📚 Limits and Derivatives | 极限与导数
In IB Mathematics, the concepts of limits and derivatives form the foundation of calculus. Understanding limits allows us to investigate the behaviour of functions as they approach specific points or infinity. The derivative, defined as a limit, measures the instantaneous rate of change and provides powerful tools for analysing graphs, motion, and optimisation.
在 IB 数学中，极限与导数是微积分的基础。理解极限使我们能够研究函数在趋近特定点或无穷大时的行为。导数被定义为一个极限，用来衡量瞬时变化率，并为分析图像、运动与优化提供了强有力的工具。
1. Introduction to Limits | 极限简介
ly close to a given number a. The notation is written as limₓ→ₐ f(x) = L, which is read as "the limit of f of x as x approaches a is L". Limits are essential for defining continuity, derivatives, and integrals.
极限描述的是当变量 x 无限趋近于某个给定数值 a 时，函数 f(x) 所趋向的值。其符号记作 limₓ→ₐ f(x) = L，读作“当 x 趋近于 a 时，f(x) 的极限为 L”。极限对于定义连续性、导数和积分至关重要。
2. Informal Definition and Notation | 非正式定义与符号
that we are interested in the behaviour near a, not exactly at a. For example, limₓ→₂ (x² – 4)/(x – 2) = 4, even though the function is undefined at x = 2.
非正式地说，如果通过让 x 足够接近 a（但不必等于 a）可以使 f(x) 的值任意接近 L，我们就说 limₓ→ₐ f(x) = L。极限符号强调的是在 a 附近的行为，而不是恰好在 a 点。例如，limₓ→₂ (x² – 4)/(x – 2) = 4，尽管该函数在 x = 2 处无定义。
3. One-Sided Limits | 单侧极限
ist, both one-sided limits must exist and be equal. If the left and right limits differ, the limit does not exist.
有时极限取决于 x 趋近于 a 的方向。左极限记作 limₓ→ₐ⁻ f(x)，右极限记作 limₓ→ₐ⁺ f(x)。只有当左右极限都存在并且相等时，总的极限才存在。如果左右极限不相等，则极限不存在。
For instance, consider f(x) = |x|/x. As x approaches 0 from the left, x is negative, so |x|/x = –1, giving limₓ→0⁻ f(x) = –1. From the right, x > 0, so |x|/x = 1, giving limₓ→0⁺ f(x) = 1. Since these differ, limₓ→0 f(x) does not exist.
例如，考虑 f(x) = |x|/x。当 x 从左侧趋近 0 时，x 为负，所以 |x|/x = –1，从而 limₓ→0⁻ f(x) = –1。从右侧趋近时，x > 0，所以 |x|/x = 1，得 limₓ→0⁺ f(x) = 1。因为两者不同，limₓ→0 f(x) 不存在。
4. Limits at Infinity | 无穷远处的极限
We also study limits as x approaches infinity (∞) or negative infinity (– describe the end behaviour of functions. For a rational function like f(x) = (2x² + 3)/(x² – ominator by x² gives limₓ→∞ f(x) = 2. Horizontal asymptotes are determined by such limits.
我们也研究当 x 趋近于无穷大 (∞) 或负无穷大 (–∞) 时的极限，这描述了函数的末端走势。对于有理函数如 f(x) = (2x² + 3)/(x² – 1)，分子分母同除以 x² 可得 limₓ→∞ f(x) = 2。水平渐近线由这类极限确定。
5. Computing Limits Algebraically | 代数方法计算极限
Common algebraic techniques for evaluating limits include direct substitution, factoring, rationalising numerators or denominators, and using trigonometric limits. If direct substitution yields an indeterminate form such as 0/0, we simplify the expression. Key limit properties allow breaking sums, products, and quotients.
计算极限的常见代数方法包括直接代入、因式分解、分子或分母有理化以及使用三角极限。若直接代入得到诸如 0/0 的不定式，则需化简表达式。极限的基本性质允许对和、积和商进行拆分。
Limit of a sum: limₓ→ₐ [f(x)+g(x)] = limₓ→ₐ f(x) + limₓ→ₐ g(x)
和的极限：limₓ→ₐ [f(x)+g(x)] = limₓ→ₐ f(x) + limₓ→ₐ g(x)
Limit of a product: limₓ→ₐ [f(x)·g(x)] = limₓ→ₐ f(x) · limₓ→ₐ g(x)
积的极限：limₓ→ₐ [f(x)·g(x)] = limₓ→ₐ f(x) · limₓ→ₐ g(x)
a quotient: limₓ→ₐ [f(x)/g(x)] = limₓ→ₐ f(x) / limₓ→ₐ g(x), provided denominator limit ≠ 0
商的极限：limₓ→ₐ [f(x)/g(x)] = limₓ→ₐ f(x) / limₓ→ₐ g(x)，前提是分母极限不为 0
Constant multiple: limₓ→ₐ [c·f(x)] = c·limₓ→ₐ f(x)
常数倍：limₓ→ₐ [c·f(x)] = c·limₓ→ₐ f(x)
ric limit is limₓ→0 (sin x)/x = 1, which is used to derive derivatives of sine and cosine.
