This is a beta course, so its structure, chapters, and examples may continue to change.
Comprehensive Exercises on Important Limits
This section contains comprehensive exercises on important limits, covering various types of limit computation, to help consolidate the understanding and application of important limits.
Exercises Exercise 1
Reference Answer (1 个标签)
important limit
Idea :
Use the generalized form of the first important limit.
Detailed steps :
lim x → 0 sin 5 x x = lim x → 0 5 ⋅ sin 5 x 5 x \lim_{x \to 0} \frac{\sin 5x}{x} = \lim_{x \to 0} 5 \cdot \frac{\sin 5x}{5x} lim x → 0 x s i n 5 x = lim x → 0 5 ⋅ 5 x s i n 5 x
= 5 ⋅ lim x → 0 sin 5 x 5 x = 5 ⋅ 1 = 5 = 5 \cdot \lim_{x \to 0} \frac{\sin 5x}{5x} = 5 \cdot 1 = 5 = 5 ⋅ lim x → 0 5 x s i n 5 x = 5 ⋅ 1 = 5
Answer : The limit is 5.
Exercise 2
Find the limit lim x → ∞ ( 1 + 3 x ) x \lim_{x \to \infty} \left(1 + \frac{3}{x}\right)^x lim x → ∞ ( 1 + x 3 ) x
Reference Answer (1 个标签)
important limit
Idea :
Use the generalized form of the second important limit.
Detailed steps :
lim x → ∞ ( 1 + 3 x ) x = e 3 \lim_{x \to \infty} \left(1 + \frac{3}{x}\right)^x = e^3 lim x → ∞ ( 1 + x 3 ) x = e 3 Answer : The limit is e 3 e^3 e 3
Exercise 3
Find the limit lim x → 0 e x − 1 sin x \lim_{x \to 0} \frac{e^x - 1}{\sin x} lim x → 0 s i n x e x − 1
Reference Answer (1 个标签)
important limit
Idea :
Use equivalent infinitesimal substitution.
Detailed steps :
As x → 0 x \to 0 x → 0 e x − 1 ∼ x e^x - 1 \sim x e x − 1 ∼ x sin x ∼ x \sin x \sim x sin x ∼ x
lim x → 0 e x − 1 sin x = lim x → 0 x x = 1 \lim_{x \to 0} \frac{e^x - 1}{\sin x} = \lim_{x \to 0} \frac{x}{x} = 1 lim x → 0 s i n x e x − 1 = lim x → 0 x x = 1
Answer : The limit is 1.
Exercise 4
Find the limit lim x → 0 ln ( 1 + x 2 ) x 2 \lim_{x \to 0} \frac{\ln(1 + x^2)}{x^2} lim x → 0 x 2 l n ( 1 + x 2 )
Reference Answer (1 个标签)
important limit
Idea :
Use the logarithmic limit.
Detailed steps :
As x → 0 x \to 0 x → 0 ln ( 1 + x 2 ) ∼ x 2 \ln(1 + x^2) \sim x^2 ln ( 1 + x 2 ) ∼ x 2
lim x → 0 ln ( 1 + x 2 ) x 2 = 1 \lim_{x \to 0} \frac{\ln(1 + x^2)}{x^2} = 1 lim x → 0 x 2 l n ( 1 + x 2 ) = 1
Answer : The limit is 1.
Exercise 5
Find the limit lim x → 0 ( 1 + x ) 3 − 1 x \lim_{x \to 0} \frac{(1 + x)^3 - 1}{x} lim x → 0 x ( 1 + x ) 3 − 1
Reference Answer (1 个标签)
important limit
Idea :
Use the power limit formula.
Detailed steps :
lim x → 0 ( 1 + x ) 3 − 1 x = 3 \lim_{x \to 0} \frac{(1 + x)^3 - 1}{x} = 3 lim x → 0 x ( 1 + x ) 3 − 1 = 3 Answer : The limit is 3.
Exercise 6
Find the limit lim x → 0 tan 2 x x \lim_{x \to 0} \frac{\tan 2x}{x} lim x → 0 x t a n 2 x
Reference Answer (1 个标签)
important limit
Idea :
Use the tangent limit and a change of variable.
Detailed steps :
lim x → 0 tan 2 x x = lim x → 0 2 ⋅ tan 2 x 2 x = 2 ⋅ 1 = 2 \lim_{x \to 0} \frac{\tan 2x}{x} = \lim_{x \to 0} 2 \cdot \frac{\tan 2x}{2x} = 2 \cdot 1 = 2 lim x → 0 x t a n 2 x = lim x → 0 2 ⋅ 2 x t a n 2 x = 2 ⋅ 1 = 2 Answer : The limit is 2.
