LESSON 01 / LIMITS
Lesson 1: The Idea of Limits应用微积分课程要点第一课：极限的引入与基础性质
2026 年 9 月 29 日课堂（09:51–11:25）极限部分整理与补充 / The limits material from the session of 29 September 2026, filtered and expanded
LESSON CONTENT / 本课内容
Limits: Introduction and Core Properties极限的引入与基础性质
2026.09.29 · CN / EN
COURSE
Applied Calculus应用微积分本课属于应用微积分的极限单元，是单元的第一课，也是后续导数与积分的入口。The first lesson of the limits unit, and the doorway to derivatives and integrals.
SESSION
29 Sep 2026, 09:51-11:252026 年 9 月 29 日 第 3-5 节原课堂同时讲了函数分类、函数变换与复合函数；本课只收录其中的极限内容。The session also covered function types, transformations and composition; this lesson keeps only the limits part.
KEY IDEA
A limit is about the trend, not the value极限看趋势，不看该点取值当 x 充分接近 a 时 f(x) 趋近 L；函数在 a 点是否有定义、取值多少，都不影响极限。Whether f is defined at a, and what f(a) equals, has no effect on the limit.
EXISTENCE
Left and right limits must agree左右极限存在且相等一维情形下，极限存在的充要条件是左右极限都存在且相等；阶跃函数在 0 点是反例。In one dimension the two one-sided limits must exist and be equal.
INFINITY
Infinite limits mark vertical asymptotes无穷极限对应垂直渐近线x 趋于 a 时 |f(x)| 无限增大，则 x=a 是垂直渐近线，例如 tan x 在 x=π/2+kπ。If |f(x)| grows without bound near a, the line x=a is a vertical asymptote.
CAUTION
Oscillation and undefined points震荡与无定义点sin(1/x) 在 x 趋于 0 时极限不存在；而 (x-1)/(x²-1) 在 x=1 无定义，极限却是 1/2。sin(1/x) has no limit at 0, while a function can have a limit where it is undefined.
Why limits come first极限为什么放在最前面
中学数学处理的是有限场景的运算，高等数学开始处理无穷，而极限正是处理无穷的核心工具。这门课的入门教材先用数值和图像建立直观，再给出严谨定义；俄罗斯教材与法国布尔巴基学派偏好直接给出抽象定义，美国教材则大量使用实例与图像。两种风格各有代价，共同点是：严谨的语言体系是学科能够持续演化的基础。
School mathematics works with finite processes; calculus begins with the infinite, and limits are the tool that makes it precise. The textbook used here builds intuition from numbers and graphs before giving the formal definition, while Russian and Bourbaki-style texts prefer abstract definitions first and U.S. texts prefer examples. Either way, a rigorous language is what lets a subject keep evolving.
| 教材风格 / Style | 引入方式 / Entry point | 特点 / Trade-off |
|---|
| 本课教材 / This courseU.S.-Russian hybrid | 先数值与图像，再给定义Numbers and graphs first | 对初学者友好，直观与严谨兼顾Beginner-friendly |
| 俄罗斯 · 布尔巴基学派Russian and Bourbaki | 直接给出抽象定义Abstract definition first | 严谨紧凑，需要较强数学基础Rigorous, demanding |
| 美国教材U.S. textbooks | 大量实例与图像Examples and graphs | 直观易入门，严谨证明需要自己补Intuitive, proofs left to the reader |
结论：先建立直观，再用严谨定义把它固定下来。十一之后的课程会给极限的 ε-δ 严格定义。
Conclusion: build intuition first, then pin it down with a formal definition. The ε-δ definition comes after the holiday.
Two problems that force the idea of a limit两个逼出极限的经典问题
课堂用两个问题引入极限，它们的共同结构都是：先算一段"平均量"的比值，再让距离或时间趋于 0。
Both motivating problems share one structure: form a ratio of average quantities, then let the distance or time interval go to zero.
| 问题 / Problem | 平均量 / Average quantity | 取极限 / Take the limit | 结果 / Result |
|---|
| 切线斜率Slope of a tangent | 割线斜率 (f(x)-f(a))/(x-a)Secant slope | x → a | 该点切线斜率Tangent slope |
| 瞬时速度Instantaneous velocity | 平均速度 Δs/ΔtAverage velocity | Δt → 0 | 该时刻瞬时速度Instantaneous velocity |
例 1（切线斜率）：曲线 y = x² 在点 (1, 1) 处，割线斜率为 (x²-1)/(x-1)。当 x ≠ 1 时该式等于 x+1，因此 x 趋于 1 时割线斜率趋于 2，即切线斜率为 2。
lim (x → 1) (x²-1)/(x-1) = lim (x → 1) (x+1) = 2
Example 1: for y = x² at (1, 1) the secant slope is (x²-1)/(x-1), which equals x+1 for x ≠ 1, so the slope tends to 2.
例 2（瞬时速度）：以多伦多 450 米高的电视塔自由落体实验为例，第 5 秒的瞬时速度无法用单点直接算出，只能取该点附近极短时间段的平均速度再取极限，最终结果为 49 m/s。
Example 2: a free-fall experiment from the 450 m tower in Toronto. The instantaneous velocity at t = 5 s cannot be read off a single point; it comes from the limit of average velocities over shrinking time intervals, and equals 49 m/s.
