Limits and Derivatives | 极限与导数

📚 Limits and Derivatives | 极限与导数

The concept of the limit is the foundation upon which all of calculus is built. In IB Mathematics (particularly Analysis & Approaches), understanding limits leads naturally to the derivative, which measures instantaneous rates of change. This article provides a comprehensive, exam-focused review of limits and derivatives, covering definitions, techniques, and common pitfalls.

极限是整个微积分大厦的基石。在IB数学（尤其是分析与方法）中，理解极限会自然引出导数——它衡量瞬时变化率。本文将为各位提供一份全面且紧扣考点的极限与导数复习指南，涵盖定义、技巧和常见失分点。

1. What Is a Limit? | 什么是极限？

A limit asks the question: what value does a function f(x) approach as x gets arbitrarily close to some number a? We write this as lim(x→a) f(x) = L, which reads “the limit of f(x) as x approaches a equals L.” Importantly, the limit does not require f(a) to be defined; it only cares about the behaviour of f near a.

极限回答的问题是：当x无限接近某个数a时，函数f(x)趋近于什么值？我们记为lim(x→a) f(x) = L，读作”当x趋近a时，f(x)的极限等于L”。关键在于，极限并不要求f(a)有定义；它只关心f在a附近的行为。

For example, consider the function f(x) = (x² − 1)/(x − 1). Although f(1) is undefined (division by zero), we can simplify: x² − 1 = (x − 1)(x + 1), so f(x) = x + 1 for x ≠ 1. As x approaches 1, x + 1 approaches 2. Hence lim(x→1) (x² − 1)/(x − 1) = 2.

例如，考虑函数f(x) = (x² − 1)/(x − 1)。虽然f(1)无定义（除零），但我们可以化简：x² − 1 = (x − 1)(x + 1)，所以当x ≠ 1时f(x) = x + 1。当x趋近1时，x + 1趋近2。因此lim(x→1) (x² − 1)/(x − 1) = 2。

lim(x→a) f(x) = L ⟺ f(x) 可以任意接近 L，只要 x 充分接近 a（但 x ≠ a）

In IB exams, you must be able to estimate limits graphically, numerically (using tables of values), and algebraically. A table with x = 0.9, 0.99, 0.999 on the left and x = 1.1, 1.01, 1.001 on the right is a reliable numerical approach.

在IB考试中，你必须能够通过图像、数值表格和代数方法估计极限。一个典型数值方法是取x = 0.9, 0.99, 0.999（左侧逼近）和x = 1.1, 1.01, 1.001（右侧逼近）列表计算。

2. Evaluating Limits Algebraically | 代数方法求极限

When asked to evaluate a limit algebraically, the first step is always direct substitution. If this yields a finite number, that number is the limit. If it yields 0/0 (an indeterminate form), you must manipulate the expression.

求极限的第一步永远是直接代入。如果得到有限数值，那么该数值就是极限。如果得到0/0（不定式），则必须对表达式进行变形。

The three most common algebraic techniques are:

三种最常见的代数技巧是：

Factorization: Factor the numerator and denominator, cancel common factors, then substitute again.
Factorization（因式分解）：对分子分母因式分解，消去公因子后再次代入。
Rationalization: Multiply by the conjugate to eliminate radicals, especially when square roots are present.
有理化：乘以共轭表达式消除根号，尤其当表达式含平方根时。
Common denominator: Combine fractional terms into a single fraction before simplifying.
通分：将分式项合并为单一分式后再化简。

Example: Evaluate lim(x→4) (√x − 2)/(x − 4). Direct substitution gives 0/0. Multiply by the conjugate:

例题：求lim(x→4) (√x − 2)/(x − 4)。直接代入得0/0。乘以共轭式：

lim(x→4) (√x − 2)/(x − 4) = lim(x→4) [(√x − 2)(√x + 2)] / [(x − 4)(√x + 2)] = lim(x→4) (x − 4)/[(x − 4)(√x + 2)] = lim(x→4) 1/(√x + 2) = 1/4

Note the pattern: whenever you see a radical minus a constant, multiply top and bottom by the conjugate (the same terms with the opposite sign). This is one of the most tested techniques in IB Paper 1 (no calculator) limit questions.

注意这个套路：凡是见到根号减常数，就在分子分母同时乘以共轭式（符号相反的相同表达式）。这是IB Paper 1（不允许计算器）极限题中最常考的技法。

3. One-Sided Limits and Continuity | 单侧极限与连续性

A one-sided limit considers approach from only one direction. We write lim(x→a⁻) f(x) for the left-hand limit (x approaches a from below) and lim(x→a⁺) f(x) for the right-hand limit (x approaches a from above). The two-sided limit lim(x→a) f(x) exists if and only if both one-sided limits exist and are equal.