Exercise 7
Find the limit lim x → 0 1 − cos x x 2 \lim_{x \to 0} \frac{1 - \cos x}{x^2} lim x → 0 x 2 1 − c o s x
Reference Answer (1 个标签)
important limit
Idea :
Use the cosine limit.
Detailed steps :
lim x → 0 1 − cos x x 2 = 1 2 \lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2} lim x → 0 x 2 1 − c o s x = 2 1 Answer : The limit is 1 2 \frac{1}{2} 2 1
Exercise 8
Find the limit lim x → 0 ( 1 + 2 x ) 1 x \lim_{x \to 0} (1 + 2x)^{\frac{1}{x}} lim x → 0 ( 1 + 2 x ) x 1
Reference Answer (1 个标签)
important limit
Idea :
Use a change of variable and the second important limit.
Detailed steps :
Let t = 1 x t = \frac{1}{x} t = x 1 x → 0 x \to 0 x → 0 t → ∞ t \to \infty t → ∞
lim x → 0 ( 1 + 2 x ) 1 x = lim t → ∞ ( 1 + 2 t ) t = e 2 \lim_{x \to 0} (1 + 2x)^{\frac{1}{x}} = \lim_{t \to \infty} \left(1 + \frac{2}{t}\right)^t = e^2 lim x → 0 ( 1 + 2 x ) x 1 = lim t → ∞ ( 1 + t 2 ) t = e 2
Answer : The limit is e 2 e^2 e 2
Exercise 9
Find the limit lim x → 0 sin ( e x − 1 ) x \lim_{x \to 0} \frac{\sin(e^x - 1)}{x} lim x → 0 x s i n ( e x − 1 )
Reference Answer (1 个标签)
important limit
Idea :
Use the equivalent infinitesimal of a composite function.
Detailed steps :
As x → 0 x \to 0 x → 0 e x − 1 ∼ x e^x - 1 \sim x e x − 1 ∼ x
Hence sin ( e x − 1 ) ∼ e x − 1 ∼ x \sin(e^x - 1) \sim e^x - 1 \sim x sin ( e x − 1 ) ∼ e x − 1 ∼ x
lim x → 0 sin ( e x − 1 ) x = lim x → 0 x x = 1 \lim_{x \to 0} \frac{\sin(e^x - 1)}{x} = \lim_{x \to 0} \frac{x}{x} = 1 lim x → 0 x s i n ( e x − 1 ) = lim x → 0 x x = 1
Answer : The limit is 1.
Exercise 10
Find the limit lim x → ∞ ( 1 + 1 x + 1 ) x \lim_{x \to \infty} \left(1 + \frac{1}{x+1}\right)^x lim x → ∞ ( 1 + x + 1 1 ) x
Reference Answer (1 个标签)
important limit
Idea :
Use a change of variable and the second important limit.
Detailed steps :
Let t = x + 1 t = x + 1 t = x + 1 x → ∞ x \to \infty x → ∞ t → ∞ t \to \infty t → ∞
lim x → ∞ ( 1 + 1 x + 1 ) x = lim t → ∞ ( 1 + 1 t ) t − 1 \lim_{x \to \infty} \left(1 + \frac{1}{x+1}\right)^x = \lim_{t \to \infty} \left(1 + \frac{1}{t}\right)^{t-1} lim x → ∞ ( 1 + x + 1 1 ) x = lim t → ∞ ( 1 + t 1 ) t − 1
= lim t → ∞ ( 1 + 1 t ) t ⋅ ( 1 + 1 t ) − 1 = e ⋅ 1 = e = \lim_{t \to \infty} \left(1 + \frac{1}{t}\right)^t \cdot \left(1 + \frac{1}{t}\right)^{-1} = e \cdot 1 = e = lim t → ∞ ( 1 + t 1 ) t ⋅ ( 1 + t 1 ) − 1 = e ⋅ 1 = e
Answer : The limit is e e e
Exercise 11
Find the limit lim x → 0 e sin x − 1 x \lim_{x \to 0} \frac{e^{\sin x} - 1}{x} lim x → 0 x e s i n x − 1
Reference Answer (1 个标签)
important limit
Idea :
Use the equivalent infinitesimal of a composite function.
