From 4c44b50e161eaa7abba136d3cdf48e02ec86ad11 Mon Sep 17 00:00:00 2001 From: Cym10x Date: Fri, 26 Dec 2025 07:03:23 +0800 Subject: [PATCH 01/13] minor edit --- 编写小组/讲义/子数列问题&考试易错点汇总.md | 1 + 1 file changed, 1 insertion(+) diff --git a/编写小组/讲义/子数列问题&考试易错点汇总.md b/编写小组/讲义/子数列问题&考试易错点汇总.md index 338c02d..a28402c 100644 --- a/编写小组/讲义/子数列问题&考试易错点汇总.md +++ b/编写小组/讲义/子数列问题&考试易错点汇总.md @@ -466,6 +466,7 @@ $\forall \delta>0, \exists x \in \mathring{U}(x_0,\delta)$有 $|f(x)|>M$,称$f > [!example] 例2 > 双子数列$\lim\limits_{n\rightarrow \infty}{n\cos n\pi}$ +这个也是无界但不是无穷大,请自行证明。 总结:无穷大是在邻域内“一直都很大”,无界是邻域内“有很大的” -- 2.34.1 From 94192852ed1871de6a8ab4bc742b371a257a7edf Mon Sep 17 00:00:00 2001 From: unknown <2974730459@qq.com> Date: Fri, 26 Dec 2025 10:39:53 +0800 Subject: [PATCH 02/13] vault backup: 2025-12-26 10:39:53 --- .../证明题方法:单调有界定理,介值定理.md | 4 ++++ ...法:单调有界定理,介值定理(解析版).md | 4 ++++ 2 files changed, 8 insertions(+) diff --git a/编写小组/讲义/证明题方法:单调有界定理,介值定理.md b/编写小组/讲义/证明题方法:单调有界定理,介值定理.md index 98d80bd..3eb43b3 100644 --- a/编写小组/讲义/证明题方法:单调有界定理,介值定理.md +++ b/编写小组/讲义/证明题方法:单调有界定理,介值定理.md @@ -16,6 +16,10 @@ tags: - **函数单调有界性质**: 若函数 $f(x)$ 在区间 $I$ 上单调且有界,则 $f(x)$ 在 $I$ 的端点或无穷远处存在单侧极限。 +- **这类题的基本思路**: + 先看有界性:尝试用不等式或者使用数学归纳法 + 再看单调性:尝试作差、作商或者使用数学归纳法 + ## 适用情况 适用于证明数列或函数极限的存在性,尤其是: diff --git a/编写小组/讲义/证明题方法:单调有界定理,介值定理(解析版).md b/编写小组/讲义/证明题方法:单调有界定理,介值定理(解析版).md index 4bc4449..d85674a 100644 --- a/编写小组/讲义/证明题方法:单调有界定理,介值定理(解析版).md +++ b/编写小组/讲义/证明题方法:单调有界定理,介值定理(解析版).md @@ -15,6 +15,10 @@ tags: - **函数单调有界性质**: 若函数 $f(x)$ 在区间 $I$ 上单调且有界,则 $f(x)$ 在 $I$ 的端点或无穷远处存在单侧极限。 +- **这类题的基本思路**: + 先看有界性:尝试用不等式或者使用数学归纳法 + 再看单调性:尝试作差、作商或者使用数学归纳法 + ## 适用情况 适用于证明数列或函数极限的存在性,尤其是: -- 2.34.1 From f70a26d769997e0b2da4a9a509aafa17f5c7a61e Mon Sep 17 00:00:00 2001 From: unknown <2974730459@qq.com> Date: Fri, 26 Dec 2025 10:43:18 +0800 Subject: [PATCH 03/13] vault backup: 2025-12-26 10:43:18 --- .../证明题方法:单调有界定理,介值定理.md | 4 +++- ...法:单调有界定理,介值定理(解析版).md | 4 +++- 2 files changed, 6 insertions(+), 2 deletions(-) diff --git a/编写小组/讲义/证明题方法:单调有界定理,介值定理.md b/编写小组/讲义/证明题方法:单调有界定理,介值定理.md index 3eb43b3..8c4903b 100644 --- a/编写小组/讲义/证明题方法:单调有界定理,介值定理.md +++ b/编写小组/讲义/证明题方法:单调有界定理,介值定理.md @@ -17,7 +17,9 @@ tags: 若函数 $f(x)$ 在区间 $I$ 上单调且有界,则 $f(x)$ 在 $I$ 的端点或无穷远处存在单侧极限。 - **这类题的基本思路**: - 先看有界性:尝试用不等式或者使用数学归纳法 + + 先看有界性:尝试用不等式或者使用数学归纳法,注意在上下界不好找的时候,可以先取极限,来确定一个大致范围,再对猜测的有界性尝试证明 + 再看单调性:尝试作差、作商或者使用数学归纳法 ## 适用情况 diff --git a/编写小组/讲义/证明题方法:单调有界定理,介值定理(解析版).md b/编写小组/讲义/证明题方法:单调有界定理,介值定理(解析版).md index d85674a..e0467e9 100644 --- a/编写小组/讲义/证明题方法:单调有界定理,介值定理(解析版).md +++ b/编写小组/讲义/证明题方法:单调有界定理,介值定理(解析版).md @@ -16,7 +16,9 @@ tags: 若函数 $f(x)$ 在区间 $I$ 上单调且有界,则 $f(x)$ 在 $I$ 的端点或无穷远处存在单侧极限。 - **这类题的基本思路**: - 先看有界性:尝试用不等式或者使用数学归纳法 + + 先看有界性:尝试用不等式或者使用数学归纳法,注意在上下界不好找的时候,可以先取极限,来确定一个大致范围,再对猜测的有界性尝试证明 + 再看单调性:尝试作差、作商或者使用数学归纳法 ## 适用情况 -- 2.34.1 From 411a1b268a57c468e040191737fcf3957724358d Mon Sep 17 00:00:00 2001 From: unknown <2974730459@qq.com> Date: Fri, 26 Dec 2025 10:57:15 +0800 Subject: [PATCH 04/13] vault backup: 2025-12-26 10:57:15 --- ...方法:单调有界定理,介值定理(解析版).md | 1 + 1 file changed, 1 insertion(+) diff --git a/编写小组/讲义/证明题方法:单调有界定理,介值定理(解析版).md b/编写小组/讲义/证明题方法:单调有界定理,介值定理(解析版).md index e0467e9..8c868c6 100644 --- a/编写小组/讲义/证明题方法:单调有界定理,介值定理(解析版).md +++ b/编写小组/讲义/证明题方法:单调有界定理,介值定理(解析版).md @@ -4,6 +4,7 @@ tags: --- **内部资料,禁止传播** **编委会(不分先后,姓氏首字母顺序):程奕铭 韩魏 刘柯妤 卢吉辚 王轲楠 支宝宁 郑哲航 + # 单调有界准则 ## 原理 -- 2.34.1 From cfc14f1ec859826b4334f9fd8be9a7a971974623 Mon Sep 17 00:00:00 2001 From: unknown <2974730459@qq.com> Date: Fri, 26 Dec 2025 11:01:14 +0800 Subject: [PATCH 05/13] vault backup: 2025-12-26 11:01:14 --- .../讲义/极限计算的基本方法(解析版2).md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/编写小组/讲义/极限计算的基本方法(解析版2).md b/编写小组/讲义/极限计算的基本方法(解析版2).md index e91db4c..f63d7d9 100644 --- a/编写小组/讲义/极限计算的基本方法(解析版2).md +++ b/编写小组/讲义/极限计算的基本方法(解析版2).md @@ -343,7 +343,8 @@ $$ ### 4.3 使用原则 1. **乘除运算可直接替换** 2. **加减运算需谨慎**:只有同阶无穷小相加减时,不能随意替换 -3. **复合函数可整体替换** +3. **无穷乘方运算一般也不行**(即乘方上下均有变量时):一对等价量可以写成差一个无穷小的形式,经过无穷乘方后无法得知其情况 +4. **复合函数可整体替换** ### 4.4 典型例题 -- 2.34.1 From 0018f177c28ce101275de1ad7f6cec2cdf7dca33 Mon Sep 17 00:00:00 2001 From: unknown <2974730459@qq.com> Date: Fri, 26 Dec 2025 11:03:45 +0800 Subject: [PATCH 06/13] vault backup: 2025-12-26 11:03:45 --- .../讲义/极限计算的基本方法.md | 22 +++++++++- ...