单侧极限只考虑从一个方向逼近。lim(x→a⁻) f(x)表示左极限（x从小于a的方向逼近），lim(x→a⁺) f(x)表示右极限（x从大于a的方向逼近）。双侧极限lim(x→a) f(x)存在的充要条件是左右极限都存在且相等。

A function f is continuous at x = a if and only if all three of the following hold:

函数f在x = a处连续当且仅当以下三条同时成立：

f(a) is defined (a is in the domain of f).
f(a)有定义（a在f的定义域内）。
lim(x→a) f(x) exists (left and right limits agree).
lim(x→a) f(x)存在（左右极限相等）。
lim(x→a) f(x) = f(a) (the limit equals the function value).
lim(x→a) f(x) = f(a)（极限值等于函数值）。

Common continuous functions on their domains: polynomials (everywhere), rational functions (except where denominator = 0), trigonometric functions, exponential functions, and logarithmic functions (on x > 0). In IB questions, you may be asked to find the value of a parameter k that makes a piecewise function continuous.

在其定义域上常见的连续函数有：多项式（处处连续）、有理函数（分母不为0处）、三角函数、指数函数和对数函数（x > 0）。IB题中常要求求参数k的值使分段函数连续——方法是令分段点处的左右极限与函数值相等。

Example: Find k such that f(x) = { x² if x < 2; kx − 1 if x ≥ 2 } is continuous at x = 2. Left limit: lim(x→2⁻) x² = 4. Right limit: lim(x→2⁺) (kx − 1) = 2k − 1. Set equal: 4 = 2k − 1, so k = 5/2.

例题：求k使分段函数f(x) = { x²（x < 2）；kx − 1（x ≥ 2）}在x = 2处连续。左极限：lim(x→2⁻) x² = 4。右极限：lim(x→2⁺) (kx − 1) = 2k − 1。令其相等：4 = 2k − 1，得k = 5/2。

4. Limits at Infinity | 无穷远处的极限

Limits at infinity describe the end behaviour of a function — what happens to f(x) as x → +∞ or x → −∞. These limits determine horizontal asymptotes. For a rational function f(x) = P(x)/Q(x) where P has degree m and Q has degree n, three cases arise:

无穷远处的极限描述函数的端部行为——当x → +∞或x → −∞时f(x)的走向。这些极限决定水平渐近线。对于有理函数f(x) = P(x)/Q(x)，设P的次数为m，Q的次数为n，有三种情形：

If m < n, then lim(x→∞) f(x) = 0 (horizontal asymptote y = 0).
若m < n，则lim(x→∞) f(x) = 0（水平渐近线y = 0）。
If m = n, then the limit equals the ratio of leading coefficients, a/b.
若m = n，则极限等于首项系数之比a/b。
If m > n, then the limit is infinite (no horizontal asymptote; an oblique asymptote may exist).
若m > n，则极限不存在（无穷大）（无水平渐近线；可能存在斜渐近线）。

Example: lim(x→∞) (3x² − 2x + 1)/(5x² + 4) = 3/5, since the degrees of the numerator and denominator are both 2, and the leading coefficients are 3 and 5.

例题：lim(x→∞) (3x² − 2x + 1)/(5x² + 4) = 3/5，因为分子分母次数均为2，首项系数分别为3和5。

A powerful technique is to divide every term by the highest power of x present in the denominator. This converts the expression into one whose limit is obvious, since terms like 1/x → 0 as x → ∞.

一个通用技巧是：将每一项除以分母中x的最高次幂。这会把表达式转化为极限显而易见的形式，因为1/x这类项在x → ∞时趋近于0。

5. The Derivative from First Principles | 从第一性原理定义导数

The derivative of a function f at a point x, denoted f'(x) or dy/dx, is defined as the limit of the average rate of change as the interval shrinks to zero:

函数f在点x处的导数，记作f'(x)或dy/dx，定义为平均变化率在区间收缩到零时的极限：

f'(x) = lim(h→0) [f(x + h) − f(x)] / h

This is known as differentiation from first principles (also called the delta method). Geometrically, it represents the slope of the tangent line to the curve y = f(x) at the point (x, f(x)). In IB AA, you must be able to derive the derivative of simple functions (quadratics, cubics, 1/x, √x) using this limit definition.

