Infinitesimal multiplication, the product of an infinite infinitesimal is not infinitesimal

First of all, it must be clear: infinitesimal and 0 are two completely different things. So"Multiply a finite infinite amount = 0"This statement is inherently wrong. You can start with the definition of limits.

Okay, now that you've revised the title, I'll talk a little bit about my understanding, do you know the principle of infinite comparison? A finite infinity multiplies or infinity (for real numbers) and only when an infinite infinity multiplies is it greater than an infinity. The proof of this is also quite interesting, but it is relatively long, so I will talk about the idea, you look at the image of tanx, it is a mapping of a finite field and an infinite field, one by one, to prove that two infinities are multiplied by an infinity, and then any finite infinity is decentralized, so as to prove.

The infinitesimal situation should be similar, and you can irresponsibly interpret it as the reciprocal of infinity, which may be easier to understand.

Since the multiplication of infinitely infinite quantities is no longer infinity, then the multiplication of infinite infinitesimal quantities in the same way does not necessarily = infinitesimal small.

As for the example, I'm really sorry, I can't cite it because I am not knowledgeable, but I don't think it will be a concept in a general sense.

I would like to say that the addition of several "infinitely small quantities" is not greater than one "infinitely small", the multiplication of several "infinitely large quantities" is not greater than one "infinitesimal large", infinity is just a comparison, as long as it is a quantity, it is not infinite, and infinity can only be understood abstractly and cannot be a number. Multiplying infinite infinitesimal quantities, it is not a number, but can only be understood abstractly. So it's not equal to 0.

The product of two infinitesimal is infinitesimal, so the product of an infinite infinitesimal is infinitesimal.

For example, let the function fn(x)=1 (0 x n-1).

fn(x)=x^(n-1) (n-1＜x≤n, n=1,2,3,…)fn(x)=1/x (n≤x＜+∞

Then when n +, for every natural number n there is fn(x) 0, i.e. fn(x) is an infinitesimal quantity. But their product is f(x) = 1, )fn(x) = 1, (0 x +.)

When x +, the function f(x) is also not an infinitesimal quantity. So the product of an infinitesimal is not necessarily infinitesimal.

1. An 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.

Yes. The product of two infinitesimal is infinitesimal and infinitesimal is infinitesimal and infinitesimal is infinitesimal product. 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.

An infinitesimal quantity is a variable with the number 0 as the limit, and the infinite allows the front to approach 0. 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.

Properties: 1. The infinitesimal quantity is not a sail 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.

Infinitesimal quantities have the following properties: 1. A finite infinitesimal algebra is still an infinitesimal quantity. 2. The product of a finite infinitesimal quantity is still an infinitesimal quantity.

3. The product of the bounded function and the infinitesimal quantity is the infinitesimal quantity. 4. The product of the constant and the infinitesimal is also an infinitesimal quantity. 5. The reciprocal of an infinitesimal quantity that is never zero is infinite, and the reciprocal of infinity is infinitesimal.

Here's an example.

An infinite number of sequences.

The n-th term is 1 before n-1, the nth term is n (n-1), and after the nth term is 1 (n+1) 1 (n+2)...

So the limit of n sequences is 0, which is infinitesimal, but if you multiply them, you can see that the product of each of them is 1, so the limit of the product is 1, not infinitesimal.

TwoInfinitesimalThe product of the must be infinitesimal in terms of bad luck.

If n-> infinity, a(n)=0, b(n)=0, then a(n)*b(n)=0*0=0

Two eggplants accompany an infinitesimal trembling eggplant merchant is not necessarily infinitesimal.

a(n)=1/n;b(n)=1/n^2

When n-> infinite, a(n)=0, b(n)=0 but a(n) b(n)=n, when n-> infinite, a(n) b(n)-> infinite.

Infinity and Sunless Town are poor and small.

The product can be converted into infinity, infinity, or infinitesimal infinitesimus, and then the law of Lopida can be used to feast on the coarse rule.

Solving. It is impossible to determine the auspiciousness.

For example, f(x)=x, g(x)=1 sinx, when x 0, limf(x) *limf(y)=1

f(x)=2x, g(x)=1 sinx, when x 0, limf(x) *limf(y)=2

f(x)=x, g(x)=1 sinx, when x 0, limf(x) *limf(y)=0

f(x)=sinx, g(x)=1 x, when x 0, limf(x) *limf(y)=