Introduction to Limits and Calculus极限与微积分入门

As $x$ gets arbitrarily close to $c$ from either side (but never equal to $c$), the value of $f(x)$ gets arbitrarily close to $L$. The limit does not care what $f(c)$ is , only what $f$ does on a small neighbourhood around $c$.当 $x$ 从两侧任意接近 $c$（但不等于 $c$）时，$f(x)$ 任意接近 $L$。极限不在乎 $f(c)$ 本身的值 , 它只关心 $f$ 在 $c$ 附近邻域上的行为。

• Two ways to read a limit.读极限的两种方式。 Graphical: zoom in on the graph near $x = c$; the $y$-value the curve approaches is the limit. Tabular: tabulate $f(x)$ at $x$-values that creep up on $c$ from both sides; the shared trend value is the limit.图像法：在 $x = c$ 附近放大图像，曲线趋近的 $y$ 值即极限。数表法：列出从两侧逼近 $c$ 的 $x$ 处的 $f(x)$，两侧共同趋近的值即极限。
• "Hole" example."洞"的例子。 For $f(x) = (x^2 - 1)/(x - 1)$, the value $f(1)$ is undefined ($0/0$), but the limit $\lim_{x \to 1} f(x) = 2$ exists because $f(x) = x + 1$ for $x \ne 1$.对 $f(x) = (x^2 - 1)/(x - 1)$，$f(1)$ 未定义（$0/0$），但极限 $\lim_{x \to 1} f(x) = 2$ 存在，因为 $x \ne 1$ 时 $f(x) = x + 1$。
• Why this matters.为何重要。 The derivative ($\S 4$) and the definite integral ($\S 7$) are both limits. Without limits, no calculus. MCV4U Strand A spells this out: "make connections between average rate of change over an interval and instantaneous rate of change at a point, using the slopes of secants and tangents and the concept of the limit."导数（§4）与定积分（§7）都是极限。没有极限就没有微积分。MCV4U 单元 A 明确："通过把区间上的平均变化率与某点的瞬时变化率联系起来，并使用割线、切线斜率与极限概念。"

1. Direct substitution.直接代入。 If $f$ is "nice" (polynomial, or rational with non-zero denominator, etc.), then $\lim_{x \to c} f(x) = f(c)$. Always try this first.若 $f$ "好"（多项式，或分母不为零的有理式等），则 $\lim_{x \to c} f(x) = f(c)$。先试此法。
2. Factor and cancel因式分解并约分 when substitution gives the indeterminate form $0/0$. Common when the numerator and denominator share a factor $(x - c)$.当代入得未定式 $0/0$ 时。常见于分子分母含公因式 $(x - c)$ 的情形。
3. Conjugate technique共轭法 when the expression contains a $\sqrt{\,}$ that creates a $0/0$ form. Multiply numerator and denominator by the conjugate to rationalise.当表达式含 $\sqrt{\,}$ 且产生 $0/0$ 时。分子分母同乘共轭以有理化。
4. Rationalise / common denominator通分 / 化为同分母 for expressions like $\lim_{x \to 0} (1/x)(1/(x+1) - 1)$ , combine the fractions, simplify, then substitute.对于 $\lim_{x \to 0} (1/x)(1/(x+1) - 1)$ 等表达式 , 先通分化简，再代入。

• One-sided limits.单侧极限。 $\lim_{x \to c^-} f(x)$ means $x$ approaches $c$ from the left (smaller values); $\lim_{x \to c^+} f(x)$ from the right. The two-sided limit $\lim_{x \to c} f(x)$ exists if and only if both one-sided limits exist and are equal.$\lim_{x \to c^-} f(x)$ 表示 $x$ 从左侧（更小值）逼近 $c$；$\lim_{x \to c^+} f(x)$ 从右侧。双侧极限 $\lim_{x \to c} f(x)$ 存在当且仅当两个单侧极限都存在且相等。
• Continuity at a point.在某点的连续性。 $f$ is continuous at $x = c$ when all three conditions hold: (1) $f(c)$ is defined; (2) $\lim_{x \to c} f(x)$ exists; (3) $\lim_{x \to c} f(x) = f(c)$.$f$ 在 $x = c$ 处连续，须三件事同时成立：(1) $f(c)$ 有定义；(2) $\lim_{x \to c} f(x)$ 存在；(3) $\lim_{x \to c} f(x) = f(c)$。
• Limits at infinity无穷处的极限 describe end behaviour: $\lim_{x \to \infty} f(x) = L$ means $f(x)$ approaches $L$ for arbitrarily large $x$. Geometrically, $y = L$ is a horizontal asymptote.描述尾部行为：$\lim_{x \to \infty} f(x) = L$ 表示 $x$ 任意大时 $f(x)$ 趋近 $L$。几何上 $y = L$ 是水平渐近线。
• Rational-function trick.有理函数技巧。 For $f(x) = (a_n x^n + \ldots)/(b_m x^m + \ldots)$ as $x \to \infty$: if $n < m$ the limit is $0$; if $n = m$ it is $a_n / b_m$; if $n > m$ it is $\pm \infty$.$x \to \infty$ 时对有理函数 $f(x) = (a_n x^n + \ldots)/(b_m x^m + \ldots)$：若 $n < m$ 极限为 $0$；若 $n = m$ 为 $a_n / b_m$；若 $n > m$ 为 $\pm \infty$。

Pick a point $(x, f(x))$ and a nearby point $(x + h, f(x + h))$ on the graph. The line through them is the secant; its slope is the average rate of change:在图像上取一点 $(x, f(x))$ 与邻近点 $(x + h, f(x + h))$。两点连线为割线，其斜率即平均变化率：

