Is infinitesimal a size or not

An infinitesimal is an infinitely close to 0 quantity.

Within the range of the field of real numbers, the infinitesimal size is not considered. That is, infinitesimal is not a "number", it is a changing concept.

When comparing infinitesimal infinitesims, you can consider the infinitesimal increase and decrease speed, for example, when x->0, the square of x and x are close to 0, but the square of x, when x is close to 0, is close to 0 the fastest.

chengongqpzm I was wrong, infinitesimal is a number that is infinitely close to 0, but not 0, so infinity and 0 are reciprocal to each other. There are also two kinds of infinite approximation to 0, the right approximation and the left approximation, the reciprocal from the right is positive infinity, and the reciprocal from the left is negative infinity. Infinity cannot be compared or calculated.

But infinitesimal really doesn't have a size

No, only definite numbers can be compared, and uncertain numbers cannot be compared.

Swipe it first. Actually, I've discussed this issue with *** a lot of people).

It has to do with the limit, which can be said to be big and small and certainly comparable.

For example, x>0 always has this infinitesimal - minimum (in fact, the problem here is that you know -8 [infinitely]?).If so, it must not be size) less than x is comparable.

To sum up, it depends on which one you are asking.

There is a theory that infinitesimally small or large numbers are calculated as a number, that is, there is a size, but the size is generally not determined.

It tends to be none, but it is not without size, and it can only be imagined, and it cannot be expressed in exact mathematical numbers.

Infinitesimal is an important concept in calculus. It is not equal to 0 and can be the denominator, but it is smaller than any positive number and can be ignored.

No! Infinitesimals are infinitesimal!!

The numbers are endless.

No, the number is never-ending.

It's kind of like a matter of logic ...

In real life, 0 is the smallest, and in mathematics, there seems to be no size.

It's bigger than the smallest, hehe, I seem to be more abstract. There's also no such thing as a minimum, which is just a conceptual problem of mathematics, and mathematics is rigorous.

Invincible watermelon, did you have calculus and passed?

An infinitesimal quantity is an amount that is infinitely close to 0.

Something that approaches negative infinity or an infinite quantity.

Because the focus of our research is to see if the sequence is close to or far from the x-axis.

If you don't believe me, go and flip through the book.

There is no size that our math analysis teacher said.

The ratio of two infinitesimal is 1. This is because two infinitesimal numbers can be regarded as the same large number, so the ratio of two identical numbers is 1, which is the most basic mathematical theorem, and it is also the most basic mathematical concept defense problem, because the ratio of the same two numbers is one, so two infinitesimal numbers can be regarded as two numbers of the same size, so their ratio is 1.

Infinitesimal conceptual properties.

1. Infinitesimal messy stove shouting.

No is a number, it's a variable.

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

3. Infinitesimal quantities and independent variables.

trend correlation.

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 infinite.

The reciprocal of infinity is infinitesimal and infinitesimal.

The symbol (x)=o( (x)) indicates that the function allows trace (x) to be infinitesimal of order higher than function (x), or (x) is infinity of order lower than function (x).

The symbol (x)=o*( x)) indicates that (x) and the ratio function (x) are infinitesimal of the same order, or infinity.

Let and be both functions of x, and lim = 0 and lim = 0, i.e., , are infinitesimal. 、

If lim( 0, then is said to be infinitesimal of the higher order, i.e., 0 is faster than 0;

If lim ( is said to be the infinitesimal of the lower order, i.e., 0 is slower than 0;

If lim( c≠0 is said to be infinitesimal of the same order, i.e. 0 and 0 are the same degree;

If lim( 1, then it is said that is the infinitesimal of the order of the order and denoted as

If lim( k) = c≠0 and k >0 then it is said that is infinitesimal of k order .

Equivalent infinitesimal :

1、e^x-1～x (x→0)

2、 e^(x^2)-1～x^2 (x→0)

cosx～1/2x^2 (x→0)

cos(x^2)～1/2x^4 (x→0)

5、sinx~x (x→0)

6、tanx~x (x→0)

7、arcsinx~x (x→0)

8、arctanx~x (x→0)

cosx~1/2x^2 (x→0)

10、a^x-1~xlna (x→0)

11、e^x-1~x (x→0)

Infinity and infinitesimal are values that cannot be calculated, but the difference is as follows:

A positive number divided by an infinitesimal number becomes infinitesimal, divided by infinity becomes infinitesimal, and a negative number is reversed;

x 1-, e x-1 is neither infinity nor infinitesimal at x-1.

ln(1-x) is infinity.

sin(x-1) is infinitesimal.

1 cos(x-1) is neither infinity nor infinitesimal x 0+.

The limit of sinx 1+tanx is 0

The limit of e -x is equal to 1

The limit of 2 -x is equal to 1

The limit of e (1 x) is equal to +

Infinity: Infinity is a variable or function in which the absolute value of an independent variable increases infinitely during a certain change. It is mainly divided into positive infinity, negative grinding infinity and infinity (which can be positive or negative), which are denoted as + and , respectively, and are very widely used in mathematics.

