First of all, this kind of thinking spirit is worth encouraging. The landlord is much better than many questioners. Many questioners just ask a very simple question, and add "Kuaikuai Kuai ~" after the topic to seek the answer, probably to cope with the homework. But the landlord is different. He knows how to think.

However, some really don't need to get in touch so early in high school, such as how the derivative formula is derived ... I won't elaborate on these technical problems, but mainly talk about ideas.

The landlord's problems can be roughly summarized into three.

Does the (1) variable approach the limit of numbers within or outside the domain?

② How to calculate the limit (including how to derive the derivative formula)?

When you see a (complex) function, how do you deduce it?

Say it one by one.

Take the example that the landlord said, lim x→0 1/x=? You said that approaching zero from two different directions has different results, one is positive infinity and the other is negative infinity. There is no limit to this, but it is not the reason that the landlord said. Even if y =1| x | (absolute value), there is no limit when x→0, because it cannot fall on a specific number. However, this process of approaching infinity is different from the general infinite limit. For example, sin( 1/x) oscillates back and forth between 1 when x→0, and it can never decide where it is, which is essentially different from1| x | when x→0. First introduce the definition of limit, and then discuss it back.

The landlord must first understand a problem. The limit of lim(x→a) f(x)=b is not that x is completely equal to a, so a is not necessarily within the definition, which is natural. All limit expressions are actually talking about a "skill". For example, lim(x→a) f(x)=b talks about "the function of f(x), which has the ability to make its value infinitely close to b by making x infinitely close to a". Translating this sentence into mathematical language is to give the number ε >; 0 is the degree I want it to be close to B, which requires it to find a degree δ close to A (that is, | x-a |)

We'll talk about infinity later. Approaching infinity is also a skill, that is, "the function of f(x) has the ability to make its value as large as possible by infinitely approaching a", which can also be defined by the strict mathematical language above:

Give it to M>0 (how big do you want it to be), with δ >; 0, so as long as | x-a |m (adjusted to meet your requirements). This ability is called "generalized limit". If you have to distinguish between positive and negative, then remove the absolute value defined earlier: so f (x) >; M is positive infinity, f (x)

This is the definition of limit, and see how much the landlord can understand.

2 how to calculate the limit. This problem is divided into several steps. First of all, some basic limits, such as sinx/x approximation 1, must be proved by the definition I gave above. After these proofs are completed, we prove that the limit has the nature of four operations, that is, a function +B function takes the limit, that is, A and B take the limit first, then add it, and so on, so that we can calculate the combination of addition, subtraction, multiplication and division of these limits that we have proved. Then we can sum up a good rule, that is, people have proved that there is a large class of functions (elementary functions) that are "continuous", that is, if A is in the definition domain, lim(x→a)f(x) is the function value f(a), and many limits can be calculated at one time. Finally, besides some basic definitions, there are some useful theorems that are not in the definition domain. For example, if you meet a lim(x→a)f(x)/g(x), but both f and g are 0 when x→a, what should you do? This is L'H?pital's law. You can derive f and g first, and then take the limit. Lim (x → a) f (x)/g (x) = lim (x→ a) f' (x)/g' (x) may not be zero after derivation.

How to prove the derivative formula is based on some proved limits. For example, limδx→0[2(δx+x)-2x]/δx = lnx. The landlord can understand what that definition proves. In short, this is a technical problem ... (and the landlord said that Δ x can't be equal to 0, so why is there a limit? It's understandable now. This is a trick, but Δ x is not really equal to 0).

③ How to calculate the derivative of a function? First of all, we should recite some derivative formulas of functions, for example, the derivative of sinx is cosx and so on, which are all in these textbooks. Then remember some rules for calculating derivatives, such as f+g, then derivative equals derivative and then addition, fg multiplication and derivative equals f×g'+g×f' and so on. Then we can use these to find the derivatives of all the functions encountered in high school.

For example, f(x)=√( 1-x? )-x

Derivation f'=[√( 1-x? )]'-x '

X'= 1 This is the formula, [√( 1-x? )]' As a function of √u, compound u= 1-x? This function takes the derivative of √u to u, regardless of its "internal structure", and has a formula equal to1(2 √ u). Then take the derivative of x to its "internal structure", which is -2x, and multiply it to get [√ (1-x).

These formulas will be learned by the landlord in senior three, so don't worry if you can't remember them now.