This is a beta course, so its structure, chapters, and examples may continue to change.

The Definition of Limits

The definition of a limit has both an intuitive and a rigorous form. Understanding both is important for mastering the concept of limits.

The Intuitive Definition

The intuitive definition of a limit

If, as $n$ increases without bound, the terms $a_n$ of the sequence $\{a_n\}$ get infinitely close to a certain constant $A$ , then the sequence $\{a_n\}$ is said to converge to $A$ , and $A$ is called the limit of the sequence.

Written as: $\lim_{n \to \infty} a_n = A$ or $a_n \to A \quad (n \to \infty)$

符号说明

符号	类型	读音/说明	在本文中的含义
$\lim$	运算符号	limit（极限）	The limit operator
$\infty$	符号	infinity（无穷大）	Infinity; $n \to \infty$ means $n$ tends to infinity

The Rigorous Definition (the ε-N Definition)

The ε-N definition of a limit

Let $\{a_n\}$ be a sequence and $A$ a constant. If for any given positive number $\varepsilon$ (however small), there always exists a positive integer $N$ such that whenever $n > N$ ,

$|a_n - A| < \varepsilon$

then the sequence $\{a_n\}$ is said to converge to $A$ , written as $\lim_{n \to \infty} a_n = A$ .

符号说明

符号	类型	读音/说明	在本文中的含义
$\varepsilon$	希腊字母	Epsilon（伊普西隆）	An arbitrarily small positive number specifying the degree of closeness
$N$	变量	N	A positive integer indicating from which term the condition holds

Why do we need a rigorous definition?

The intuitive definition is easy to understand, but not precise enough:

What does “infinitely close” mean?
To what extent does “increase without bound” go?

The rigorous ε-N definition resolves these issues:

$|a_n - A| < \varepsilon$ quantifies “closeness”
$n > N$ quantifies “large enough”

This rigor is the foundation of mathematical analysis, allowing us to carry out rigorous proofs.

Analogy: just as laws need precise provisions rather than merely “the general meaning”.

Convergence and Divergence

Convergent: the sequence has a limit
Divergent: the sequence has no limit

Examples of Convergence

Example 1: $\lim_{n \to \infty} \frac{1}{n} = 0$

Proof (using the ε-N definition):

For any $\varepsilon > 0$ , to make $|\frac{1}{n} - 0| < \varepsilon$ , i.e., $\frac{1}{n} < \varepsilon$ , it suffices to take $n > \frac{1}{\varepsilon}$ .

Take $N = \lceil \frac{1}{\varepsilon} \rceil$ (the smallest integer greater than $\frac{1}{\varepsilon}$ ). Then when $n > N$ , $|\frac{1}{n} - 0| < \varepsilon$ .

Therefore $\lim_{n \to \infty} \frac{1}{n} = 0$ .

Example 2: $\lim_{n \to \infty} \frac{2n+1}{n} = 2$

$\frac{2n+1}{n} = 2 + \frac{1}{n} \to 2$

Examples of Divergence

Example 1: $a_n = n$ diverges (tends to infinity)

Example 2: $a_n = (-1)^n$ diverges (oscillates)

Example 3: $a_n = \sin n$ diverges (irregular oscillation)

Practice Problems

Exercise 1

Use the ε-N definition to prove: $\lim_{n \to \infty} \frac{1}{n^2} = 0$

Reference Answer(1 个标签)

limit definition of sequences

Proof:

For any $\varepsilon > 0$ , to make $|\frac{1}{n^2} - 0| < \varepsilon$ , i.e., $\frac{1}{n^2} < \varepsilon$ , it suffices to have $n^2 > \frac{1}{\varepsilon}$ , i.e., $n > \frac{1}{\sqrt{\varepsilon}}$ .

Take $N = \lceil \frac{1}{\sqrt{\varepsilon}} \rceil$ . Then when $n > N$ , $|\frac{1}{n^2} - 0| < \varepsilon$ .

Therefore $\lim_{n \to \infty} \frac{1}{n^2} = 0$ .

Exercise 2

Determine whether the following sequences converge. If they converge, find their limits:

$a_n = \frac{3n+2}{n+1}$
$a_n = \frac{n^2}{n+1}$
$a_n = \frac{(-1)^n}{n}$

Reference Answer(1 个标签)

limit definition of sequences

Solution:

$a_n = \frac{3n+2}{n+1} = \frac{3 + \frac{2}{n}}{1 + \frac{1}{n}} \to \frac{3}{1} = 3$ , convergent, with limit 3
$a_n = \frac{n^2}{n+1} = \frac{n}{1 + \frac{1}{n}} \to \infty$ , divergent
$a_n = \frac{(-1)^n}{n}$ : since $|a_n| = \frac{1}{n} \to 0$ , we have $a_n \to 0$ , convergent, with limit 0

Exercise 3

Prove: if $\lim_{n \to \infty} a_n = A$ , then $\lim_{n \to \infty} |a_n| = |A|$

Reference Answer(1 个标签)

limit definition of sequences

Proof:

From $\lim_{n \to \infty} a_n = A$ , for any $\varepsilon > 0$ , there exists an $N$ such that when $n > N$ , $|a_n - A| < \varepsilon$ .

Using the inequality $||a_n| - |A|| \leq |a_n - A|$ (the triangle inequality for absolute values),

when $n > N$ , $||a_n| - |A|| \leq |a_n - A| < \varepsilon$ .

Therefore $\lim_{n \to \infty} |a_n| = |A|$ .

Summary

Symbols Used in This Article

符号	类型	读音/说明	在本文中的含义
$\lim$	运算符号	limit	Limit
$\infty$	符号	infinity	Infinity
$\varepsilon$	希腊字母	epsilon	An arbitrarily small positive number
$N$	变量	N	A certain positive integer

中英对照

中文术语	英文术语	音标	说明
极限	limit	/ˈlɪmɪt/	The eventual trend of a sequence
收敛	convergent	/kənˈvɜːdʒənt/	The sequence has a limit
发散	divergent	/daɪˈvɜːdʒənt/	The sequence has no limit
ε-N定义	epsilon-N definition	-	The rigorous definition of a limit

Limits of Sequences

Properties of Limits