这叫做第一性原理求导（也称delta方法）。几何上，它代表曲线y = f(x)在点(x, f(x))处切线的斜率。在IB AA中，你必须能够用极限定义推导简单函数（二次、三次、1/x、√x）的导数。

Example: Find f'(x) for f(x) = x² from first principles.

例题：用第一性原理求f(x) = x²的导函数。

f'(x) = lim(h→0) [(x + h)² − x²] / h = lim(h→0) (x² + 2xh + h² − x²) / h = lim(h→0) (2xh + h²) / h = lim(h→0) (2x + h) = 2x

Key steps: expand, cancel f(x), divide by h, then let h → 0. Remember that as long as h ≠ 0, we may cancel it; the limit process then gives the final value.

关键步骤：展开、消去f(x)、除以h、令h → 0。记住只要h ≠ 0就可以约分；极限过程给出最终值。

When computing the derivative at a single point, an alternative equivalent formula is:

计算某一点处的导数时，另一个等价公式是：

f'(a) = lim(x→a) [f(x) − f(a)] / (x − a)

Use whichever form is more convenient for simplification.

选用更便于化简的那个形式即可。

6. Basic Differentiation Rules | 基本求导法则

The power rule states that if f(x) = xⁿ (where n is any real constant), then f'(x) = nxⁿ⁻¹. This rule works for positive, negative, and fractional exponents alike. In your final answer, rewrite negative exponents as fractions and fractional exponents as radicals unless instructed otherwise.

幂法则指出：若f(x) = xⁿ（n为任意实常数），则f'(x) = nxⁿ⁻¹。该法则对正指数、负指数和分数指数都适用。除非题目另有要求，最终答案中应将负指数改写为分式、分数指数改写为根式。

The constant multiple and sum/difference rules complete the core toolkit:

常数倍法则和和差法则构成核心工具包：

d/dx [c·f(x)] = c·f'(x), where c is a constant.
d/dx [c·f(x)] = c·f'(x)，其中c为常数。
d/dx [f(x) ± g(x)] = f'(x) ± g'(x).
d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。

These rules allow you to differentiate any polynomial term by term. For example, f(x) = 5x³ − 4x² + 7x − 2 gives f'(x) = 15x² − 8x + 7.

这些法则允许你逐项求导任意多项式。例如，f(x) = 5x³ − 4x² + 7x − 2的导数为f'(x) = 15x² − 8x + 7。

Standard derivatives you must know for IB:

IB必须掌握的标准导数公式：

d/dx (sin x) = cos x; d/dx (cos x) = −sin x; d/dx (tan x) = sec²x
d/dx (sin x) = cos x；d/dx (cos x) = −sin x；d/dx (tan x) = sec²x
d/dx (eˣ) = eˣ; d/dx (aˣ) = aˣ ln a; d/dx (ln x) = 1/x
d/dx (eˣ) = eˣ；d/dx (aˣ) = aˣ ln a；d/dx (ln x) = 1/x

7. Product and Quotient Rules | 积法则与商法则

When differentiating a product of two functions, you cannot simply differentiate each factor separately. The product rule states:

求两个函数乘积的导数时，不能简单地对每个因子分别求导。积法则如下：

若 y = uv，则 dy/dx = u·(dv/dx) + v·(du/dx)

In shorthand: (uv)’ = u’v + uv’. The order matters only in the sense that each derivative multiplies the other original function.

简写为：(uv)’ = u’v + uv’。注意每个导数与另一个原函数相乘。

Example: Differentiate y = x²·sin x. Let u = x², v = sin x. Then u’ = 2x, v’ = cos x. Hence y’ = 2x·sin x + x²·cos x.

例题：求y = x²·sin x的导数。令u = x²，v = sin x。则u’ = 2x，v’ = cos x。因此y’ = 2x·sin x + x²·cos x。

The quotient rule handles ratios of functions:

商法则处理函数的比值：

若 y = u/v，则 dy/dx = (v·(du/dx) − u·(dv/dx)) / v²

In shorthand: (u/v)’ = (u’v − uv’) / v². A helpful mnemonic: “low d-high minus high d-low, over low squared.” Keep v² in the denominator the whole time.

简写为：(u/v)’ = (u’v − uv’) / v²。助记口诀：”上导乘下减下导乘上，除以分母的平方”。分母的v²始终保留。

Example: Differentiate y = x/(x² + 1). Let u = x, v = x² + 1. Then u’ = 1, v’ = 2x. So y’ = [(1)(x² + 1) − x(2x)] / (x² + 1)² = (x² + 1 − 2x²)/(x² + 1)² = (1 − x²)/(x² + 1)².

