[calculus]The clarification of ε-δ definition

I believe that you learned the simple concepts of limit, integral and differential when you met calculus for the first time. However, when we want to learn deeper, there is a strange thing called $$\varepsilon -\delta$$ definition in the chapter 1. We might memorize this stuff but don’t understand clearly. This article will help you to clarify the concept about the definition.

Preconcept

function

In short, function is a thing that you give something to it, and something else will come out from it, like some kind of converter.

Notice:If I put the same thing into this function, I must get the same result.

Now we have a function $$f(x)$$, and want to know what will happen when $$x$$ is close to $$c$$. We’ll write this situation like

$$\displaystyle{ \lim_{x\to c}f(x)}$$

There is some characteristics to the limit:

• The value what we discuss is the point that “very close” to some other point (say c). So the existence of \f(c) is not important.
• If we get the same point when we approach c from both left and right, then we’ll say the limit of this function exists.
• In other words, when the left limit and the right limit exist and the left limit equals to the right limit, we’ll say the limit of this function exists.
• What we focus on is the scope of a certain point. That is, we only care about this range. No matter what the function looks like that far from the point we focus, it’s not our business.

The graph below is an example of limit：
Take $$f(x)=x^2$$ for example. You can pull the tie rod in the graph and observe the changes between $$\varepsilon$$ and $$\delta$$.

Some symbols need to know

♦ $$\delta$$
We use the letter $$\delta$$ (delta) to get closed to the value $$c$$ along the x-axis.

$$\left| x-c \right|<\delta$$

♦ $$\varepsilon$$
We use the letter $$\varepsilon$$ (epsilon) to get closed to the value $$L$$ along the y-axis.

$$\left| f(x)-L \right|<\varepsilon$$

♦ $$\forall$$
The symbol is from the A of “for all” and put it upside down. We called it “for all”.

♦ $$\exists$$
The symbol is from the e of “exists”. It is called “exists”.

Oh, by the way, when you search this definition or read books about math, you might see this type \epsilon (written as “epsilon” in Latex) more often. I prefer write epsilon like this \varepsilon (we write it as “varepsilon” in Latex). So when I wrote the article, I used the latter. These are the same thing, so there is no right or wrong. All depends on your habit.

If you are interested in or have a doubts about Greek letters, you can order this main course: Math material-greek alphabets.

The definition of the limit

We have laid out the preliminary concept above. The next step is simple, what we need to do is to combine these concepts with mathematical symbols.

“limit” means very closed, super closed, that is, as closed as you want. We got the expression of distance on x-axis and y-axis, and now we want to know how to express the concept “as closed as you want” by math.

First, you need to know that the concept of limit is dynamic, so the number isn’t fixed.

And we add something on the concept above:

for every “closed” of $$\varepsilon$$ ，there exists the “closed” $$\delta$$ ，such that for every $$x$$ in $$0<\left| x-c \right| < \delta$$ satisfies $$\left| f(x)-L \right| < \varepsilon$$

And we use some mathematics symbol

$$\varepsilon -\delta$$ definition

We say that $$\displaystyle{\lim_{x \to c}f(x)=L}$$

$$\forall \varepsilon > 0, \exists \delta > 0$$ such that $$\forall x \; in \; 0<\left|x-c\right|<\delta, \; \left|f(x)-c\right|<\varepsilon$$

I believe that it is easy for you.

I have mentioned that the corresponding value by approaching from the left and the right need to be the same. Then can we just discuss the value approach from the left? Of course! And there is other name called “left limit”, and the value approach from the right called “right limit”.

The next is about the definition of left limit and right limit:

left limit definition

We say that $$\displaystyle{\lim_{x \to c^-}f(x)=L}$$
$$\forall \varepsilon > 0, \exists \delta > 0$$ such that $$\forall x \; in \; (c-\delta,c), \; \left|f(x)-c\right|<\varepsilon$$

right limit definition

We say that $$\displaystyle{\lim_{x \to c^+}f(x)=L}$$
$$\forall \varepsilon > 0, \exists \delta > 0$$ such that $$\forall x \; in \; (c,c+\delta), \; \left|f(x)-c\right|<\varepsilon$$

The above is the rough explanation of  $$\varepsilon -\delta$$ definition. I recommand courses of 高淑蓉 for those who speak Chinese. Maybe you can recommend calculus courses (in English or other languages) to everyone for this post. I will be very grateful to you for helping me complete this post!

Seasoned with Chinese

◊ limit – 極限
◊ calculus – 微積分
◊ greek alphabets – 希臘字母
◊ definition – 定義