Limit introduction summary

1 Basic concept of limit

Function as y=f(x),x There is a point on the axis a, When x Very close to a, But it doesn't mean a when ,y What kind of ？
such as ：

y=x,x It's not equal to 1

The graph of this function is number 1、3 The median of the quadrant , But in （1,1） The is empty .
that , You can imagine when x Very close to 1,y How much is the ？

I want to solve this problem , Obviously , Substitute function ,y=x=1, But it's meaningless , because ,x！=1
But then again , We let x=0.9999999, Will find y=0.9999999, Give Way x=1.0000001,y will =1.0000001, in other words , function y=x stay x=1 Nearby , Its value will really approach 1, This is an important concept , Let's talk about limits , It's about approaching , Instead of being equal to .
The limit formula is as follows ：

$$\underset{x\to 1}{\lim }x=1;Whenx!=1$$

emphasize , Above mentioned =, It's not equal to , But approach .

2 The value of the limit

We put y=x;x!=1 Under transformation ：

y=x; When x=1,y=2

Now, the graph of this function is number 1、3 The median of the quadrant , stay （1,1） The is empty , stay （1,2） The top is solid .

that , When x Very close to 1, How much closer ？
The difference between our function and the previous function is , stay x=1 When , This function is meaningful , When x=1,y=2.
So when x stay 1 Nearby ,y Tend to be 2 Do you ？ Of course not. , By looking at the image , We learned that , Even though （1,1） non-existent , and （1,2） There is , But in terms of approach , When x stay 1 Nearby , function y In fact, it is close to 1 Of .

$$\underset{x\to 1}{\lim }x=1$$
So the function y stay x=a The limit value and x=a Sometimes it doesn't matter .

3 Left limit and right limit

Now there are functions $y=1/x$ Here's the picture

When x=0 When ,y What is the limit of ？
This function has two parts , Look at the picture and you can see , stay 0 Nearby , Part on the left , Will tend to $-\infty$, And the part on the right , Will tend to $+\infty$
Their limit values are different , We can write this way ：
$$\underset{x\to 0^{-}}{\lim }y=-\infty and\underset{x\to 0^{+}}{\lim }y=\infty$$
0 The following minus and plus signs represent the left and right limits respectively .
But it must not be written like this ：${\lim }_{x\to 0}y=\infty$
Because this means that both the left and right limits are positive infinity , This obviously does not conform to the function , We can say ${\lim }_{x\to 0}yNosavestay$

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So the function in the second section above

y=x; When x=1,y=2

When x=1 The limit of time can also be written in this way ：
$$\underset{x\to 1^{-}}{\lim }y=1and\underset{x\to 1^{+}}{\lim }y=1$$
This means that both the left limit and the right limit are 1, This sum ${\lim }_{x\to 1}y=1$ It is equivalent.

4 There are no limits

Can a function have no limit at all ？ yes , we have , Let's look at a function

y=sin(1/x), When x Tend to 0 when ,y There are no limits

You can see on the image , When x Whether from the right or the left 0 When moving , The image vibrates faster and faster , When x The closer to 0,y Does not approach any value , Quantum superposition ？ This function is in x Tend to 0 There is no limit when .

5 $-\infty and\infty$ Limit at

Sometimes we are right , When x When it approaches positive infinity or negative infinity ,y Interested in the question of the limits of .
$$\underset{x\to -\infty }{\lim }y=Lorperson\underset{x\to \infty }{\lim }y=L$$
For example 4 Function diagram of section , It can be seen that
$$\underset{x\to -\infty }{\lim }y=0WithAnd\underset{x\to \infty }{\lim }y=0$$

6 Sandwich Theorem （ Pinch theorem ）

The name NIMA , Remember when I was in high school , When you learn this chapter , The class burst into laughter , Looking back, it has been more than ten years .

Theorem ： If a function y Caught in a function g And the function h middle , When x trend a When , These two functions g and h All converge to the same limit L, that y It must be x trend a When , Also converges to the limit L
Nothing is as good as looking at pictures , Two pictures show you the theorem .

This picture is g Clip in h and f middle , All converge to L

This schema y=xsin(1/x) Clip in y=x and y=-x middle , stay x Tend to 0 When , All converge to 0