Sections 1 to 6: limit laws, the squeeze theorem, limits at infinity, the epsilon-delta definition, continuity and the IVT第 1 至 6 节：极限法则、夹逼定理、无穷极限、epsilon-delta 定义、连续性与介值定理CALC I
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PART I · CORE TECHNIQUES核心技法Computational fluency计算熟练度 · 28 marks分
Evaluating Limits极限的计算
Show all working. A limit that fails to exist must be justified, not merely asserted. State the technique (factor, conjugate, one-sided, leading-term) you use at each step.请展示全部解题过程。若极限不存在，必须给出论证，不可仅作断言。在每一步注明所用技法（因式分解、有理化共轭、单侧极限、最高次项比较）。
Q1MEDIUMCOREindeterminate forms: factor and conjugate不定式：因式分解与共轭有理化[6 marks]
Evaluate each limit, or show that it does not exist.求各极限，若不存在请加以证明。
(b)Find the values of $a$ and $b$ that make $f$ continuous at $x=2$.求使 $f$ 在 $x=2$ 处连续的 $a$ 和 $b$ 的值。[4]
(c)The expression $\dfrac{x^{2}-4}{x-2}$ on its own is undefined at $x=2$. Classify the discontinuity it has there, and explain in one sentence why it can be "repaired".表达式 $\dfrac{x^{2}-4}{x-2}$ 在 $x=2$ 处无定义。分类该点的间断类型，并用一句话说明为何可以"修复"。[2]
PART II · DEFINITIONS AND PROOF定义与证明Rigorous arguments严格论证 · 26 marks分
Epsilon-Delta and the IVTEpsilon-Delta 与介值定理
These items are graded on the logic of the argument, not just the final line. In an epsilon-delta proof, state your choice of $\delta$ explicitly and then verify it. In an IVT argument, check every hypothesis before invoking the conclusion.本部分按论证的逻辑评分，而非仅看最终结论。在 epsilon-delta 证明中，须明确写出 $\delta$ 的选取，并加以验证。在介值定理论证中，须逐一验证所有前提条件，方可引用结论。
Q5HARDPROOFepsilon-delta definition of a limit极限的 epsilon-delta 定义[8 marks]
Use the precise ($\varepsilon$-$\delta$) definition of a limit throughout. A correct proof must produce a $\delta$ in terms of $\varepsilon$ and then verify the implication.全程使用极限的精确定义（$\varepsilon$-$\delta$）。完整证明须给出以 $\varepsilon$ 表示的 $\delta$，并验证蕴含关系。
(a)Prove that $\displaystyle\lim_{x\to 4}(2x+3)=11$.证明 $\displaystyle\lim_{x\to 4}(2x+3)=11$。[3]
(b)Prove that $\displaystyle\lim_{x\to 3} x^{2}=9$. (You will need to bound the factor $|x+3|$ by first restricting $\delta\le 1$.)证明 $\displaystyle\lim_{x\to 3} x^{2}=9$。（需先限制 $\delta\le 1$，以控制因子 $|x+3|$ 的大小。）[5]
Q6HARDPROOFIntermediate Value Theorem and fixed points介值定理与不动点[10 marks]
In each part, state which function you apply the IVT to, verify continuity and the sign change, and only then state the conclusion.每小题须说明对哪个函数应用介值定理，验证连续性和符号变化，再陈述结论。
(a)Show that the equation $x^{3}-4x+1=0$ has at least one solution in the interval $(0,1)$.证明方程 $x^{3}-4x+1=0$ 在区间 $(0,1)$ 内至少有一个实根。[3]
(b)Show that the equation $\cos x = x$ has a solution in $\left(0,\tfrac{\pi}{2}\right)$.证明方程 $\cos x = x$ 在 $\left(0,\tfrac{\pi}{2}\right)$ 内有解。[3]
(c)Let $f:[0,1]\to[0,1]$ be continuous. Prove that $f$ has a fixed point: that is, there exists $c\in[0,1]$ with $f(c)=c$.设 $f:[0,1]\to[0,1]$ 连续。证明 $f$ 存在不动点，即存在 $c\in[0,1]$ 使得 $f(c)=c$。[4]
Q7HARDPROOFderiving the fundamental trig limit推导基本三角极限[8 marks]