补充：由自由落体 s = ½gt² 得 v = gt，取 g ≈ 9.8 m/s²、t = 5 s 即 49 m/s，与课堂结果一致。
The intuitive definition朴素定义与极限到底在看什么
当自变量 x 充分接近 a（但 x ≠ a）时，函数值 f(x) 可以任意接近某个常数 L，就说 x 趋于 a 时 f(x) 的极限是 L，记作 lim (x → a) f(x) = L。这个定义只关心"趋近的趋势"，不关心 a 点本身。
If f(x) can be made arbitrarily close to a constant L by taking x sufficiently close to a (but not equal to a), the limit of f at a is L, written lim (x → a) f(x) = L. Only the trend matters, not the point a itself.
| f 在 a 点的情况 / At the point | 例子 / Example at a = 1 | f(a) | 极限 / Limit |
|---|
| 有定义且等于极限Defined and equal | f(x) = x + 1 | 2 | 2 |
| 有定义但不等于极限Defined but different | 同上，只把 f(1) 单独定义为 5Same, with f(1) = 5 | 5 | 2 |
| 在 a 点无定义Not defined | f(x) = (x-1)/(x²-1) = 1/(x+1)for x ≠ 1 | 无定义 | 1/2 |
课堂上特别强调：多数同学已经会算这类极限，但只停留在直观层面，没有严谨的理论框架。从应用角度看，朴素认知够用；要往深处走，必须掌握严谨定义。
Most students can already compute such limits intuitively, but without a formal framework. That is enough for applications, not for going deeper.
One-sided limits and when a limit exists单侧极限与极限存在的条件
左极限指 x 从小于 a 的方向趋近 a，记 x → a⁻；右极限指 x 从大于 a 的方向趋近 a，记 x → a⁺。在一维情形下，函数在 a 点极限存在，当且仅当该点的左右极限都存在且相等。
The left limit uses x approaching a from below (x → a⁻); the right limit approaches from above (x → a⁺). In one dimension, the limit at a exists exactly when both one-sided limits exist and are equal.
| 情形 / Case | 左极限 | 右极限 | 结论 / Verdict |
|---|
| Heaviside 阶跃函数：x < 0 取 0，x > 0 取 1Step function | 0 | 1 | x = 0 处极限不存在No limit at 0 |
| x + 1 在 a = 1Polynomial | 2 | 2 | 极限存在，等于 2Limit exists |
| (x-1)/(x²-1) 在 a = 1Undefined at a | 1/2 | 1/2 | 极限存在，等于 1/2Limit exists despite f(1) undefined |
补充：阶跃函数在 0 点的取值通常约定为 1/2 或 1，但无论取多少都不影响左右极限，也不影响"极限不存在"的结论。
在二维或更高维空间，趋近某点的方向有无穷多个，"左右两侧"的判定不再适用，必须沿任意路径都趋于同一个值，极限才存在。
In two or more dimensions there are infinitely many directions of approach, so the two-sided test no longer applies; every path must lead to the same value.
When a limit does not exist极限不存在的典型情形
课堂给出的典型例子是 sin(1/x)：当 x 趋于 0 时，函数值在 -1 到 1 之间无限加快地震荡，自变量的极小变化就会让函数值完成多次周期震荡，不存在统一的趋近目标，因此该点极限不存在。
The lecture's standard example is sin(1/x): as x approaches 0 the value oscillates ever faster between -1 and 1, so there is no single value being approached and the limit does not exist.
补充：证明思路是构造两条都趋于 0 的序列，看它们对应的函数值。x(n) = 1/(2nπ + π/2) → sin(1/x(n)) = 1
x(n) = 1/(2nπ) → sin(1/x(n)) = 0两条序列的函数值极限不同，所以极限不存在。这也是后续用序列刻画极限（海涅定理）的伏笔。
The usual proof takes two sequences that both tend to 0 but give different values of sin(1/x), so no limit exists.
| 情形 / Case | 例子 / Example | 说明 / Note |
|---|
| 左右极限不相等One-sided limits differ | Heaviside 阶跃函数在 0 点Step function at 0 | 两侧趋势不同，没有共同目标Two different trends |
| 有界震荡Bounded oscillation | sin(1/x) 在 x → 0Near zero | 函数值被限制在 [-1, 1] 内，但没有极限Bounded but no limit |
| 无界震荡Unbounded oscillation | (1/x) · sin(1/x) 在 x → 0Near zero | 既震荡又无界，也不存在极限Neither bounded nor convergent |
| 趋于无穷大Unbounded growth | 1/x² 在 x → 0Near zero | 按实数极限定义属于不存在，写作 ∞ 只表示趋势Not a real limit, only a trend |
Neighbourhoods, infinite limits and vertical asymptotes邻域、无穷极限与垂直渐近线
点 a 的 δ 邻域指以 a 为中心、δ 为半径的开区间 (a-δ, a+δ)；去掉中心点 a 后就是去心邻域，写作 0 < |x - a| < δ。讨论极限时用的正是去心邻域，因为 a 点本身不参与。
The δ-neighbourhood of a is the open interval (a-δ, a+δ); removing a leaves the punctured neighbourhood 0 < |x - a| < δ, which is what limits actually use.
