The following is a record of my two answers.
The first answer
A, limit calculation:
Is to calculate when x infinitely approaches a certain value x, and when the function f(x) infinitely approaches what value?
In general, it is directly substituted.
Some situations can't be directly substituted, that is, seven types of infinitives, such as numerator and denominator tending to 0,
We can't substitute all the numerator and denominator into 0. But in order to find out that their ratio is getting closer and closer,
Therefore, we have produced various methods to calculate the limit.
Second, the proof of limit theory.
This part is not easy to understand, please take a closer look at the following explanation, and it will suddenly become clear.
1, the earliest budding concept of limit, existed in our ancestors, but it was buried as sophistry.
Today, there are still most math teachers who are dogmatic in sophistry.
The bottom denies that there is no room for rational speculation.
2, the devil's ancestors, also have sophistry, they carefully studied the paradox, thus
The limit theory holds. Limit theory is a bridge, with elementary mathematics on one side and elementary mathematics on the other.
It's calculus, it's advanced mathematics. Our theoretical contribution is limited to the bridge side and the bridge side.
In the construction of the world, we are almost completely incompetent, and our backwardness in scientific research is
Start here.
3. What is the limit theory?
First, the proof theory of limit.
This is our first heavy heart at the beginning of freshman university.
Sap. Limit theory is actually a theory of argument, an endless process of argument, and also
Theoretical process of infinite enumeration method. For example:
(1), I said that when X infinitely approaches 2, X? Infinitely close to 4.
If you don't believe me, you want me to prove it to you.
(3), I said, then you just give a very small number, you give 0.5.
(4) Through calculation, I said that as long as x = 2. 10.
(5) You went back on your word and changed it to 0.4.
(6), I recalculated, I said as long as x = 2.09.
(7) You went back on your word and changed it to 0.3.
(8), I recalculated, I said as long as x = 2.07.
(9) If you go back on your word, change it to 0.2.
(10), I calculated again, and I said as long as x = 2.04.
,,,, You keep going back on your word and putting forward more and more demanding data, and I keep calculating.
Give a specific number that is getting closer and closer to 2, that is, limit the process of X approaching 2 more and more.
Degree,,,,,
As a result, we were all bored
(1 1), I said, stop it, please give a symbol number that can represent tiny.
(12), you gave a code ε.
(13), according to your code ε, after some calculation, I found another digital code δ.
I'm telling you, you can find a number that is not more than δ from 2.
Forget it, forget it, I'll give you the formula. You can calculate ε by yourself and find δ by yourself.
What do you have to say?
The quarrel ended like this, and the infinite list became a theoretical calculation process, and the result was proved.
The logical thinking of this proof is:
As long as you give a smaller number, ε;
I can calculate a delta according to your ε;
As long as the value of x is limited within the range of δ, the difference between the function value and the limit value is less than ε.
Since ε can be arbitrarily small, the difference between them can be infinitely small, which proves the limit.
δ is calculated according to ε, I calculate a δ, and you can limit the range of X with a δ smaller than me.
So ε is arbitrary, and δ is calculated according to ε, but δ is not unique, and there can be countless δ.
Stricter and smaller values. So, there is always a delta, but this delta must be decided by us.
Find out according to ε.
Second, the limit calculation
The first part of calculus is to find various calculation methods, the most typical of which is Robida's law.
Third, restrictions on use
It can be said that limit is the basis of calculus, and calculus is the application of limit theory.
If you can't understand the theoretical proof of limit,
Well, we must congratulate you! You really know our traditional excellent history of mathematics, and when it comes to algebra,
Why did you suddenly fall behind? Contemporary theory, we didn't participate in the establishment, so far, we also
Don't belong to third-rate.
If you understand the theoretical proof of limit,
Well, we must congratulate you! You really began to realize the true meaning of modern mathematics and modern science. Learn from experience
To our traditional, qualitative, ambiguous and indifferent style of study, more modern mathematics, modern science,
The gap between modern medicine, medicine, medicine, medicine, medicine, medicine and medicine is so deep, so wide and so unparalleled.
The second answer
1 and ε are arbitrarily given, but they are not sure!
ε can change at any time and get smaller and smaller, but ε is not infinitesimal;
ε is just a symbolic small positive number, which can be changed at will.
Any meaning:
Can be arbitrarily small; Can be changed at will;
For any given case of ε, find δ or n,
This is the core of limit proof!
That is to say,
δ or n is a function of ε, which is determined by ε;
Change ε, δ or n at will.
2. As far as ε-N proof method is concerned,
According to ε, calculate an n, which is not fixed:
The values of a and n are closely related to ε, or n is determined by ε;
B However, in the specific proof, in order to prove the smooth progress of the process,
You can take different n, that is, according to ε, solve inequalities,
Suppose it is n, you can solve n? It may be difficult to solve this problem.
Can we enlarge this and make it n? ，N？ & gtn？ , in order to
Strictly prove that we take N = N? .
It may also be written as N = max{N? ，N？ ，N？ ，N？ , 、、、}。
Then, when n > when n, the difference between the value calculated by the limit formula and the limit value,
Less than ε, the proof is over.
3. The process of limit proof is actually:
First, a quarrel process; The theoretical process of infinite enumeration;
B, an endless cheating process, ε can be given at will, that is, it can be changed.
The process of finding n according to ε is a theoretical process. No matter how ε changes,
No matter how rogue, just give ε and you will find n.
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This process is both theoretical and tendentious.
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It's just that our teaching is too ostentatious and ignores the tendency.
Focus only on the limits and limitations.
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If you don't realize this, in the end, it is impossible to get real feelings.
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People who study limit proof theory are in Qian Qian every year, and most of them just come to join in the fun.
They will never understand the true meaning, including most math teachers, who are echoing ideas and don't know what to say.
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Come on!
Limit theory has matured for hundreds of years. The establishment and perfection of limit theory,
It has nothing to do with us, we have nothing to contribute!
Limit theory, for us, is completely imported! It's totally fake!
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The limit theory is established by the devil, integrated by the devil and perfected by the devil;
We just, we just learn, we just wave flags and shout, we just sing praises,
It's just parrot talk, it's just an imitation show, nothing else.
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So far,
A, our teacher will subconsciously hint at students when teaching, as if the limit theory is established.
We also played a role!
B, the limit theory seems to have just been established, and it seems to be still in the process of establishment!
These are all deliberately misleading! Deliberately fool!
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Students often ask:
1, what is the research status of limit theory?
2. What is the current situation of limit research in China?
、、、、、、、、、
These problems have repeatedly shown that the poor child was treated as an idiot by the hateful teacher!
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Come on! A long way to go!
The heavy responsibility is to avenge the shame that teachers have made no contribution to contemporary science!
It is a long way to go to correct the teachers' deep-rooted misleading intentionally or unintentionally!