限计算的基本方法(解析版).md | 42 ++++++++++++++++++- 2 files changed, 61 insertions(+), 3 deletions(-) diff --git a/编写小组/讲义/极限计算的基本方法.md b/编写小组/讲义/极限计算的基本方法.md index 86a0f37..e227386 100644 --- a/编写小组/讲义/极限计算的基本方法.md +++ b/编写小组/讲义/极限计算的基本方法.md @@ -192,8 +192,26 @@ $$\lim\limits_{x \to \infty} x(e^\tfrac{1}{x}-1)=\lim\limits_{x \to \infty } \tf # 等价无穷小求极限法 - -本质是利用已知公式进行等价无穷小因子的代换将复杂函数化简 +## 4.1 基本原理 + +在乘积和商的极限运算中,可以将复杂的无穷小量替换为等价的简单无穷小量,简化计算。 + +## 4.2 常用等价无穷小($x\to 0$) +| 等价形式 | 条件 | +|---------|------| +| $x \sim \sin x \sim \tan x \sim \arcsin x \sim \arctan x$ | 基础等价 | +| $x \sim e^x-1 \sim \ln(1+x)$ | 指数对数等价 | +| $1-\cos x \sim \frac{1}{2}x^2$ | 三角函数等价 | +| $(1+x)^a-1 \sim ax$ | 幂函数等价 | +| $a^x-1 \sim x\ln a\ (a>0)$ | 指数函数等价 | +| $x - \sin x \sim \frac{1}{6}x^3$ | 高阶等价 | +| $\tan x - x \sim \frac{1}{3}x^3$ | 高阶等价 | + +## 4.3 使用原则 +1. **乘除运算可直接替换** +2. **加减运算需谨慎**:只有同阶无穷小相加减时,不能随意替换 +3. **无穷乘方运算一般也不行**(即乘方上下均有变量时):一对等价量可以写成差一个无穷小的形式,经过无穷乘方后无法得知其情况 +4. **复合函数可整体替换** > [!example] 例1 diff --git a/编写小组/讲义/极限计算的基本方法(解析版).md b/编写小组/讲义/极限计算的基本方法(解析版).md index b49dde7..d8e0b7e 100644 --- a/编写小组/讲义/极限计算的基本方法(解析版).md +++ b/编写小组/讲义/极限计算的基本方法(解析版).md @@ -371,7 +371,47 @@ $$= \frac{2\tan t}{1 - \tan t}.$$ # 等价无穷小求极限法 -本质是利用已知公式进行等价无穷小因子的代换将复杂函数化简 +## 4.1 基本原理 + +在乘积和商的极限运算中,可以将复杂的无穷小量替换为等价的简单无穷小量,简化计算。 + +## 4.2 常用等价无穷小($x\to 0$) +| 等价形式 | 条件 | +|---------|------| +| $x \sim \sin x \sim \tan x \sim \arcsin x \sim \arctan x$ | 基础等价 | +| $x \sim e^x-1 \sim \ln(1+x)$ | 指数对数等价 | +| $1-\cos x \sim \frac{1}{2}x^2$ | 三角函数等价 | +| $(1+x)^a-1 \sim ax$ | 幂函数等价 | +| $a^x-1 \sim x\ln a\ (a>0)$ | 指数函数等价 | +| $x - \sin x \sim \frac{1}{6}x^3$ | 高阶等价 | +| $\tan x - x \sim \frac{1}{3}x^3$ | 高阶等价 | + +## 4.3 使用原则 +1. **乘除运算可直接替换** +2. **加减运算需谨慎**:只有同阶无穷小相加减时,不能随意替换 +3. **无穷乘方运算一般也不行**(即乘方上下均有变量时):一对等价量可以写成差一个无穷小的形式,经过无穷乘方后无法得知其情况 +4. **复合函数可整体替换** + + +> [!example] 例1 +> 计算 $\displaystyle \lim_{x\to 0}\frac{\sin mx}{\sin nx}$ + + + + + + + +> [!example] 例2 +> 计算 $$\displaystyle \lim_{x\to 0}(\cos x)^{\frac{1+x}{\sin^2 x}}$$ + + + + + + +熟悉常见的极限,灵活运用好极限的四则运算法则和等价无穷小,已经能够解决绝大多数的极限计算题了,无论是简单题还是难题。很多看上去很复杂的所谓难题,无非是在四则运算和等价无穷小之间反复套娃而已,如果可以熟练的分离这些特征,其实不需要泰勒公式等“高级工具”就能快速准确地得出极限值。当然,不是贬低其他的工具,只是说不要学习了所谓的一些高级工具之后,就不重视这些初级的工具和结论。 + 引入练习:计算 $$\lim\limits_{x\to0}\frac{\sqrt{1+2tanx}-\sqrt{1+2sinx}}{xln(1+x)-x^2}$$     -- 2.34.1 From a40ea30f91218c6ee54bdc1ac4d9352d54083029 Mon Sep 17 00:00:00 2001 From: unknown <2974730459@qq.com> Date: Fri, 26 Dec 2025 11:10:21 +0800 Subject: [PATCH 07/13] vault backup: 2025-12-26 11:10:21 --- .../讲义/极限计算的基本方法(解析版).md | 6 +----- 1 file changed, 1 insertion(+), 5 deletions(-) diff --git a/编写小组/讲义/极限计算的基本方法(解析版).md b/编写小组/讲义/极限计算的基本方法(解析版).md index d8e0b7e..534af70 100644 --- a/编写小组/讲义/极限计算的基本方法(解析版).md +++ b/编写小组/讲义/极限计算的基本方法(解析版).md @@ -350,9 +350,7 @@ $$t = x - \frac{\pi}{4} \quad \Rightarrow \quad x = t + \frac{\pi}{4},\ t \to 0. $$= \frac{2\tan t}{1 - \tan t}.$$ 代入原极限:$$\lim\limits_{x\to\pi/4} \frac{\tan x - 1}{x - \pi/4} - = \lim\limits_{t\to 0} \frac{\frac{2\tan t}{1 - \tan t}}{t} - = \lim\limits_{t\to 0} \frac{2\tan t}{t(1 - \tan t)}.$$ 利用重要极限 $\displaystyle \lim_{u\to 0} \frac{\tan u}{u} = 1:\frac{2\tan t}{t(1 - \tan t)} = 2 \cdot \frac{\tan t}{t} \cdot \frac{1}{1 - \tan t}$ @@ -360,7 +358,6 @@ $$= \frac{2\tan t}{1 - \tan t}.$$ 当 $t \to 0$ 时,$\tan t \to 0,\lim\limits_{t\to 0} \frac{\tan t}{t} = 1,\quad \lim\limits_{t\to 0} (1 - \tan t) = 1.$ 所以$$\lim\limits_{t\to 0} 2 \cdot \frac{\tan t}{t} \cdot \frac{1}{1 - \tan t} - = 2 \cdot 1 \cdot \frac{1}{1 - 0} = 2$$ (4) @@ -558,5 +555,4 @@ $$\alpha n + \ln \left( a + \frac{\cdots}{e^{\alpha n}} \right)$$ 小练习: - -$$\lim_{x \to 0} \frac{\csc(x) - \cot(x)}{x}=$$ \ No newline at end of file +$$\lim_{x \to 0} \frac{\csc(x) - \cot(x)}{x}= $$ -- 2.34.1 From d0bf5aedb11e08aa82732f0fe8980acbb6a97b9e Mon Sep 17 00:00:00 2001 From: unknown <2974730459@qq.com> Date: Fri, 26 Dec 2025 11:23:54 +0800 Subject: [PATCH 08/13] vault backup: 2025-12-26 11:23:54 --- ...&考试易错点汇总(解析版).md | 29 +++++++++++++++---- 1 file changed, 23 insertions(+), 6 deletions(-) diff --git a/编写小组/讲义/子数列问题&考试易错点汇总(解析版).md b/编写小组/讲义/子数列问题&考试易错点汇总(解析版).md index bf45c72..2d80338 100644 --- a/编写小组/讲义/子数列问题&考试易错点汇总(解析版).md +++ b/编写小组/讲义/子数列问题&考试易错点汇总(解析版).md @@ -226,12 +226,29 @@ $$D(x)= \begin{cases}1, & x\text{为有理数时}, \\ 0, & x\text{为无理数 **解析** -1. 构造第一个数列$\{x_n^{(1)}\}$: - 取$x_n^{(1)} = x_0 + \frac{1}{n}$(有理数),则$$\lim\limits_{n \to \infty} x_n^{(1)} = x_0$$由于$x_n^{(1)}$是有理数,所以$D(x_n^{(1)}) = 1$,因此$$\lim\limits_{n \to \infty} D(x_n^{(1)}) = 1$$ -2. 