For $0<|x|<\tfrac{\pi}{2}$ the geometric area argument gives the inequality $\cos x \le \dfrac{\sin x}{x}\le 1$.当 $0<|x|<\tfrac{\pi}{2}$ 时，几何面积论证给出不等式 $\cos x \le \dfrac{\sin x}{x}\le 1$。
(a)Use this inequality and the squeeze theorem to prove that $\displaystyle\lim_{x\to 0}\frac{\sin x}{x}=1$.利用此不等式及夹逼定理，证明 $\displaystyle\lim_{x\to 0}\frac{\sin x}{x}=1$。[4]
(b)Hence, working only from the result in (a) and algebra, evaluate $\displaystyle\lim_{x\to 0}\frac{1-\cos x}{x}$.由此，仅利用 (a) 的结论和代数运算，计算 $\displaystyle\lim_{x\to 0}\frac{1-\cos x}{x}$。[4]
PART III · APPLICATIONS AND SYNTHESIS应用与综合Extended problems综合题 · 28 marks分
Asymptotes, the Derivative as a Limit, and Parameters渐近线、导数作为极限，以及参数
Set up each problem cleanly. Carry exact values through intermediate steps and simplify only at the end. Diagrams of asymptotic behaviour earn method credit.每题须清晰建立解题框架。中间步骤保留精确值，最后再化简。画出渐近线行为示意图可获方法分。
Q8HARDAPPLIEDfull asymptotic analysis of a rational function有理函数的完整渐近线分析[10 marks]
(a)Factor numerator and denominator and state the domain of $f$.对分子和分母分别因式分解，并写出 $f$ 的定义域。[2]
(b)Show that $f$ has a removable discontinuity at one of the excluded points, and give the coordinates of the resulting hole.证明 $f$ 在某个被排除点处存在可去间断点，并给出"空洞"的坐标。[3]
(c)Identify the vertical asymptote and describe the behaviour of $f$ on each side of it using one-sided limits.确定铅直渐近线，并用单侧极限描述 $f$ 在其两侧的行为。[3]
(d)Find the horizontal asymptote by evaluating $\displaystyle\lim_{x\to\pm\infty} f(x)$.通过计算 $\displaystyle\lim_{x\to\pm\infty} f(x)$ 求水平渐近线。[2]
Q9HARDAPPLIEDthe derivative as a limit of difference quotients导数作为差商的极限[10 marks]
Let $f(x)=\sqrt{x}$, defined for $x\ge 0$.设 $f(x)=\sqrt{x}$，定义域为 $x\ge 0$。
(a)Using the limit definition $f'(a)=\displaystyle\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$, show that $f'(a)=\dfrac{1}{2\sqrt{a}}$ for $a>0$.利用极限定义 $f'(a)=\displaystyle\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$，证明当 $a>0$ 时 $f'(a)=\dfrac{1}{2\sqrt{a}}$。[4]
(b)Hence find the equation of the tangent line to $y=\sqrt{x}$ at the point where $a=9$.由此求 $y=\sqrt{x}$ 在 $a=9$ 处的切线方程。[3]
(c)Confirm the value $f'(9)$ using the alternative form $f'(a)=\displaystyle\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$.利用另一种形式 $f'(a)=\displaystyle\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$ 确认 $f'(9)$ 的值。[3]
Q10HARDAPPLIEDparameter selection for a finite limit使极限有限的参数选取[8 marks]
Consider $\displaystyle g(x)=\frac{\sqrt{x+c}-3}{x-4}$ for $x\ne 4$, where $c$ is a constant.设 $\displaystyle g(x)=\frac{\sqrt{x+c}-3}{x-4}$（$x\ne 4$），其中 $c$ 为常数。
(a)Find the value of $c$ for which $\displaystyle\lim_{x\to 4} g(x)$ exists as a finite number, and explain why no other value of $c$ works.求使 $\displaystyle\lim_{x\to 4} g(x)$ 存在且为有限值的 $c$，并说明其他 $c$ 值均不满足条件的原因。[3]
(b)For that value of $c$, evaluate $\displaystyle\lim_{x\to 4} g(x)$.对该 $c$ 值，计算 $\displaystyle\lim_{x\to 4} g(x)$。[3]
(c)State the value that $g(4)$ must be assigned so that the completed function is continuous at $x=4$.写出 $g(4)$ 应赋予的值，使完整函数在 $x=4$ 处连续。[2]