As $h \to 0$, the secant rotates and limits onto the tangent line at $x$. The slope of that tangent is the derivative:当 $h \to 0$ 时，割线旋转并以 $x$ 处的切线为极限。该切线斜率即导数：

• Notation.记号。 $f'(x)$, $\dfrac{df}{dx}$, $\dfrac{dy}{dx}$, $Df(x)$ , all mean the same thing.$f'(x)$、$\dfrac{df}{dx}$、$\dfrac{dy}{dx}$、$Df(x)$ , 含义相同。
• Geometric meaning.几何意义。 $f'(c)$ is the slope of the tangent line to $y = f(x)$ at $x = c$. Positive: curve rising. Negative: falling. Zero: horizontal tangent (often a max / min).$f'(c)$ 是 $y = f(x)$ 在 $x = c$ 处切线的斜率。正：曲线上升；负：下降；零：水平切线（常对应极值）。
• "From first principles""从定义出发" means using this limit directly. MCV4U Strand A and AP Calc Unit 2 both require students to compute at least one derivative this way before reaching for shortcut rules.指直接用此极限求导。MCV4U 单元 A 与 AP Calc Unit 2 都要求至少有一次"从定义出发"求导，才能套用法则。

• Power rule.幂法则。 Bring the exponent down as a coefficient, subtract one from the exponent. Works for every real $n$ (MCV4U verifies it for natural $n$; the AP Calc / IB feeders prove the general case).把指数变为系数，指数减一。对任意实数 $n$ 都成立（MCV4U 仅对自然数 $n$ 验证；AP Calc / IB 衔接证明一般情形）。
• Constant multiple.常数倍法则。 Multiplying $f$ by a constant multiplies its derivative by the same constant. Constants pass through differentiation untouched.$f$ 乘常数后，导数也乘以同一常数。常数对求导是"透明"的。
• Sum / difference rule.求和 / 差法则。 Differentiate term by term. Powerful in combination with the power rule for any polynomial.逐项求导。与幂法则结合可对任意多项式求导。
• Derivative of a constant is zero.常数的导数为零。 $\dfrac{d}{dx}[k] = 0$. Geometrically, a horizontal line has slope zero everywhere.$\dfrac{d}{dx}[k] = 0$。几何上水平直线处处斜率为零。

An antiderivative of $f(x)$ is any function $F(x)$ with $F'(x) = f(x)$. Antiderivatives are unique only up to an additive constant: if $F'(x) = f(x)$, then $(F + C)'(x) = f(x)$ too. The notation $\int f(x) \, dx$ stands for the entire family.$f(x)$ 的反导数是任何满足 $F'(x) = f(x)$ 的函数 $F(x)$。反导数仅在差一常数的意义下唯一：若 $F'(x) = f(x)$，则 $(F + C)'(x) = f(x)$ 也成立。记号 $\int f(x) \, dx$ 表示整族。

• Reverse the power rule.幂法则倒过来。 Add one to the exponent, divide by the new exponent. Then add $+ C$.指数加一，除以新指数，再加 $+ C$。
• Why $n \ne -1$.为何 $n \ne -1$。 If $n = -1$, the formula gives $x^0 / 0$ , undefined. The antiderivative of $1/x$ is $\ln |x| + C$, an exception that AP Calc Unit 6 and IB Math HL E3 treat carefully.若 $n = -1$，公式给出 $x^0 / 0$ , 未定义。$1/x$ 的反导数是 $\ln |x| + C$，AP Calc Unit 6 与 IB Math HL E3 专门处理此例外。
• Linearity carries over.线性性同样保留。 $\int (c \cdot f) \, dx = c \int f \, dx$ and $\int (f \pm g) \, dx = \int f \, dx \pm \int g \, dx$.$\int (c \cdot f) \, dx = c \int f \, dx$；$\int (f \pm g) \, dx = \int f \, dx \pm \int g \, dx$。
• Never forget $+ C$.永远别漏 $+ C$。 An indefinite integral without the constant is wrong by a whole infinite family of functions. AP Calc / IB markers deduct for this.不带常数的不定积分错了整整一族函数。AP Calc / IB 评卷会扣分。

"Signed" means areas above the $x$-axis count positive; areas below count negative. The notation comes from "summing infinitesimal slices": $\int$ is a stretched-out $S$ for "sum," and $dx$ is the width of each slice."带符号"指 $x$ 轴上方的面积为正，下方为负。记号源自"无穷小切片求和"：$\int$ 是拉长的 $S$（"sum"），$dx$ 是每片的宽度。

• Fundamental Theorem of Calculus (intuitive form).微积分基本定理（直观形式）。 If $F$ is any antiderivative of $f$, then若 $F$ 是 $f$ 的任一反导数，则 $$\int_a^b f(x) \, dx \;=\; F(b) - F(a).$$ In words: definite integration and antidifferentiation are inverse operations. This is the theorem of all of calculus.即：定积分与反导数互为逆运算。这是整个微积分的核心定理。
• No constant on the definite integral.定积分不带常数。 The $+ C$ cancels because $F(b) + C - (F(a) + C) = F(b) - F(a)$. Indefinite integrals carry $+ C$; definite integrals do not.$+ C$ 在差中消去：$F(b) + C - (F(a) + C) = F(b) - F(a)$。不定积分带 $+ C$，定积分不带。
• Reversal sign-flip.交换上下限变号。 $\int_a^b f \, dx = -\int_b^a f \, dx$. Swapping the bounds flips the sign.$\int_a^b f \, dx = -\int_b^a f \, dx$。交换上下限改变符号。