构造第二个数列$\{x_n^{(2)}\}$: - 取$x_n^{(2)} = x_0 + \frac{\sqrt{2}}{n}$(无理数),则$$\lim\limits_{n \to \infty} x_n^{(2)} = x_0$$由于x_n^{(2)}是无理数,所以$D(x_n^{(2)}) = 0$,因此$$\lim\limits_{n \to \infty} D(x_n^{(2)}) = 0$$由于$$\lim\limits_{n \to \infty} D(x_n^{(1)}) = 1 \neq 0 = \lim\limits_{n \to \infty} D(x_n^{(2)})$$根据海涅定理,$\lim\limits_{x \to x_0} D(x)$不存在。 - 由于$x_0$是任意一点,所以狄利克雷函数在任何点处都不存在极限。 +1. 构造第一个数列 $\{x_n^{(1)}\}$: + + 取 $x_n^{(1)} = x_0 + \frac{1}{n}$(有理数),则 + $$\lim\limits_{n \to \infty} x_n^{(1)} = x_0$$ + + 由于 $x_n^{(1)}$ 是有理数,所以 $D(x_n^{(1)}) = 1$,因此 + $$\lim\limits_{n \to \infty} D(x_n^{(1)}) = 1$$ + +2. 构造第二个数列 $\{x_n^{(2)}\}$: + + 取 $x_n^{(2)} = x_0 + \frac{\sqrt{2}}{n}$(无理数),则 + $$\lim\limits_{n \to \infty} x_n^{(2)} = x_0$$ + + 由于 $x_n^{(2)}$ 是无理数,所以 $D(x_n^{(2)}) = 0$,因此 + $$\lim\limits_{n \to \infty} D(x_n^{(2)}) = 0$$ + + 由于 + $$\lim\limits_{n \to \infty} D(x_n^{(1)}) = 1 \neq 0 = \lim\limits_{n \to \infty} D(x_n^{(2)})$$ + + 根据海涅定理,$\lim\limits_{x \to x_0} D(x)$ 不存在。 + + 由于 $x_0$ 是任意一点,所以狄利克雷函数在任何点处都不存在极限。 # 考试易错点总结 @@ -298,7 +315,7 @@ $$ \lim_{x \to \infty} y = \lim_{x \to \infty} \frac{1 + \frac{2}{x} - \frac{3}{ **错误做法示范**:观察级数 $\sum_{n=1}^{\infty} \frac{(-1)^{n} n^2}{3^n}$,若直接对其使用比值判别法: - $$ \lim_{n \to \infty} \left| \frac{u_{n+1}}{u_n} \right| = \lim_{n \to \infty} \frac{(n+1)^2 / 3^{n+1}}{n^2 / 3^n} = \frac{1}{3} < 1 $$ +$$ \lim_{n \to \infty} \left| \frac{u_{n+1}}{u_n} \right| = \lim_{n \to \infty} \frac{(n+1)^2 / 3^{n+1}}{n^2 / 3^n} = \frac{1}{3} < 1 $$ 若由此断言“原级数收敛”,则犯了**滥用判别法**的错误。因为比值判别法(及比较、根值判别法)仅 -- 2.34.1 From 0ab5ce154b9649911ff53dbeef1359528f809819 Mon Sep 17 00:00:00 2001 From: unknown <2974730459@qq.com> Date: Fri, 26 Dec 2025 11:24:29 +0800 Subject: [PATCH 09/13] vault backup: 2025-12-26 11:24:29 --- .../子数列问题&考试易错点汇总(解析版).md | 1 + 1 file changed, 1 insertion(+) diff --git a/编写小组/讲义/子数列问题&考试易错点汇总(解析版).md b/编写小组/讲义/子数列问题&考试易错点汇总(解析版).md index 2d80338..1b46175 100644 --- a/编写小组/讲义/子数列问题&考试易错点汇总(解析版).md +++ b/编写小组/讲义/子数列问题&考试易错点汇总(解析版).md @@ -268,6 +268,7 @@ $$ dy = \frac{\sin 2x}{1+\sin^2 x} dx $$ 注意:不要漏写 $dx$ ! ## **Vol.2等价无穷小问题** + 注意,等价无穷小只能用于乘除,用于加减虽然有时也会得到正确的答案,但这并不是有保证的。归根到底这是因为等价无穷小是一种**近似**,在乘除中它的近似程度还可以用,但在加减中就未必了,加减中我们需要更精确的近似方法:泰勒展开。