Calculus review 1

Lesson one Functional limit and continuity

This series of articles refer to 《 Zhang Yu high number 18 speak 》, In order to summarize the knowledge of calculus in higher mathematics , This series of articles is divided into 18 A section , See the table of contents in this section below ：

• Introduction
• Definition and use of function limit
• Calculation of function limit
• The existence of functional limits
• Application of function limit

The following will be explained according to the above contents

refer:

1. 《 Zhang Yu high number 18 speak 》

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1.1 Introduction

The limit calculation of function is the basis of calculus . This chapter mainly starts from the concept 、 Calculation 、 Analysis, proof and application , Explain the limit of function . The thought framework map is as follows ：

The Greek alphabet is common. It's best to remember something like epsilon These . The common ones in the figure have been marked in red , It is best to be able to remember, write, read and distinguish .

1.2 Definition and use of function limit

1.2.1 Definition

$$\underset{x\to \cdot }{\lim }f(x)=A\iff \forall \epsilon >0,x\to \cdot when,|f(x)-A|<\epsilon$$

There are six trends , They tend to be a number , Tends to the left and right limits of numbers , Towards infinity , Plus or minus infinity .

1.2.2 Use

Under the condition that the limit exists , Yes 5 A test site , this 5 It can also be understood as the existence of limit , Because the existence of the limit can immediately deduce this 5 Nature .

1. Is constant
   A It's a constant , Always remember $$\underset{x\to \cdot }{\lim }f(x)=A$$ The derivative that follows 、 Variable upper limit integral 、 definite integral 、 Double integral 、 Triple integral exists , It is also a constant .
2. Uniqueness
   A only , The left limit = Right limit . This is also Definition , That is, there is a mutual push relationship ：$$\underset{x\to x_0}{\lim }f(x)\exists \iff \underset{x\to x_0^{+}}{\lim }f(x)=\underset{x\to x_0^{-}}{\lim }f(x)$$
3. Local boundedness
   $$x\to \cdot when,\exists M>0,|f(x)|\le M$$
4. Local number preservation
   $$x\to \cdot when,ifA>0\Rightarrow f(x)>0;iff(x)>0\Rightarrow A>0;$$ In fact, there is a strict greater than or equal sign , For the convenience of memory, there is no equal sign here .
5. Equation decapitation
   $$f(x)=A+\alpha ,Itsin\underset{x\to \cdot }{\lim }\alpha =0$$

1.2.3 Extension

One 、 Integral function

For rounding functions [x], We need to remember its relation $x-1<[x]\le x.$

Two 、 Judgment and properties of function continuity

First we need to know ,

1. Elementary function refers to the elementary function and the function obtained by the finite number of four operations and coincidence of the elementary function .
2. All elementary functions must be continuous in their definition region .（ Not a domain , Is to define the area ）
3. If the function is continuous, the value of the function is equal to the limit value （ The existence of the limit is not necessarily continuous , But a certain limit of continuity exists . For example, the left and right limits are equal but not equal to the function value ）

3、 ... and 、 Function bounded judgment

It is divided into two cases: closed interval and open interval

1. if f(x) stay [a,b] Continuous on , be f(x) stay [a,b] bounded
2. if f(x) stay (a,b) Internal continuity , And the right limit exists at the left endpoint 、 The left limit of the right endpoint exists （ Local boundedness ）, be f(x) stay (a,b) There is a boundary inside

1.3 Calculation of function limit

When calculating the function limit , First of all, we need to make it clear that our research object is seven kinds of infinitive , Divided into three groups , Namely ：
$\frac{0}{0},\frac{\infty }{\infty },\infty \ast 0$ Commonly used lobida ,0 Multiply by infinity to invert numerator and denominator
$\infty -\infty$ To create a denominator and then to divide
${\infty }^0,0^0,1^{\infty }$ Commonly used $u^v=e^{v\ln u}$

We can notice that when calculating the limit of a function , Sometimes the four operations can be disassembled , But detachability depends on Both limits exist after disassembly .

1.3.1 Simplify first

One 、 The equivalent infinitesimal Replace

The following commonly used infinitesimals need to be kept in mind , When $x\to 0when,$
$$\sin x\sim x,\tan x\sim x,\arcsin x\sim x,e^x-1\sim x,\ln (1+x)\sim x,a^x-1\sim x\ln a$$
$$1-\cos x\sim \frac{1}{2}{x}^2,(1+x{)}^{\alpha }-1\sim \alpha x$$

Find the leading brother ： if a and b Are infinitesimal quantities in the trend process of the same independent variable , And a = o(b), be a+b Equivalent to a

Two 、 Identical deformation

Extract the common factor
Exchange element
General points （ Constant harmony $1^{\infty }$ combination ）

Power index
Use formula

The mean value theorem （ Lagrange —— Function difference and derivative relation ; Newton Leibniz —— Relation between integral and difference of original function ; Mean value theorem of integral —— The relation between integral and function difference ; Taylor formula —— Higher derivative ）

3、 ... and 、 Timely propose that the limit exists and is not 0 The factor of

1.3.2 The law of lophida

One 、 What is lobida's law

Lobida is used in the form of limit division , Simplification of numerator and denominator , There are three forms of lobida's law , They are as follows ：

Two 、 The applicable conditions of lobida's law

Can lobida's law work , Let's talk about it .

1.3.3 Taylor formula

One 、 What is Taylor formula

You need to memorize at least the following ten Taylor formulas , The first six of them can be connected with infinite series , It's easy to remember . Note that at this time, they are all in x0=0 It's unfolding at a point .

Two 、 The applicable principle of Taylor formula

There are generally two cases of Taylor formula expansion , Namely a/b and a-b, The principles followed by the two are also different , They are the same order principle and the power minimum principle . Note that at this time, they are all in x0=0 It's unfolding at a point .

The principle that the upper and lower levels are of the same order

Power minimum principle

1.3.4 Infinitesimal order

1.4 The existence of functional limits

1.4.1 Concreteness

When lobida's law fails, it is often considered that Pinch rule , Find out more than fx And less than fx Function of , Make them all tend to a little , Use zoom to achieve ：

1.4.2 Abstract type

For abstract functions , consider The trend is endless Whether the limit exists or not is calculated , Often consider using Monotone bounded criteria Prove that the limit of the function exists .
Monotonically increasing with upper bound leads to the existence of positive infinite limit .

1.5 Application of function limit —— Continuity and discontinuity

One 、 Study location

All elementary functions must be continuous in their definition interval , So for continuity and discontinuity , We only study two kinds of points , Namely Undefined point （ Interrupted ） and Segment point .
Suspicious point ： The point used to find the extreme value , Points containing stationary points and derivatives that do not exist .
Best value point ： Suspicious point + Endpoint .
Stagnation point ： Derivative is 0 The point of .

Two 、 Judge whether a point is continuous

The left limit of the function at this point = Right limit = Function value $\iff$ Then the point is continuous .
For a specific endpoint , If the left limit = Function value , Then left continuous .

Here, （1） in , If the limit exists, the left and right limits are equal to the function limit , It's about one thing .

3、 ... and 、 Interrupted

It is divided into 4 A kind of discontinuity , They are jumping 、 Go to 、 infinite 、 Oscillate .