Limit summary (under update)

To clarify a knowledge point is nothing more than three angles ： Concept , nature , application

1. Concept

The basic concept of calculus , The generalized “ limit ” Refer to “ To approach infinitely and never reach ” It means .

The concept of limit is divided into function and sequence

1.1 Function limit

a.
When the independent variable X To a point or infinity ∞ when , Its function value is infinitely close to a value .

b.

c.

d.

e

1.2 sequence limit

a. When the independent variable n Towards infinity ∞ when , Its sequence {xn} Infinite approaching a value .

notes ：
1. there ∞ It must be infinite , And in the function ∞ It means infinity and infinitesimal
2.n As a positive integer
3. The sequence {xn} Is a set of infinite real numbers

b.

c.

2. nature

1. Uniqueness

2. Number guarantee

3. Boundedness

3. Rules 、 Theorems and rules

It is equivalent to the skill of application

3.1 Rules

Monotone bounded criteria

Pinch rule

3.2 Theorem

3.3 The laws of

a. The law of lophida

The conditions given by the title correspond to up to n Derivative,

b. Rational algorithm


3.4 other
a. Taylor formula


b. The maximum term limit is the multinomial sum limit

c. Propose an infinite factor

d. The concept of limit

f. Infinitesimal

g. An infinite number of

h.

i.

j.

k.

l.

4. application

There are three types of limit parts , Concept , Find the limit , Comparison of infinitesimal orders . There are different techniques for different scenarios

1 The limits of abstraction

There is no specific expression or the expression includes the limit of exponential function

Corresponding method ： The necessary and sufficient condition of limit

1.1.1.d

Key points ： Number guarantee

2 The limit of concrete type

in the light of 5.1 The opposite scenario of , Specifically, there is a specific function expression （ The sequence ）
notes ： The limit of summation sequence is excluded here , Because there are other methods for it （5.3）

Corresponding method ： Taylor's formula and 3 Various laws and theorems of the big point

https://zhuanlan.zhihu.com/p/137746002
https://zhuanlan.zhihu.com/p/32871279

3.3.b. and

Key points ： Find a way to keep all the removed parts

3.3.b.(3)

1.2.c. notes .4

3.3.a

3 The limit of summation form

Corresponding method ： Pinch rule , The maximum term limit is the multinomial sum limit

https://zhuanlan.zhihu.com/p/133704948

4 The limit of recurrence form sequence

Corresponding method ： Monotone bounded criteria （ There are many ways to achieve this , Such as induction ）

https://zhuanlan.zhihu.com/p/134074488

Key points ： The red part

5. Comparison of infinitesimal orders

Key points ： Equivalent substitution

https://zhuanlan.zhihu.com/p/137794260
https://baike.baidu.com/item/%E6%9E%81%E9%99%90/3564509