一个重要的三角极限是 limₓ→0 (sin x)/x = 1，用于推导正弦和余弦的导数。
6. Continuity and Limits | 连续性与极限
graph without lifting your pen. Polynomials, exponentials, and trigonometric functions are continuous on their domains. Discontinuities occur at points where these conditions fail.
若满足三个条件，则函数 f 在 x = a 处连续：f(a) 有定义，limₓ→ₐ f(x) 存在，且 limₓ→ₐ f(x) = f(a)。直观上，可以不抬笔画出图像。多项式、指数函数和三角函数在其定义域上连续。不连续点出现在这些条件不满足的地方。
7. Introduction to Derivatives | 导数简介
The derivative of a function at a point represents the instantaneous rate of change or the slope of the tangent line. It is denoted by f'(x), dy/dx, or df/dx. The derivative of position with respect to time gives velocity, of velocity gives acceleration.
函数在某点的导数表示瞬时变化率，或者说该点切线的斜率。导数用 f'(x)、dy/dx 或 df/dx 表示。位置关于时间的导数是速度，速度关于时间的导数是加速度。
8. Derivative as a Limit: The Definition | 导数作为极限：定义
Formally, the derivative of f at x is defined by the limit: f'(x) = limₕ→₀ (f(x+h) – is called the first principle or definition from first principles. An equivalent form is f'(a) = limₓ→ₐ (f(x) – f(a)) / (x – a).
正式地，f 在 x 处的导数由以下极限定义：f'(x) = limₕ→₀ (f(x+h) – f(x)) / h，前提是该极限存在。这称为第一原理或从基本原理出发的定义。等效形式为 f'(a) = limₓ→ₐ (f(x) – f(a)) / (x – a)。
For f(x) = x², we compute f(x+h) = (x+h)² = x² + 2xh + h². Then (f(x+h) – f(x))/h = (2xh + h²)/h = 2x + h. Taking h→0 gives f'(x) = 2x.
对于 f(x) = x²，计算得 f(x+h) = (x+h)² = x² + 2xh + h²。那么 (f(x+h) – f(x))/h = (2xh + h²)/h = 2x + h。令 h→0 得到 f'(x) = 2x。
9. Differentiation Rules: Power Rule | 微分法则：幂法则
on each time is inefficient; we develop rules. The power rule states: if f(x) = xⁿ, then f' eeds up differentiation. We also have the constant multiple rule and sum/difference rules.
每次使用极限定义效率很低，因此我们推导出运算法则。幂法则指出：若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹，其中 n 为任意实数。这条法则大大加快了求导速度。此外还有常数倍法则以及和差法则。
Constant rule: d/dx (c) = 0
常数法则：d/dx (c) = 0
Constant multiple: d/dx [c f(x)] = c f'(x)
常数倍法则：d/dx [c f(x)] = c f'(x)
Sum/difference: d/dx [f(x)±g(x)] = f'(x) ± g'(x)
和差法则：d/dx [f(x)±g(x)] = f'(x) ± g'(x)
Power rule: d/dx (xⁿ) = n xⁿ⁻¹
幂法则：d/dx (xⁿ) = n xⁿ⁻¹
10. Tangent Lines and Instantaneous Rate of Change | 切线与瞬时变化率
The derivative f' of the slope of the tangent line to the curve y = f(x) at x = a. The equation of the tangent line is y – f(a) = f'(a)(x – ion of the function near that point. Instantaneous velocity is the derivative of position.
导数 f'(a) 是曲线 y = f(x) 在 x = a 处切线的斜率。切线方程为 y – f(a) = f'(a)(x – a)。这可用于函数在该点附近的线性近似。瞬时速度就是位置函数的导数。
Example: f(x) = x³, f'(x) = 3x², so f'(1) = 3 and f(1)=1. Tangent line: y – 1 = 3(x – 1) → y = 3x – 2.
例如：f(x) = x³，f'(x) = 3x²，所以 f'(1) = 3，f(1)=1。切线为 y – 1 = 3(x – 1) → y = 3x – 2。
11. Differentiability and Smoothness | 可微性与平滑性
differentiable at a corner, cusp, vertical tangent, or discontinuity. For instance, f(x)=|x| is continuous at x=0 but not differentiable because the left and right derivatives differ.
如果一个函数在某点的导数存在，则称该函数在该点可微。可微一定连续，但连续不一定可微。函数可能在尖点、尖拐点、垂直切线或不连续点处连续但不可微。例如，f(x)=|x| 在 x=0 处连续但不可微，因为左右导数不相等。
The derivative from the left is –1 and from the right is +1, so f'(0) does not exist.
左侧导数为 –1，右侧为 +1，所以 f'(0) 不存在。
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