Basic Concepts of Limits

Limits are the soul of calculus. They describe how a function behaves near a point and form the bedrock for learning calculus.

极限的定义

函数极限

Definition of a function limit

When $x$ approaches $x_0$, $f(x)$ approaches $A$, denoted as:

$$\lim_{x \to x_0} f(x) = A$$

Mathematical wording: For any $\varepsilon > 0$, there exists a $\delta > 0$ such that whenever $0 < \lvert x - x_0 \rvert < \delta$, we have $\lvert f(x) - A \rvert < \varepsilon$.

Geometric meaning: Near $x_0$, the graph of $f(x)$ squeezes toward the horizontal line $y = A$.

数列极限

Definition of a sequence limit

When $n$ approaches infinity, $x_n$ approaches $A$, denoted as:

$$\lim_{n \to \infty} x_n = A$$

Mathematical wording: For any $\varepsilon > 0$, there exists an integer $N$ such that when $n > N$, we have $\lvert x_n - A \rvert < \varepsilon$.

Examples：

• $\lim_{n \to \infty} \frac{1}{n} = 0$
• $\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e$

左极限与右极限

左极限

Left-hand limit

When $x$ approaches $x_0$ from the left, the limit of $f(x)$ is written as:

$$\lim_{x \to x_0^-} f(x)$$

Mathematical wording: For any $\varepsilon > 0$, there exists a $\delta > 0$ such that whenever $x_0 - \delta < x < x_0$, we have $\lvert f(x) - A \rvert < \varepsilon$.

右极限

Right-hand limit

When $x$ approaches $x_0$ from the right, the limit of $f(x)$ is written as:

$$\lim_{x \to x_0^+} f(x)$$

Mathematical wording: For any $\varepsilon > 0$, there exists a $\delta > 0$ such that whenever $x_0 < x < x_0 + \delta$, we have $\lvert f(x) - A \rvert < \varepsilon$.

极限存在的充要条件

Theorem: The limit of $f(x)$ exists at $x_0$ if and only if its left-hand and right-hand limits both exist and are equal.

Criterion for the existence of a limit

$\lim_{x \to x_0} f(x) = A \Leftrightarrow \lim_{x \to x_0^-} f(x) = \lim_{x \to x_0^+} f(x) = A$

Examples：

• For $f(x) = \frac{|x|}{x}$ at $x = 0$, the left-hand limit is $-1$ and the right-hand limit is $1$, so the limit does not exist.

极限的性质

唯一性

Property: If the limit exists, it is unique.

Proof idea: Assume $\lim_{x \to x_0} f(x) = A$ and $\lim_{x \to x_0} f(x) = B$; then $A = B$.

有界性

Property: If $\lim_{x \to x_0} f(x) = A$, then there is a neighborhood of $x_0$ on which $f(x)$ is bounded.

Corollary: If a function has a limit at a point, it must be bounded in some neighborhood of that point.

保号性

Property: If $\lim_{x \to x_0} f(x) = A > 0$, then there is a neighborhood of $x_0$ where $f(x) > 0$.

Corollary: If $\lim_{x \to x_0} f(x) = A < 0$, then there is a neighborhood of $x_0$ where $f(x) < 0$.

极限的几何解释

函数极限的几何意义

• As $x$ gets arbitrarily close to $x_0$, $f(x)$ gets arbitrarily close to the constant $A$.
• The graph near $x_0$ clusters around the line $y = A$.
• From either side of $x_0$, $f(x)$ tends to the same value.

数列极限的几何意义

• Points of the sequence on the number line get arbitrarily close to $A$.
• Beyond some term, all points fall inside any neighborhood of $A$.
• The “tail” of the sequence gets closer and closer to its limit.

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练习题

练习 1

Determine whether the limit of $f(x) = \frac{x^2 - 1}{x - 1}$ exists at $x = 1$.

参考答案

Idea： Compute left-hand and right-hand limits and see if they are equal.

Steps：

1. Right-hand limit: $\lim_{x \to 1^+} \frac{x^2 - 1}{x - 1} = \lim_{x \to 1^+} \frac{(x-1)(x+1)}{x-1} = \lim_{x \to 1^+} (x+1) = 2$
2. Left-hand limit: $\lim_{x \to 1^-} \frac{x^2 - 1}{x - 1} = \lim_{x \to 1^-} \frac{(x-1)(x+1)}{x-1} = \lim_{x \to 1^-} (x+1) = 2$
3. Since the left-hand and right-hand limits match, the limit exists.

Answer：The limit exists and equals $2$.

练习 2

Prove that the sequence $x_n = \frac{n}{n+1}$ has limit $1$.

参考答案

Idea： Use the definition of a limit: for any $\varepsilon > 0$, find $N$ such that $|x_n - 1| < \varepsilon$ for $n > N$.

Steps：

1. $|x_n - 1| = \left|\frac{n}{n+1} - 1\right| = \left|\frac{n-(n+1)}{n+1}\right| = \frac{1}{n+1}$
2. To make $\frac{1}{n+1} < \varepsilon$, we need $n+1 > \frac{1}{\varepsilon}$, i.e., $n > \frac{1}{\varepsilon} - 1$
3. Let $N = \left\lfloor \frac{1}{\varepsilon} - 1 \right\rfloor + 1$; then for $n > N$, $|x_n - 1| < \varepsilon$

Answer：The sequence limit is $1$.

练习 3

Determine whether the limit of $f(x) = \frac{1}{x}$ exists at $x = 0$.

参考答案

Idea： Compute left-hand and right-hand limits.

Steps：

1. Right-hand limit: $\lim_{x \to 0^+} \frac{1}{x} = +\infty$
2. Left-hand limit: $\lim_{x \to 0^-} \frac{1}{x} = -\infty$
3. Since they differ, the limit does not exist.

Answer：The limit does not exist.

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总结

本文出现的符号

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）	Arbitrarily small positive number
$\delta$	希腊字母	Delta（德尔塔）	Positive number depending on $\varepsilon$
$N$	数学符号	Positive integer	A sufficiently large positive integer
$\lim$	数学符号	Limit	Denotes the limit of a function or sequence
$\to$	数学符号	Tends to	Indicates approaching a value
$\infty$	数学符号	Infinity	Represents infinity
$\vert x - x_0 \vert$	数学符号	Absolute value	Distance between $x$ and $x_0$

中英对照

中文术语	英文术语	音标	说明
极限	limit	/ˈlɪmɪt/	函数或数列在某个点或无穷远处的极限值
函数极限	limit of a function	/ˈlɪmɪt əv ə ˈfʌŋkʃən/	函数在某点的极限
数列极限	limit of a sequence	/ˈlɪmɪt əv ə ˈsiːkwəns/	数列在无穷远处的极限
左极限	left-hand limit	/left hænd ˈlɪmɪt/	从左侧趋向于某点的极限
右极限	right-hand limit	/raɪt hænd ˈlɪmɪt/	从右侧趋向于某点的极限
唯一性	uniqueness	/juːˈniːknəs/	极限值唯一的性质
有界性	boundedness	/ˈbaʊndɪdnəs/	函数在邻域内有界的性质
保号性	sign preservation	/saɪn ˌprezəˈveɪʃən/	极限值符号在邻域内保持不变
邻域	neighborhood	/ˈneɪbəhʊd/	某点附近的区间
充要条件	necessary and sufficient condition	/nɪˈsesəri ənd səˈfɪʃənt kənˈdɪʃən/	既是必要条件又是充分条件

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