另外重要极限也是等价无穷小的两种特殊情况。 >[!example] 例题 >求极限$$\lim\limits_{x\to0}\frac{tanx-sinx}{x^3}$$. -- 2.34.1 From de5bd95f9d5cc8ce488c2ae970062a385acf1c40 Mon Sep 17 00:00:00 2001 From: unknown <2974730459@qq.com> Date: Fri, 26 Dec 2025 11:24:33 +0800 Subject: [PATCH 10/13] vault backup: 2025-12-26 11:24:33 --- .../子数列问题&考试易错点汇总(解析版).md | 1 + 1 file changed, 1 insertion(+) diff --git a/编写小组/讲义/子数列问题&考试易错点汇总(解析版).md b/编写小组/讲义/子数列问题&考试易错点汇总(解析版).md index 1b46175..db5fe03 100644 --- a/编写小组/讲义/子数列问题&考试易错点汇总(解析版).md +++ b/编写小组/讲义/子数列问题&考试易错点汇总(解析版).md @@ -311,6 +311,7 @@ $$ \lim_{x \to \infty} y = \lim_{x \to \infty} \frac{1 + \frac{2}{x} - \frac{3}{ **结论**:曲线的渐近线为 $x = 2$ 和 $y = 1$。 ## Vol. 4:正项级数的判别法勿滥用 + >[!example] 例3 >判断级数 $\sum_{n=1}^{\infty} \frac{(-1)^{n} n^2}{3^n}$ 的敛散性(绝对收敛、条件收敛或发散)。 -- 2.34.1 From 841d7aeb986ec77cdba8ad5d746bee1f0a86ec5a Mon Sep 17 00:00:00 2001 From: unknown <2974730459@qq.com> Date: Fri, 26 Dec 2025 11:30:33 +0800 Subject: [PATCH 11/13] vault backup: 2025-12-26 11:30:33 --- .../子数列问题&考试易错点汇总.md | 9 +++++---- ...&考试易错点汇总(解析版).md | 20 +++++++++++++------ 2 files changed, 19 insertions(+), 10 deletions(-) diff --git a/编写小组/讲义/子数列问题&考试易错点汇总.md b/编写小组/讲义/子数列问题&考试易错点汇总.md index a28402c..e176882 100644 --- a/编写小组/讲义/子数列问题&考试易错点汇总.md +++ b/编写小组/讲义/子数列问题&考试易错点汇总.md @@ -232,7 +232,7 @@ $$D(x)= \begin{cases}1, & x\text{为有理数时}, \\ 0, & x\text{为无理数 ## Vol. 1:补药漏写dx口牙! ->[!example] 例1 +>[!example] 例题 >设函数 $y = \ln(1 + \sin^2 x)$,求其微分 $dy$。 ``` @@ -268,7 +268,7 @@ $$D(x)= \begin{cases}1, & x\text{为有理数时}, \\ 0, & x\text{为无理数 ## Vol. 3:可去间断点的说明 ->[!example] 例2 +>[!example] 例题 >求曲线 $y = \frac{x^2 + 2x - 3}{x^2 - 3x + 2}$ 的所有渐近线。 ``` @@ -284,7 +284,8 @@ $$D(x)= \begin{cases}1, & x\text{为有理数时}, \\ 0, & x\text{为无理数 ## Vol. 4:正项级数的判别法勿滥用 ->[!example] 例3 + +>[!example] 例题 >判断级数 $\sum_{n=1}^{\infty} \frac{(-1)^{n} n^2}{3^n}$ 的敛散性(绝对收敛、条件收敛或发散)。 ``` @@ -434,7 +435,7 @@ $y=f^{-1}(x)$,即$x = f(y)$,求$f^{-1'}$就是在求$\frac{dy}{dx}$,而$\f 所以,我们还需要将$y = f^{-1}(x)$代入,即: $\frac{dy}{dx}=\frac{1}{f'(y)}=\frac{1}{f'(f^{-1}(x))}$ -> [!example] 例1 +> [!example] 例题 > 求$d(\arcsin x)$ ``` diff --git a/编写小组/讲义/子数列问题&考试易错点汇总(解析版).md b/编写小组/讲义/子数列问题&考试易错点汇总(解析版).md index db5fe03..01f067c 100644 --- a/编写小组/讲义/子数列问题&考试易错点汇总(解析版).md +++ b/编写小组/讲义/子数列问题&考试易错点汇总(解析版).md @@ -254,7 +254,7 @@ $$D(x)= \begin{cases}1, & x\text{为有理数时}, \\ 0, & x\text{为无理数 ## Vol. 1:补药漏写dx口牙! ->[!example] 例1 +>[!example] 例题 >设函数 $y = \ln(1 + \sin^2 x)$,求其微分 $dy$。 **解**: @@ -270,8 +270,9 @@ $$ dy = \frac{\sin 2x}{1+\sin^2 x} dx $$ ## **Vol.2等价无穷小问题** 注意,等价无穷小只能用于乘除,用于加减虽然有时也会得到正确的答案,但这并不是有保证的。归根到底这是因为等价无穷小是一种**近似**,在乘除中它的近似程度还可以用,但在加减中就未必了,加减中我们需要更精确的近似方法:泰勒展开。另外重要极限也是等价无穷小的两种特殊情况。 + >[!example] 例题 ->求极限$$\lim\limits_{x\to0}\frac{tanx-sinx}{x^3}$$. +>求极限$$\lim\limits_{x\to0}\frac{tanx-sinx}{x^3}$$ **解**:如果直接用$tanx\sim x,sinx\sim x(x\to0)$的话,分子就会变成$0$,从而极限为$0$. 然而从另一个角度看,$$\lim\limits_{x\to0}\frac{tanx-sinx}{x^3}=\lim\limits_{x\to0}\frac{sinx-sinxcosx}{x^3cosx}=\lim\limits_{x\to0}\frac{sinx(1-cosx)}{x^3}\cdot \frac{1}{cosx}=\lim\limits_{x\to0}\frac{x\cdot\frac{1}{2}x^2}{x^3}=\frac{1}{2}.$$ @@ -281,7 +282,7 @@ $$ dy = \frac{\sin 2x}{1+\sin^2 x} dx $$ ## Vol. 3:可去间断点的说明 ->[!example] 例2 +>[!example] 例题 >求曲线 $y = \frac{x^2 + 2x - 3}{x^2 - 3x + 2}$ 的所有渐近线。 **解**: @@ -312,7 +313,7 @@ $$ \lim_{x \to \infty} y = \lim_{x \to \infty} \frac{1 + \frac{2}{x} - \frac{3}{ ## Vol. 4:正项级数的判别法勿滥用 ->[!example] 例3 +>[!example] 例题 >判断级数 $\sum_{n=1}^{\infty} \frac{(-1)^{n} n^2}{3^n}$ 的敛散性(绝对收敛、条件收敛或发散)。 **错误做法示范**:观察级数 $\sum_{n=1}^{\infty} \frac{(-1)^{n} n^2}{3^n}$,若直接对其使用比值判别法: @@ -451,7 +452,9 @@ $$L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$$ - 所以$a_n$ 发散。 ## **Vol.7 分段函数分段点处求导问题** + 分段函数分段点处无论是求导还是判断连续都必须从**左右两边的极限**分别去算,而且计算的时候一定只能**用定义**。 + >[!example] 例题 >设$f(x)=\begin{cases} \frac{2}{3}x ,\ \ x\le1 \\ x^2, \ \ x>1,\end{cases}$则$f(x)$在$x=1$处的\[ \]. >(A)左右导数都存在 (B)左导数存在,右导数不存在 @@ -472,7 +475,7 @@ $y=f^{-1}(x)$,即$x = f(y)$,求$f^{-1'}$就是在求$\frac{dy}{dx}$,而$\f 所以,我们还需要将$y = f^{-1}(x)$代入,即: $\frac{dy}{dx}=\frac{1}{f'(y)}=\frac{1}{f'(f^{-1}(x))}$ -> [!example] 例1 +> [!example] 例题 > 求$d(\arcsin x)$ 解:设$y=\arcsin x$,即$x=\sin y$,$dx=\cos y\ dy$,即$dy=\frac{dx}{\cos y}$ @@ -482,14 +485,19 @@ $\frac{dy}{dx}=\frac{1}{f'(y)}=\frac{1}{f'(f^{-1}(x))}$ 得$\cos y=\sqrt{1-x^2}$,综上,$dy=\frac{dx}{\sqrt{1-x^2}}$,即$d(\arcsin x)=\frac{dx}{\sqrt{1-x^2}}$ ## Vol. 10: 无界与无穷大的辨析 + 很多人都觉得无界和无穷大是同一个概念,因为它们的实在是太像了:画在坐标系上都是“直指苍穹🚀”或者“飞流直下三千尺”嘛!