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Limits of Inverse Trigonometric Functions

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

Limit of the Arcsine Function

Limit of the arcsine function

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

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

Limit of the Arctangent Function

Limit of the arctangent function

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

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

Practice Problems

Exercise 1

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

Reference Answer

inverse trigonometric limit

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

Detailed steps:

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

Answer: The limit is 2.

Exercise 2

Find the limit $\lim_{x \to 0} \frac{\arctan 3x}{x}$ .

Reference Answer

inverse trigonometric limit

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

Detailed steps:

$\lim_{x \to 0} \frac{\arctan 3x}{x} = \lim_{x \to 0} 3 \cdot \frac{\arctan 3x}{3x} = 3 \cdot 1 = 3$

Answer: The limit is 3.

Exercise 3

Find the limit $\lim_{x \to 0} \frac{\arcsin x}{\arctan x}$ .

Reference Answer

inverse trigonometric limit

Idea: Use equivalent infinitesimal substitution.

Detailed steps:

As $x \to 0$ , $\arcsin x \sim x$ and $\arctan x \sim x$
$\lim_{x \to 0} \frac{\arcsin x}{\arctan x} = \lim_{x \to 0} \frac{x}{x} = 1$

Answer: The limit is 1.

Summary

Symbols Used in This Article

符号	类型	读音/说明	在本文中的含义
$\arcsin$	数学符号	arcsine function	The inverse of the sine function
$\arctan$	数学符号	arctangent function	The inverse of the tangent 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
反三角函数	inverse trigonometric function	/ɪnˈvɜːs ˌtrɪɡənəˈmetrɪk ˈfʌŋkʃən/	The inverse of trigonometric functions
反正弦函数	arcsine function	/ˈɑːksaɪn ˈfʌŋkʃən/	The inverse of the sine function
反正切函数	arctangent function	/ˈɑːktændʒənt ˈfʌŋkʃən/	The inverse of the tangent 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
反函数	inverse function	/ɪnˈvɜːs ˈfʌŋkʃən/	A function that is the inverse operation of the original one

Limits of Exponential Functions

Limits of Logarithmic Functions

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Limits of Inverse Trigonometric Functions

Limit of the Arcsine Function

Limit of the Arctangent Function

Practice Problems

Exercise 1

Exercise 2

Exercise 3

Summary

Symbols Used in This Article

中英对照

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