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

The Squeeze Theorem

The squeeze theorem is one of the most intuitive and practical criteria for the existence of limits. It determines the limit of a target function by “sandwiching” it between two other functions.

Other Names

The squeeze theorem is also known under different names in various references:

Sandwich Theorem (夹值定理)
Sandwich Theorem (三明治定理)
Squeeze Theorem (挤压定理)
Pinching Theorem (夹逼定理)
Intermediate Value Theorem for Limits (中间值定理)

All of these names vividly describe the core idea of this criterion: determine the limit of a target function by “sandwiching” it between two functions with known limits.

The Theorem

The squeeze theorem

If in a neighborhood of a point, $g(x) \leq f(x) \leq h(x)$ always holds, and $\lim g(x) = \lim h(x) = A$ , then

$\lim f(x) = A$ .

几何解释

If the function $f(x)$ is “sandwiched” between two functions $g(x)$ and $h(x)$ , and the limits of both of these functions tend to the same value $A$ , then the limit of $f(x)$ must also be $A$ .

证明

Since $\lim g(x) = A$ , for any $\varepsilon > 0$ , there exists $\delta_1 > 0$ such that when $0 < \vert x - x_0 \vert < \delta_1$ , we have $\vert g(x) - A \vert < \varepsilon$
Since $\lim h(x) = A$ , for any $\varepsilon > 0$ , there exists $\delta_2 > 0$ such that when $0 < \vert x - x_0 \vert < \delta_2$ , we have $\vert h(x) - A \vert < \varepsilon$
Let $\delta = \min\{\delta_1, \delta_2\}$ ; then when $0 < \vert x - x_0 \vert < \delta$ :
- $A - \varepsilon < g(x) \leq f(x) \leq h(x) < A + \varepsilon$
- Hence $\vert f(x) - A \vert < \varepsilon$
Therefore $\lim f(x) = A$

符号说明

Symbol	Type	Pronunciation/Description	Meaning in this context
$\varepsilon$	Greek letter	Epsilon	An arbitrarily small positive number
$\delta$	Greek letter	Delta	A positive number related to $\varepsilon$

Application Scenarios

The function is “sandwiched” between other functions
The limit is difficult to compute directly
The function has an oscillatory nature
It involves the boundedness of trigonometric functions

Practice Problems

Exercise 1

Use the squeeze theorem to find the limit $\lim_{x \to 0} x^2 \sin \frac{1}{x}$ .

Reference Answer(1 个标签)

squeeze theorem

Idea: Use the boundedness of the $\sin$ function to construct a squeeze inequality.

Detailed steps:

Since $-1 \leq \sin \frac{1}{x} \leq 1$ , we have $-x^2 \leq x^2 \sin \frac{1}{x} \leq x^2$
$\lim_{x \to 0} (-x^2) = \lim_{x \to 0} x^2 = 0$
By the squeeze theorem, $\lim_{x \to 0} x^2 \sin \frac{1}{x} = 0$

Answer: The limit is 0.

Exercise 2

Find the limit $\lim_{x \to 0} x \sin \frac{1}{x}$ .

Reference Answer(1 个标签)

squeeze theorem

Idea: Use the boundedness of the $\sin$ function to construct a squeeze inequality.

Detailed steps:

Since $-1 \leq \sin \frac{1}{x} \leq 1$ , we have $-|x| \leq x \sin \frac{1}{x} \leq |x|$
$\lim_{x \to 0} (-|x|) = \lim_{x \to 0} |x| = 0$
By the squeeze theorem, $\lim_{x \to 0} x \sin \frac{1}{x} = 0$

Answer: The limit is 0.

Exercise 3

Find the limit $\lim_{x \to 0} x^2 \cos \frac{1}{x}$ .

Reference Answer(1 个标签)

squeeze theorem

Idea: Use the boundedness of the $\cos$ function to construct a squeeze inequality.

Detailed steps:

Since $-1 \leq \cos \frac{1}{x} \leq 1$ , we have $-x^2 \leq x^2 \cos \frac{1}{x} \leq x^2$
$\lim_{x \to 0} (-x^2) = \lim_{x \to 0} x^2 = 0$
By the squeeze theorem, $\lim_{x \to 0} x^2 \cos \frac{1}{x} = 0$

Answer: The limit is 0.

Exercise 4

Find the limit $\lim_{x \to \infty} \frac{\sin x}{x}$ .

Reference Answer(1 个标签)

squeeze theorem

Idea: Use the boundedness of the $\sin$ function to construct a squeeze inequality.

Detailed steps:

Since $-1 \leq \sin x \leq 1$ , we have $-\frac{1}{x} \leq \frac{\sin x}{x} \leq \frac{1}{x}$
$\lim_{x \to \infty} \left(-\frac{1}{x}\right) = \lim_{x \to \infty} \frac{1}{x} = 0$
By the squeeze theorem, $\lim_{x \to \infty} \frac{\sin x}{x} = 0$

Answer: The limit is 0.

Summary

Symbols Used in This Article

符号	类型	读音/说明	在本文中的含义
$\lim f(x)$	数学符号	limit of f of x	The limit of a function
$f(x), g(x), h(x)$	数学符号	f/g/h of x	Target, lower bound, and upper bound functions
$A$	数学符号	A	The common limit of the upper and lower bounds
$\varepsilon$	希腊字母	Epsilon（伊普西隆）	An arbitrarily small positive number
$\delta$	希腊字母	Delta（德尔塔）	A positive number related to $\varepsilon$
$x_0$	数学符号	x-sub-zero	The limit point
$	x	$	数学符号
$\sin x, \cos x$	数学符号	sine/cosine of x	Trigonometric functions
$\frac{1}{x}$	数学符号	one over x	The reciprocal function
$\infty$	数学符号	infinity	Infinity

中英对照

中文术语	英文术语	音标	说明
夹逼准则	squeeze theorem	/skwiːz ˈθɪərəm/	Finding a limit by squeezing between upper and lower bounds
夹值定理	sandwich theorem	/ˈsænwɪtʃ ˈθɪərəm/	Another name for the squeeze theorem
三明治定理	sandwich theorem	/ˈsænwɪtʃ ˈθɪərəm/	Another name for the squeeze theorem
邻域	neighborhood	/ˈneɪbəhʊd/	The region near the limit point
有界性	boundedness	/ˈbaʊndɪdnəs/	The property that function values stay bounded
振荡	oscillate	/ˈɒsɪleɪt/	To swing back and forth within an interval
极限存在准则	criterion for existence of limits	/kraɪˈtɪəriən fɔːr ɪɡˈzɪstəns əv ˈlɪmɪts/	A method for determining whether a limit exists

Criteria for the Existence of Limits

The Monotone Convergence Criterion