例题：求y = x/(x² + 1)的导数。令u = x，v = x² + 1。则u’ = 1，v’ = 2x。因此y’ = [(1)(x² + 1) − x(2x)] / (x² + 1)² = (x² + 1 − 2x²)/(x² + 1)² = (1 − x²)/(x² + 1)²。

In IB, you may be asked to simplify the resulting expression and then evaluate at a specific point. Always simplify fully to avoid errors in subsequent steps.

在IB中，你可能需要化简结果表达式并在特定点处求值。务必化到最简，以免后续步骤出错。

8. The Chain Rule | 链式法则

The chain rule differentiates composite functions. If y = f(g(x)), then the derivative is the derivative of the outer function evaluated at g(x), multiplied by the derivative of the inner function:

链式法则用于求复合函数的导数。若y = f(g(x))，则导数为外层函数在g(x)处的导数乘以内层函数的导数：

dy/dx = f'(g(x)) · g'(x)，或 dy/dx = (dy/du) × (du/dx)

The second form, using Leibniz notation, emphasizes the cancellation analogy: you “multiply” rates of change along a chain.

第二种形式使用莱布尼茨记号，强调类比”消去”：沿着链式路径”相乘”各个变化率。

Common applications in IB include:

IB中常见的应用包括：

Powers of functions: d/dx (uⁿ) = nuⁿ⁻¹ · u’ (e.g., d/dx (2x + 1)⁵ = 5(2x + 1)⁴ · 2 = 10(2x + 1)⁴)
函数的幂：d/dx (uⁿ) = nuⁿ⁻¹ · u’（例如d/dx (2x + 1)⁵ = 5(2x + 1)⁴ · 2 = 10(2x + 1)⁴）
Exponentials: d/dx (eᵘ) = eᵘ · u’ (e.g., d/dx e^(3x) = 3e^(3x))
指数函数：d/dx eᵘ = eᵘ · u’（例如d/dx e^(3x) = 3e^(3x)）
Logarithms: d/dx (ln u) = u’/u (e.g., d/dx ln(x² + 1) = 2x/(x² + 1))
对数函数：d/dx (ln u) = u’/u（例如d/dx ln(x² + 1) = 2x/(x² + 1)）
Trigonometric: d/dx sin(2θ) = 2cos(2θ)
三角函数：d/dx sin(2θ) = 2cos(2θ)

The chain rule often combines with the product or quotient rule. Use brackets carefully and write out intermediate steps — most algebraic errors on IB papers come from rushing composite derivatives.

链式法则常与积法则或商法则结合使用。务必谨慎添加括号并写出中间步骤——IB卷面上大多数代数错误源于求复合导数时过于匆忙。

9. Higher-Order Derivatives | 高阶导数

Differentiating a function twice gives the second derivative, f”(x) or d²y/dx². Repeated differentiation yields third, fourth, and higher derivatives, denoted f⁽ⁿ⁾(x) or dⁿy/dxⁿ. In IB, you are typically expected to compute up to the second derivative, occasionally the third.

对函数求导两次得到二阶导数f”(x)或d²y/dx²。重复求导得到三阶、四阶及更高阶导数，记为f⁽ⁿ⁾(x)或dⁿy/dxⁿ。在IB中，通常要求计算到二阶导数，偶尔要求三阶。

Second derivatives have important interpretations:

二阶导数有重要的解释：

f”(x) > 0 on an interval means f is concave up (curving upward, cup-shaped ∪).
f”(x) > 0意味着在该区间f是凹向上的（向上弯曲，杯形∪）。
f”(x) < 0 on an interval means f is concave down (curving downward, cap-shaped ∩).
f”(x) < 0意味着在该区间f是凹向下的（向下弯曲，帽形∩）。
f”(x) = 0 at a point where concavity changes is an inflection point.
f”(x) = 0且凹凸性发生改变的点为拐点。

Example: If f(x) = x³ − 3x² + 2, then f'(x) = 3x² − 6x, f”(x) = 6x − 6. Setting f”(x) = 0 gives x = 1, and since the sign of f” changes from negative to positive across x = 1, there is an inflection point at x = 1.

例题：若f(x) = x³ − 3x² + 2，则f'(x) = 3x² − 6x，f”(x) = 6x − 6。令f”(x) = 0得x = 1。由于f”在x = 1两侧由负变正，x = 1处存在拐点。

When using Leibniz notation, the second derivative is d²y/dx² — note that the exponents appear in different positions. This is not (dy/dx)²; it is the operator d/dx applied twice.