Infinitesimal quantity: An infinitesimal quantity is a concept in mathematical analysis that is used to strictly define informal descriptions such as "quantity that will eventually disappear", "quantity whose absolute value is smaller than any positive number", etc. In classical calculus or mathematical analysis, infinitesimal quantities usually appear as functions, sequences, etc., e.g., a sequence a=(a n) } if it satisfies the following properties:

For a pre-given positive real number varepsilon>0 for any faith-pure, there is a positive integer displaystyle n such that |a_k|n must be true; Or use the limit symbol to abbreviate the above property as lim a n = 0 then the sequence a is called the infinitesimal quantity when n to infty.

In non-standard analysis, infinitesimal quantities are also treated as concrete "numbers" as real numbers, which are greater than zero but smaller than any positive real number. The classical method of defining infinitesimal quantities in terms of sequences is more or less difficult to deal with, while the "non-standard" infinitesimal quantities.

The key to judging infinity and infinitesimal is to find the limit. If the limit is 0, it is said to be infinitesimal, and if the limit is infinite, it is said to be infinite. The reciprocal of infinity is equal to infinitesimal, and the infinitesimal reciprocal (when it is not equal to 0, because the reciprocal is only meaningful at this point, and the infinitesimal quantity is possible to take 0) is infinitely large.

The key to judging infinity and infinitesimal is to find the limit. If the limit is 0, it is said to be infinitesimal, and if the limit is infinite, it is said to be infinite. The reciprocal of infinity is equal to infinitesimal, and the infinitesimal reciprocal (when it is not equal to 0, because the reciprocal is only meaningful at this point, and the infinitesimal quantity is possible to take 0) is infinitely large.

Infinitesimal. Infinity is still infinity.

Infinitesimal multiplying infinity by infinity doesn't make sense.

If there is a formula that appears in the form of infinitesimal multiplication to infinity, the limit cannot be directly calculated, but must first be transformed into a meaningful form.

For example, 1 x * x (x) must first be transformed into a form that has the meaning of hail town, 1 x * x = 1. It will work later, but it is no longer in the form of infinitesimal multiplication to infinity, and the problem of infinitesimal multiplication to infinity does not exist. ）

Positive infinity + positive infinity = positive infinity.

Negative infinity + negative infinity = negative infinity.

Positive infinity + negative infinity has no meaning (if it appears, it must be transformed into a meaningful form to find the limit).

Infinity multiplied by infinity is still infinity.

The infinitesimal Theravada is good to be infinitesimal and still infinitesimal.

Infinity and infinitesimal are not finite constants.

The algorithm of constants cannot be fully adhered to.

Several upstairs are. You can take a look at the real variable function of the undergraduate students of the Department of Mathematics and Loss.

Practical analysis of Kenichi. You can find these things I said (in real numbers).

Infinity: In mathematics, infinity does not refer to a specific concept, but is related to the following topics: limits, Alev numbers, and set theory.

classes in , hyperreal numbers, projective geometry, extended real axes, absolute infinity, and so on. An infinite amount is in the independent variable.

The absolute value of a certain variation process.

Infinitely large variables or functions.

Precisely defined. Let the function f(x) be in a decentric neighborhood of x0.

There is a definition (or |x|It is defined when it is greater than a certain positive number).If there is always a positive number δ(or positive number x) for any given positive number m, no matter how large it is, as long as x fits the inequality 0m, then the function f(x) is said to be infinity when x x0 (or x).

In the same process of change of independent variables, infinity and infinitesimal have a reciprocal relationship, that is, when x a f(x) is infinite, then 1 f(x) is infinitesimal; Conversely, f(x) is infinitesimal and f(x) is invariably incursive when it is not 0 in a decentered neighborhood of a, and 1 f(x) is infinite.

Infinity is not to be confused with very large numbers.

Classify. Infinity is divided into positive infinity, negative infinity, and infinity (which can be positive or negative), which are denoted as +, and , respectively, and are very widely used in mathematics.

Quality. The sum of two infinitesimal quantities is not necessarily infinity;

The product of a bounded quantity and an infinitely large quantity is not necessarily infinity (e.g., a constant 0 is considered a bounded function);

The product of two infinitely large quantities must be infinite.

In addition, just because a sequence of numbers is not infinitely large does not mean that it is bounded (e.g., sequences 1, 1 2, 3, 1 3, ,......).

Infinitesimal quantity: The infinitesimal amount of acres is a variable with the number 0 as the limit. 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).

For example, f(x)=(x 1) 2 is the infinitesimal quantity when x 1, f(n)=1 n is the infinitesimal quantity when n, and f(x)=sin(x) is the infinitesimal quantity when x 0. In particular, it is important not to confuse very small numbers with infinitesimal quantities.

Beginners should note that the infinitesimal quantity is a variable with a limit of 0 and not a quantity of 0, which means that the limit of the independent variable is quantity 0 under a certain change mode.

It cannot be said in general terms that 0 is an infinitesimal quantity. It can't be said that the infinitesimal of the town is 0

Infinitesimal quantities are usually written in lowercase Greek letters.

Representations, such as , , etc., and sometimes (x), x)[1], etc., indicate that infinitesimal quantities are functions with x as an independent variable.

Note:1An infinitesimal quantity is not a very small number, it is a variable.

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

3.The infinitesimal is related to the tendency of the independent variable.