当 x 充分接近 a 时，如果 |f(x)| 可以大于任意给定的正数 M，就说 x 趋于 a 时 f(x) 的极限为无穷，记作 ∞ 或 -∞。要紧的是：无穷不是实数，它只是刻画函数变化趋势的符号。
If |f(x)| can exceed any given positive number M once x is close enough to a, the limit is said to be infinite, written ∞ or -∞. Infinity is not a real number; it is a symbol describing a trend.
| 函数 / Function | x → 0 时的行为 / Behaviour | 记法 / Notation | x=0 是垂直渐近线？ |
|---|
| 1/x² | 无限增大Grows without bound | +∞ | 是 / Yes |
| -1/x² | 无限减小Decreases without bound | -∞ | 是 / Yes |
| 1/x | 左右两侧分别趋于 -∞ 与 +∞Left -∞, right +∞ | 两侧不同，不能写成 ∞ | 是 / Yes |
| sin(1/x) | 有界震荡Bounded oscillation | 不存在，也不是 ∞Neither finite nor infinite | 否 / No |
垂直渐近线的判定：若 x 趋于 a 时函数的极限为无穷，则直线 x = a 就是这条曲线的一条垂直渐近线。典型例子是正切函数，tan x 的垂直渐近线为 x = π/2 + kπ（k 为任意整数）。
Vertical asymptote test: if the limit at a is infinite, the line x = a is a vertical asymptote. The standard example is tan x, with asymptotes at x = π/2 + kπ.
Limit laws and a worked example补充：极限运算法则与例题
以下为补充内容，课堂只做直观引入；正式证明会在严谨定义之后给出。
设 lim f 与 lim g 都存在，则四则运算可以直接拆分：
lim (f ± g) = lim f ± lim g
lim (f · g) = lim f · lim g
lim (f / g) = lim f / lim g （要求 lim g ≠ 0）
If both limits exist, sums, products and quotients can be split term by term, provided the denominator's limit is not zero.
遇到 0/0 型未定式时，不能直接"约掉 0"，要先化简：常用手段是因式分解、分子有理化、通分。例题：
lim (x → 3) (x² - 9)/(x - 3) = lim (x → 3) (x + 3) = 6
后续课程会反复使用的两个标准极限：
lim (x → 0) sin x / x = 1
lim (n → ∞) (1 + 1/n)^n = e
提醒：运算法则的前提是极限存在；0/0 只是"未定"，需要继续化简，而不是无意义或等于 0。
Terminology check and study plan术语核对与预习安排
| 常见表述 / As said | 核对与建议 / Check |
|---|
| "Delta 邻域" | 写作 δ 邻域：δ 是希腊字母，表示邻域半径；去心邻域写成 0 < |x - a| < δ。Use the Greek letter δ for the radius. |
| 极限等于无穷大，所以极限存在。An infinite limit exists. | 按实数极限的定义，极限为无穷属于"不存在"，lim = ∞ 只是一个趋势记号。Infinite limits are not limits in the real-number sense. |
| 函数在 a 点没有定义，所以极限不存在。Undefined at a means no limit. | 反例：(x-1)/(x²-1) 在 x = 1 无定义，极限却是 1/2。A function can have a limit where it is undefined. |
| 震荡函数看一眼就知道极限不存在。Oscillation is obvious. | 结论对，但需要用两条序列或 ε-δ 证明，不能只凭图像。The conclusion is right, but it still needs a proof. |
| 左右极限都存在，极限就存在。Both one-sided limits exist. | 还需要二者相等；Heaviside 阶跃函数就是反例。They must also be equal. |
| 自由落体的 49 m/s 是直接测出来的。49 m/s was measured. | 它是由 v = gt（g ≈ 9.8 m/s²，t = 5 s）取极限得到的瞬时速度，是计算而非单点测量。It is computed as a limit, not read off a single measurement. |
预习与进度：十一假期预习极限相关内容，为后续课堂做准备；十一之后的课程正式讲解极限的严谨数学定义，随后逐步推进极限计算、连续性与导数。
Over the holiday, preview the limits material. After the holiday the course gives the rigorous definition of a limit, then moves on to limit techniques, continuity and derivatives.
资料来源：2026 年 9 月 29 日「高等数学函数与极限基础讲解」课堂纪要（09:51-11:25）中与极限相关的部分，经筛选、核对与补充；标注为"补充"的小节属于课程外的整理内容，正式表述以教师讲义与后续课件为准。原课堂中的函数分类、函数变换与复合函数内容不属于极限，未收录在本课。
Source: the limits-related parts of the 29 September 2026 lecture notes, filtered, checked and expanded. Sections marked as supplements are additions outside the lecture; the instructor's materials remain authoritative.
ETS · ENGLISH · GLOBAL PATHWAYSONE TEST.