使用莱布尼茨记号时，二阶导数为d²y/dx²——注意两个指数位置不同。这不是(dy/dx)²；而是运算符d/dx作用两次的结果。

10. Tangent Lines and Rates of Change | 切线与变化率

The derivative f'(a) gives the slope of the tangent line to y = f(x) at x = a. Therefore, the equation of the tangent line is:

导数f'(a)给出曲线y = f(x)在x = a处切线的斜率。因此切线方程为：

y − f(a) = f'(a)(x − a)

This is the point-slope form of a line. In IB, you will frequently be asked to find tangent lines, normal lines (perpendicular to the tangent, with slope −1/f'(a) when f'(a) ≠ 0), and points where the tangent is horizontal (solve f'(x) = 0).

这是直线方程的点斜式。IB中常见问题包括求切线、法线（与切线垂直，斜率为−1/f'(a)，当f'(a) ≠ 0时）以及切线水平处的点（解f'(x) = 0）。

The derivative also represents the instantaneous rate of change. In contextual problems, this could be the velocity of a particle (v = ds/dt), the growth rate of a population, or the marginal cost in economics. The average rate of change over an interval [a, b] is simply [f(b) − f(a)] / (b − a), which is the slope of the secant line.

导数还代表瞬时变化率。在应用题中，它可以是粒子的速度（v = ds/dt）、种群增长率或经济学中的边际成本。区间[a, b]上的平均变化率就是[f(b) − f(a)] / (b − a)，即割线的斜率。

Example (Kinematics): A particle moves along a straight line with displacement s(t) = t³ − 6t² + 9t (metres, t in seconds). Velocity: v(t) = s'(t) = 3t² − 12t + 9. Acceleration: a(t) = v'(t) = 6t − 12. The particle is at rest when v(t) = 0, i.e., 3(t² − 4t + 3) = 3(t − 1)(t − 3) = 0, so t = 1 s or t = 3 s. The particle changes direction at these times.

例题（运动学）：一质点沿直线运动，位移s(t) = t³ − 6t² + 9t（米，t以秒计）。速度：v(t) = s'(t) = 3t² − 12t + 9。加速度：a(t) = v'(t) = 6t − 12。质点静止时v(t) = 0，即3(t² − 4t + 3) = 3(t − 1)(t − 3) = 0，故t = 1 s或t = 3 s。质点在这两个时刻改变运动方向。

11. Common IB Exam Pitfalls | IB考试常见陷阱

Many students lose marks not from misunderstanding concepts but from small technical errors. Be aware of the following warning signs:

许多学生丢分并非因为不理解概念，而是源于技术细节错误。注意以下警示信号：

Forgetting the chain rule: d/dx (x² + 1)³ ≠ 3(x² + 1)². The correct answer is 3(x² + 1)² · 2x = 6x(x² + 1)². Always check whether the inner function has a non-trivial derivative.
忘记链式法则：d/dx (x² + 1)³ ≠ 3(x² + 1)²。正确答案是3(x² + 1)² · 2x = 6x(x² + 1)²。始终检查内层函数是否具有非平凡的导数。
Sign errors in the quotient rule: the numerator is u’v − uv’, not u’v + uv’. A minus sign mistake changes the entire result.
商法则中的符号错误：分子是u’v − uv’，不是u’v + uv’。一个负号错误会改变整个结果。
Simplifying 0/0 too early: you must factor or rationalize to cancel the problematic term before substituting. Jumping to “undefined” is incorrect when the limit is finite.
过早认为0/0无解：必须先因式分解或有理化以消去问题项，再代入。当极限有限时，直接答”无定义”是错误的。
Mixing up tan x derivative: d/dx (tan x) = sec²x, not cos²x or −sec²x.
混淆tan x的导数：d/dx (tan x) = sec²x，不是cos²x或−sec²x。
Not rewriting negative exponents: if asked for f'(x) in a simplified form, express terms like x⁻² as 1/x².
未改写负指数：若题目要求化简f'(x)，应将x⁻²这类项写成1/x²。

Always check your answer by testing whether it has the correct units and dimensions in applied problems, and whether plugging in a simple value like x = 0 or x = 1 gives a plausible slope.

在应用题中，始终用单位量纲检验答案是否合理；代入x = 0或x = 1这样的简单值，看斜率是否合理。

12. Practice Problems | 练习题目

Work through these problems, then check your answers against

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Limits and Derivatives | 极限与导数