infinitesimal.

0.The reason why the textbook definition of infinitesimal is difficult to understand is that they regard infinitesimal as a number that has value in one dimension, which contradicts existing logic, because no matter how small a number is, it must result in an infinite number after an infinite number of additions. Moreover, the test of this definition is based on an infinite number of operations, which cannot be fully realized.

1.Infinitesimal quantities should be understood as "numbers of lower dimensions". The so-called low-dimensional, for example, a square with a side length of 8, its area is 64, and the side length 8 here is a low-dimensional number relative to the area of 64, it has a value, it is 8; But its value appears to be 0 in terms of area.

That is, the edge length has no value relative to the area, but it has a value of its own.

2.In this way, an infinitesimal quantity can be defined as: a point value is a variable, and a line value is a quantity of 0. This definition is very clear and clear, and there is no vague problem as the textbook definition.

3.From the above clear definition, the operation of infinitesimal quantities has also become clear and unambiguous, and the discarding of point value variables is also easy to understand.

An infinitesimal quantity is a class of functions with 0 as the limit when the independent variable has a certain tendency, and as for whether the higher order or the lower order is naturally obtained by comparing it with other infinitesimal quantities, whether it is high or low is completely relative, and the comparison is the speed at which the value of the function tends to 0.

The closer to 0 the absolute value is smaller.

If lim( )=0, then " is said to be infinitesimal than the higher order. This means that in a certain process (x x0 or something like x), 0 is faster than 0.

When multiplication, the number of times is added, and when it is added or subtracted, the number of times is low or high. If lim x x0 f(x) g(x)=0, then f is said to be the higher-order infinitesimal quantity of g, or g is called the lower-order infinitesimal quantity of f. It is important to note that these two concepts are relative.

The two concepts of higher-order infinitesimal and low-order infinitesimal quantity are relative, and it cannot be said that a quantity is a high-order infinitesimal quantity or a low-order infinitesimal quantity, but a certain quantity should be a high-order infinitesimal quantity or a low-order infinitesimal quantity of a certain quantity. This definition has to do with the knowledge of limits, and you need to state that your variable tends to be related to a certain number or infinity, which is the condition.

An infinitesimal quantity is a variable with the number 0 as the limit, infinitely close to 0.

Infinitesimal quantities are a concept in mathematical analysis, and in classical calculus or mathematical analysis, infinitesimal quantities usually appear in the form of functions, sequences, etc.

To be precise, when the independent variable x is infinitely close to x0 (or the absolute value of x increases infinitely), and the function value f(x) is infinitely close to 0, i.e., f(x) 0 (or f(x)=0), then f(x) is said to be an infinitesimal quantity when x x0 (or x). In particular, it is important not to confuse very small numbers with infinitesimal quantities.

Property 1, infinitesimal quantity is not a number, it is a variable.

2. Zero can be the only constant for infinitesimal quantities.

3. The infinitesimal quantity is related to the trend of the independent variable.

4. The sum of finite infinitesimal quantities is still infinitesimal quantities.

5. The product of a finite infinitesimal quantity is still an infinitesimal quantity.

6. The product of the bounded function and the infinitesimal quantity is the infinitesimal quantity.

7. In particular, the product of a constant and an infinitesimal quantity is also an infinitesimal quantity.

8. The reciprocal of an infinitesimal quantity that is constant and not zero is infinitesimal and the reciprocal of infinity is infinitesimal.

The infinitesimal quantity is a function with 0 as the limit, and the speed at which the infinitesimal quantity converges to 0 can be fast or slow. Therefore, between two infinitesimal quantities, they are divided into high-order infinitesimal , low-order infinitesimal , same order infinitesimal and equivalent infinitesimal .

Infinitesimal is a process, and infinitesimal can be compared with each other, and comparison can distinguish the relatively high and low order! 0 is the infinitesimal of the highest order. The high and low order refers to the speed that approaches 0. The highest order means that if necessary, it can be exchanged for an infinitesimal of the lower order at any time.

Let b(x) be an infinitesimal quantity higher than a(x) that means that when x tends to infinity, the value of b a tends to 0, and the concept of infinity must be in the sense of the limit to be valuable.

Boring.

That is, both numbers have to become 0 eggs, and one of them becomes 0 eggs faster than the other! Then it's a high-end small!

For example, x is infinitesimal, then x 2 is the higher order infinitesimal, and x 3 is the higher order.

x 2 tends to 0 faster than x, so choose these higher orders whenever possible.

Is your question not perfect, can you complete the question, or describe the language of the sentence in detail?

o(x) is the higher-order infinitesimal.

Although two infinitesimal in the same process of change both tend to zero at the same time, the speed of their approach to zero is sometimes different, or even very different. In practical problems, it is sometimes necessary to discuss the speed of this kind of return to the basis zero.

If lim( )=0, then " is said to be infinitesimal than the higher order. This means that in a certain process (x x0 or something like x), 0 is faster than 0.

Question 1: O(x) stands for the higher-order infinitesimal of x, and what does o(x) mean (Note: "bold o" is an uppercase o) Definition.

o(x): If for any x, there is a constant k, such that x Question 2: What does the infinitesimal o(x) of the higher-order Qingsan represent? _?o(x n) means that [x degrees] are greater than or equal to n in all subsequent polynomials of x

For example: f(x) = 1 + x + x 2 + x 3 + x 4 + x 5 +

It can be expressed as:

f(x) =1 + x + x^2 + o(x^3)

Because when x approaches infinity, the larger n is, the closer x n approaches 0, so when n is large enough, x m (m n) is very, very close to 0 to ignore them, so it would be nice to just replace them with a symbol o(x n).

Question 3: How to pronounce the O Fuyu Lao's number in the higher-order infinitesimal representation The higher-order infinitesimal seems to be just a symbol, indicating that when x tends to 0, it is much smaller than the content in parentheses. It is not used for calculation, but if you divide two infinitesimal quantities, you may divide the constant.

Question 4: How to represent the high-order infinitesimal in latex Just use o(x) or something like that.

(a) lim( x+sinx) x = lim x x + limsinx x = + x+sinx) is the infinitesimal of x;

b) lim(x 3+3x) x = lim(x 2+3) = 3, x 3+3x) is an infinitesimal of the same order x;

c) lim(tanx-sinx) x = limtanx(1-cosx) x = lim(1 2) x 2 = 0,tanx-sinx) is the higher order infinitesimal of x;

d) lim[ (1+x)- 1-x)] x = lim2x = lim2 [ (1+x)+ 1-x)] = 1,[ 1+x)- 1-x)] is the equivalent infinitesimal of x.

Infinitesimals of equal order Infinitesimal: that is, when the variable tends to a certain value, the limit of the quotient of the two is 1 is a constant value.

For example, if x0 and lim x sinx=1, then at x0, sinx and x are infinitesimal of equal order.

Higher-order infinitesimal quantities: that is, when the variable tends to a certain value, the limit of the quotient of the two is 0

For example, if x0 and lim x 2 sinx=0, then x 0, x 2 is the higher order infinitesimal of sinx.