Detailed steps :
As x → 0 x \to 0 x → 0 sin x ∼ x \sin x \sim x sin x ∼ x
Hence e sin x − 1 ∼ sin x ∼ x e^{\sin x} - 1 \sim \sin x \sim x e s i n x − 1 ∼ sin x ∼ x
lim x → 0 e sin x − 1 x = lim x → 0 x x = 1 \lim_{x \to 0} \frac{e^{\sin x} - 1}{x} = \lim_{x \to 0} \frac{x}{x} = 1 lim x → 0 x e s i n x − 1 = lim x → 0 x x = 1
Answer : The limit is 1.
Exercise 12
Find the limit lim x → 0 arcsin x x \lim_{x \to 0} \frac{\arcsin x}{x} lim x → 0 x a r c s i n x
Reference Answer (1 个标签)
important limit
Idea :
Use the arcsine limit.
Detailed steps :
lim x → 0 arcsin x x = 1 \lim_{x \to 0} \frac{\arcsin x}{x} = 1 lim x → 0 x a r c s i n x = 1 Answer : The limit is 1.
Exercise 13
Find the limit lim x → 0 arctan x x \lim_{x \to 0} \frac{\arctan x}{x} lim x → 0 x a r c t a n x
Reference Answer (1 个标签)
important limit
Idea :
Use the arctangent limit.
Detailed steps :
lim x → 0 arctan x x = 1 \lim_{x \to 0} \frac{\arctan x}{x} = 1 lim x → 0 x a r c t a n x = 1 Answer : The limit is 1.
Exercise 14
Find the limit lim x → ∞ ( 1 + 1 x 2 ) x 2 \lim_{x \to \infty} \left(1 + \frac{1}{x^2}\right)^{x^2} lim x → ∞ ( 1 + x 2 1 ) x 2
Reference Answer (1 个标签)
important limit
Idea :
Use a change of variable and the second important limit.
Detailed steps :
Let t = x 2 t = x^2 t = x 2 x → ∞ x \to \infty x → ∞ t → ∞ t \to \infty t → ∞
lim x → ∞ ( 1 + 1 x 2 ) x 2 = lim t → ∞ ( 1 + 1 t ) t = e \lim_{x \to \infty} \left(1 + \frac{1}{x^2}\right)^{x^2} = \lim_{t \to \infty} \left(1 + \frac{1}{t}\right)^t = e lim x → ∞ ( 1 + x 2 1 ) x 2 = lim t → ∞ ( 1 + t 1 ) t = e
Answer : The limit is e e e
Exercise 15
Find the limit lim x → 0 sin x 3 x 3 \lim_{x \to 0} \frac{\sin x^3}{x^3} lim x → 0 x 3 s i n x 3
Reference Answer (1 个标签)
important limit
Idea :
Use a change of variable and the first important limit.
Detailed steps :
Let t = x 3 t = x^3 t = x 3 x → 0 x \to 0 x → 0 t → 0 t \to 0 t → 0
lim x → 0 sin x 3 x 3 = lim t → 0 sin t t = 1 \lim_{x \to 0} \frac{\sin x^3}{x^3} = \lim_{t \to 0} \frac{\sin t}{t} = 1 lim x → 0 x 3 s i n x 3 = lim t → 0 t s i n t = 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 sin \sin sin 数学符号 sine A trigonometric function cos \cos cos 数学符号 cosine A trigonometric function tan \tan tan 数学符号 tangent A trigonometric function arcsin \arcsin arcsin 数学符号 arcsine An inverse trigonometric function arctan \arctan arctan 数学符号 arctangent An inverse trigonometric function e e e 数学符号 natural constant The base of the natural logarithm, approximately 2.71828 ln \ln ln 数学符号 natural logarithm The logarithm with base e e e lim \lim 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 α \alpha α 希腊字母 Alpha Often used as a parameter or exponent
中英对照
中文术语 英文术语 音标 说明 第一个重要极限 first important limit /fɜːst ɪmˈpɔːtənt ˈlɪmɪt/ One of the most fundamental limits in limit computation 第二个重要极限 second important limit /ˈsekənd ɪmˈpɔːtənt ˈlɪmɪt/ Another fundamental tool in limit computation 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 极限计算 limit calculation /ˈlɪmɪt ˌkælkjʊˈleɪʃən/ Computing the limit of a function or sequence 洛必达法则 L’Hôpital’s rule /loʊˈpiːtəlz ruːl/ A method for evaluating limits
