Indeed, most of our advanced math teachers are just teaching limits.

Follow the monks to chant Buddhist scriptures, and follow the Taoist paintings. Explaining is so rigid.

In a word, even they don't make sense, so do teachers, textbooks and students.

You can only draw a gourd ladle, which is difficult to fully understand.

Let's try and see if we can figure out the problem.

1, the English word for limit is limit, which has two meanings. The Chinese translation of "extreme" is actually a bit misleading.

But we don't have a better word. The first layer of these two meanings is: limit, boundary, scope,

Limit, eventually,,, and so on. For example, we say that people's physical limit, people's life limit, people's

The limit of height, the limit of running speed and so on. That's what it means. We have already emphasized this aspect.

Many, as a result, have had a fatal impact on many students, and many students can't step through this threshold all their lives.

For example:

A, y = 1/x, x is getting bigger and bigger, and 1/x is getting smaller and smaller. Will it touch the x axis? Of course not.

However, many teachers do not understand the teaching psychology and teaching methods, and will repeatedly emphasize this figure.

"Never, never" coincides with the X axis? Does this need to be emphasized? What will happen with such emphasis?

A psychological hint? What kind of logical fatal injury will it cause? They always look down on their eyes, they won't.

To care about these. The Japanese have similar problems in teaching, and their saying is never touch it, never.

Touch, never touch. This problem is not as serious as ours. At least they have theorists. Will they?

Put forward one new theory after another, and will constantly bring forth the old one. What about us?

We don't have any quantitative theory. Without this culture, students who like to question will be scolded.

Students who memorize by rote are the most popular.

B, 0.9 is strictly equal to 1? Of course not.

Is 0.99 strictly equal to 1? Of course not.

What about three 9s? What about four 9s? What about infinite nine?

Ask people around you, is the period of 0.9 roughly equal to 1?

Or exactly equal to 1? Without the slightest mistake?

You must emphasize that there is no error, no error, and it is absolutely strictly equal.

Their answer is: about equal to 1, there is still a little error.

All right! They all got into the thief boat and were taken in!

Next, you ask them what one ninth is converted into decimal, and they say 0. 1 1 without hesitation. .........

So far, they don't know, and they don't know how to be fooled.

What if you multiply them both by 9? They are still the majority of people wandering around,

They slapped themselves without knowing it. This is our sorrow. Among our college students,

Most of them have no research ability, and the problems are on their lips. Not only can they not understand, but they will follow.

You talk about their fallacies and refuse to accept them. Such elm students are the mainstream.

When the above two examples are combined and thought carefully, you will find that although we had the concept of limit in ancient times,

There is sophistry, but we regard it as absurd theory. We are stuck and backward.

Start with the limit. The second meaning of limit is tenderness, trend and method.

It's gone. Because we put too much emphasis on the meaning of limit and ignore it.

The process ignores the trend of limit, and we always replace the infinite limit process with finite process.

It is here that ancient civilization and the west go hand in hand, which we have not yet noticed.

I also like to brag about myself.

It is precisely because we ignore the trend that we begin to fall behind. The devil made rapid progress on this basis.

The first theory is the limit theory, and the first step of the limit theory is the accurate method of our translation.

Very exaggerated, we translated it into ε -δ language (ε -δ language). The essence of this method has been known by the landlord.

Tao, this is a process of debate, a process of quarrel, and a process of infinite enumeration into mathematical induction.

Process. This inductive thinking is similar to induction, but it is not carried out by the syllogism method of induction.

But a process of mathematical calculation, so this is mathematical logic.

The argument goes like this:

When I say f(x), the final result is f(a), that is, the final difference between f(x) and f(a) should be as small as possible.

Can be as small as, you gave a very, very small number, which is ε.

In other words, the word "arbitrary" means that you can give any small number, any small number you give at will.

As long as you can give it, I can always work out an interval, when my X enters this room, f(x) and f(a).

The difference can be less than this ε.

ε is an arbitrary number, as small as what you gave. After you gave it, I calculated it carefully.

As long as you give it, I can work out a range. I can always work out a range. The determination of this range

It is calculated according to your ε, so the range of existence is not fixed. Maybe I'm based on you

ε can't find a simple range at once. In order to ensure that the difference is less than ε, I may change your ε a little.

In order to understand a range, the difference between f(x) and f(a) is less than ε, and there is always almost.

The problem of reading, whether it can be found or not, is a problem of problem-solving skills.

In fact, ε does not need to be given in detail, and the number given in detail is not arbitrarily small.

This ε is just an example in the process of argument, which can be changed and repented constantly. So, this ε

It's just a principled number. With ε, we can find an interval δ, and X enters the range of δ. I'll prove it.

The absolute value of the difference between f(x) and f(a) is less than ε.

The concept of inequality is that the difference between our f(x) and f(a) must be limited to ε.

I don't know. Did I explain the problem clearly?

Come on! Science needs to be questioned! What we lack most is the spirit of questioning.

Our generation has been completely scrapped, and hope is on you!

Welcome questions.