但是,“无界”准确来说不完全是这样。要准确辨析它们,需要回到它们的**定义**上: + 无穷大的定义:$\forall M > 0, \exists \delta>0$,当$0<|x-x_0|<\delta$时有$|f(x)|>M$,称$f(x)$是当$x\rightarrow x_0$时的无穷大量 + 无界量的定义:由有界的定义($\exists M > 0, \forall x \in D_f,|f(x)| 0$, $\forall \delta>0, \exists x \in \mathring{U}(x_0,\delta)$有 $|f(x)|>M$,称$f(x)$是当$x\rightarrow x_0$时的无界量 + **核心区别**:无穷大是**存在某**去心邻域内**任意**$x$都大于$M$,无界是需要对**任意**邻域**存在**一个$x$使得$|f(x)|>M$ + **联系**:无穷大一定是无界量,但是无界量不一定是无穷大。 -> [!example] 例1 +> [!example] 例题 > 无穷震荡$\lim\limits_{x\rightarrow 0}{\frac{1}{x}\sin\frac{1}{x}}$ ![[易错点10-1.png]] -- 2.34.1 From e611c1a03436a04c29b716cee5c0db97ebcdd47b Mon Sep 17 00:00:00 2001 From: unknown <2974730459@qq.com> Date: Fri, 26 Dec 2025 11:32:39 +0800 Subject: [PATCH 12/13] vault backup: 2025-12-26 11:32:39 --- .../子数列问题&考试易错点汇总(解析版).md | 1 + 1 file changed, 1 insertion(+) diff --git a/编写小组/讲义/子数列问题&考试易错点汇总(解析版).md b/编写小组/讲义/子数列问题&考试易错点汇总(解析版).md index 01f067c..0510944 100644 --- a/编写小组/讲义/子数列问题&考试易错点汇总(解析版).md +++ b/编写小组/讲义/子数列问题&考试易错点汇总(解析版).md @@ -463,6 +463,7 @@ $$L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$$ **解:**$f(1)=\frac{2}{3},f'_-(1)=\lim\limits_{x\to1^-}\frac{\frac{2}{3}x-\frac{2}{3}}{x-1}=\frac{2}{3},f'_+(1)=\lim\limits_{x\to1^+}\frac{x^2-\frac{2}{3}}{x-1}=+\infty$,故左导数存在,右导数不存在,选$B$. ## **Vol.8绝对收敛级数** + 绝对收敛的级数满足加法交换律,也就是说,交换各项的顺序不会导致最后结果的改变。但条件收敛的级数是不满足交换律的,改变加法的顺序可能会导致最后结果的改变,甚至可能使原本收敛的级数变成发散级数。这一点了解就行,不会出题目给大家考。 ## Vol. 9: 反函数求导 -- 2.34.1 From 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这个也是无界但不是无穷大,请自行证明。 -总结:无穷大是在邻域内“一直都很大”,无界是邻域内“有很大的” +``` + + + + + + + + + + + + + + + +``` +总结:无穷大是在邻域内“一直都很大”,无界是邻域内“有很大的” \ No newline at end of file diff --git a/编写小组/讲义/子数列问题&考试易错点汇总(解析版).md b/编写小组/讲义/子数列问题&考试易错点汇总(解析版).md index 0510944..87a184a 100644 --- a/编写小组/讲义/子数列问题&考试易错点汇总(解析版).md +++ b/编写小组/讲义/子数列问题&考试易错点汇总(解析版).md @@ -506,6 +506,7 @@ $\forall \delta>0, \exists x \in \mathring{U}(x_0,\delta)$有 $|f(x)|>M$,称$f 那这个是有界的吗?也不是。这就是典型的**不是无界量的无穷大**。 > [!example] 例2 -> 双子数列$\lim\limits_{n\rightarrow \infty}{n\cos n\pi}$ +> 双子数列$\lim\limits_{n\rightarrow \infty}{n\cos \frac{n\pi}{2}}$ +![[易错点10-2.png]] 总结:无穷大是在邻域内“一直都很大”,无界是邻域内“有很大的” -- 2.34.1