Epsilon-Delta Definition

Core Sentence / 核心句

A limit exists when every requested output accuracy can be guaranteed by restricting the input sufficiently close to the limiting point.

中文理解：无论要求输出离目标多近，总能把输入范围缩得足够小，从而保证所有输出满足要求。

Formula / 公式

$$\underset{x\to a}{\lim }f(x)=L$$

means:

$$\forall ϵ>0,\exists \delta >0:0<|x-a|<\delta \Rightarrow |f(x)-L|<ϵ$$

Memory Sentence / 记忆句

Epsilon controls output error; delta controls input distance.

Watch Out / 易错点

• 顺序是 “for every $ϵ$, there exists $\delta$”。
• $\delta$ 可以依赖 $ϵ$。
• $0<|x-a|$ 排除点本身，因为 limit 只看附近。

Search and Source Mentions / 搜索与出现位置

Mentioned In / 出现位置

• Chapter 7 · Epsilon-Delta
  • every positive epsilon must have a corresponding delta

Connected Notes / 连接

• 07 - Limits, L’Hopital’s Rule, and Epsilon-Delta Definitions
• Limits
• Derivatives
• Tiny Nudges