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Limits of Trigonometric Functions

Limits of trigonometric functions are important formulas in limit theory and are frequently used in limit computation.

Limit of the Cosine Function

Limit of the cosine function

$\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$

Equivalent infinitesimal: as $x \to 0$ , $1 - \cos x \sim \frac{x^2}{2}$

Limit of the Tangent Function

Limit of the tangent function

$\lim_{x \to 0} \frac{\tan x}{x} = 1$

Equivalent infinitesimal: as $x \to 0$ , $\tan x \sim x$

Idea of the proof: use the first important limit: $\lim_{x \to 0} \frac{\tan x}{x} = \lim_{x \to 0} \frac{\sin x}{x \cos x} = \frac{1}{1} = 1$

Basic Form

$\lim_{x \to 0} \frac{\tan x}{x} = 1$

Idea of the Proof

Use the first important limit: $\lim_{x \to 0} \frac{\tan x}{x} = \lim_{x \to 0} \frac{\sin x}{x \cos x} = \frac{1}{1} = 1$

Equivalent Infinitesimal

Using this important limit, we obtain an important equivalent infinitesimal:

As $x \to 0$ , $\tan x \sim x$

Worked Example

Find $\lim_{x \to 0} \frac{\tan 2x}{x}$

Solution: $\lim_{x \to 0} \frac{\tan 2x}{x} = \lim_{x \to 0} 2 \cdot \frac{\tan 2x}{2x} = 2 \cdot 1 = 2$

Practice Problems

Exercise 1

Find the limit $\lim_{x \to 0} \frac{\tan 2x}{x}$ .

Reference Answer

trigonometric limit

Idea: Use the tangent limit and a change of variable.

Detailed steps:

$\lim_{x \to 0} \frac{\tan 2x}{x} = \lim_{x \to 0} 2 \cdot \frac{\tan 2x}{2x} = 2 \cdot 1 = 2$

Answer: The limit is 2.

Exercise 2

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

Reference Answer

trigonometric limit

Idea: Use the cosine limit and a change of variable.

Detailed steps:

$\lim_{x \to 0} \frac{1 - \cos 2x}{x^2} = \lim_{x \to 0} 4 \cdot \frac{1 - \cos 2x}{(2x)^2} = 4 \cdot \frac{1}{2} = 2$

Answer: The limit is 2.

Exercise 3

Find the limit $\lim_{x \to 0} \frac{\tan 3x}{\sin 2x}$ .

Reference Answer

trigonometric limit

Idea: Use equivalent infinitesimal substitution.

Detailed steps:

As $x \to 0$ , $\tan 3x \sim 3x$ and $\sin 2x \sim 2x$
$\lim_{x \to 0} \frac{\tan 3x}{\sin 2x} = \lim_{x \to 0} \frac{3x}{2x} = \frac{3}{2}$

Answer: The limit is $\frac{3}{2}$ .

Summary

Symbols Used in This Article

符号	类型	读音/说明	在本文中的含义
$\sin$	数学符号	sine function	A trigonometric function
$\cos$	数学符号	cosine function	A trigonometric function
$\tan$	数学符号	tangent function	A trigonometric function
$\lim$	数学符号	limit	The limit of a function or sequence
$\to$	数学符号	tends to	A variable tending to some value
$\sim$	数学符号	equivalence sign	Denotes equivalent infinitesimals

中英对照

中文术语	英文术语	音标	说明
三角函数	trigonometric function	/ˌtrɪɡənəˈmetrɪk ˈfʌŋkʃən/	Functions related to angles
正弦函数	sine function	/saɪn ˈfʌŋkʃən/	A trigonometric function
余弦函数	cosine function	/ˈkəʊsaɪn ˈfʌŋkʃən/	A trigonometric function
正切函数	tangent function	/ˈtændʒənt ˈfʌŋkʃən/	A trigonometric function
等价无穷小	equivalent infinitesimal	/ɪˈkwɪvələnt ˌɪnfɪnɪˈtesɪməl/	Two infinitesimals whose ratio tends to 1
变量代换	variable substitution	/ˈveəriəbəl ˌsʌbstɪˈtjuːʃən/	Replacing the original variable with a new one

Limits of Power Functions

Applications of Important Limits

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Limits of Trigonometric Functions

Limit of the Cosine Function

Limit of the Tangent Function

Basic Form

Idea of the Proof

Equivalent Infinitesimal

Worked Example

Practice Problems

Exercise 1

Exercise 2

Exercise 3

Summary

Symbols Used in This Article

中英对照

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