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Unit A6 · Calculus I

Linear Approximation and L'Hopital's Rule线性近似与洛必达法则

University-Style Practice Problems大学风格练习题

MEDIUM HARD CORE PROOF APPLIED

Sections 1 to 7: linearization, differentials and error estimation, Newton's method, indeterminate forms, L'Hopital's rule, exponential indeterminate forms, and growth rates第 1 至 7 节：线性化、微分与误差估计、牛顿迭代法、不定式、洛必达法则、指数不定式及增长率CALC I

Name:姓名：Date:日期：
PART I  ·  CORE TECHNIQUES第一部分  ·  核心技法Computational fluency · 28 marks计算熟练度 · 28 分

Linearization, Differentials, and Basic L'Hopital线性化、微分与基础洛必达法则

Show all working. State the centre $a$ and the formula $L(x)=f(a)+f'(a)(x-a)$ before computing. For L'Hopital questions, verify the form is $0/0$ or $\infty/\infty$ before differentiating numerator and denominator separately.写出完整计算过程。在计算前先写明中心点 $a$ 及公式 $L(x)=f(a)+f'(a)(x-a)$。对于洛必达法则题目，在分别对分子和分母求导前，须验证极限形式为 $0/0$ 或 $\infty/\infty$。

Q1MEDIUM CORE linearization and numerical approximation线性化与数值近似 [6 marks]

Use the linearization $L(x)=f(a)+f'(a)(x-a)$ to approximate each quantity. State your choice of $f$ and $a$ in each part.利用线性化公式 $L(x)=f(a)+f'(a)(x-a)$ 近似下列各量。在每小题中写明所选的 $f$ 和 $a$。

(a) Approximate $\sqrt{4.1}$.近似计算 $\sqrt{4.1}$。 [2]
(b) Approximate $\sqrt[3]{8.1}$.近似计算 $\sqrt[3]{8.1}$。 [2]
(c) Approximate $\sin(0.1)$ (in radians).近似计算 $\sin(0.1)$（弧度制）。 [2]
Q2MEDIUM CORE differentials and propagated error微分与误差传播 [8 marks]

A sphere has a measured radius of $r=5$ cm, with a possible measurement error of $|\Delta r|\le 0.04$ cm. Use differentials throughout.一个球体的测量半径为 $r=5$ cm，测量误差满足 $|\Delta r|\le 0.04$ cm。全程使用微分方法。

(a) Write the differential $dV$ of the volume $V=\tfrac{4}{3}\pi r^{3}$ in terms of $r$ and $dr$.将体积 $V=\tfrac{4}{3}\pi r^{3}$ 的微分 $dV$ 用 $r$ 和 $dr$ 表示。 [2]
(b) Estimate the maximum absolute error $|\Delta V|$ in the computed volume when $r=5$.当 $r=5$ 时，估计体积计算中的最大绝对误差 $|\Delta V|$。 [3]
(c) Find the maximum relative (percentage) error $\left|\dfrac{\Delta V}{V}\right|$ and express it as a percentage.求最大相对（百分比）误差 $\left|\dfrac{\Delta V}{V}\right|$ 并以百分比表示。 [3]
Q3HARD CORE L'Hopital's rule: 0/0 and infty/infty洛必达法则：$0/0$ 与 $\infty/\infty$ 型 [8 marks]

Evaluate each limit. Confirm the indeterminate form before applying L'Hopital's rule, and apply the rule as many times as necessary.计算下列各极限。在使用洛必达法则之前须确认不定式类型，并根据需要多次使用该法则。

(a) $\displaystyle\lim_{x\to 0}\frac{e^{x}-1-x}{x^{2}}$ [3]
(b) $\displaystyle\lim_{x\to\infty}\frac{x^{3}}{e^{x}}$ [3]
(c) $\displaystyle\lim_{x\to 0}\frac{\tan x - x}{x^{3}}$ [2]
Q4MEDIUM CORE identifying indeterminate forms and L'Hopital setup识别不定式类型与洛必达法则的应用准备 [6 marks]

For each expression, identify the limiting form as $x$ approaches the given value. If it is indeterminate, convert it to $0/0$ or $\infty/\infty$ and apply L'Hopital's rule. If it is not indeterminate, state the limit directly without differentiation.对于下列各式，识别 $x$ 趋向给定值时的极限类型。若为不定式，将其转化为 $0/0$ 或 $\infty/\infty$ 型后使用洛必达法则；若不是不定式，直接给出极限值，无需求导。

(a) $\displaystyle\lim_{x\to 0^{+}} x\ln x$ [2]
(b) $\displaystyle\lim_{x\to 0}\frac{\sin x}{x+1}$   (state whether L'Hopital applies and evaluate)（说明洛必达法则是否适用并求值） [2]
(c) $\displaystyle\lim_{x\to\infty}\left(\sqrt{x^{2}+x}-x\right)$   (convert to a quotient first)（先转化为商的形式） [2]
PART II  ·  DEFINITIONS AND PROOF第二部分  ·  定义与证明Rigorous arguments · 26 marks严格论证 · 26 分

L'Hopital's Rule: Hypotheses, Cautions, and Exponential Forms洛必达法则：假设条件、注意事项与指数型不定式

State all hypotheses explicitly before invoking any theorem. For L'Hopital's rule, write: "the form is $[\,\cdot\,]$, $f$ and $g$ are differentiable near $a$, $g'(x)\ne 0$ near $a$." Justify every step in proof items.在使用任何定理之前，须明确写出所有假设条件。对于洛必达法则，写明："极限形式为 $[\,\cdot\,]$，$f$ 和 $g$ 在 $a$ 附近可微，$g'(x)\ne 0$ 在 $a$ 附近成立。"证明题的每一步都须给出依据。

Q5HARD PROOF statement of L'Hopital conditions and a caution example洛必达法则条件的完整表述与警示例题 [8 marks]

This question tests precise statement and critical application of L'Hopital's rule.本题考查洛必达法则的精确表述与批判性应用。

(a) State L'Hopital's rule in full for the $0/0$ form: give every hypothesis (concerning $f$, $g$, and the limit of $f'/g'$) and the conclusion.完整表述 $0/0$ 型洛必达法则：给出所有假设条件（关于 $f$、$g$ 以及 $f'/g'$ 的极限）和结论。 [3]
(b) Consider $\displaystyle\lim_{x\to 0}\frac{x^{2}\sin(1/x)}{\sin x}$. A student claims the form is $0/0$ and applies L'Hopital's rule, differentiating numerator and denominator. Explain, with reference to the hypotheses of L'Hopital's rule, why this application is invalid, and evaluate the limit by an alternative method.考虑 $\displaystyle\lim_{x\to 0}\frac{x^{2}\sin(1/x)}{\sin x}$。某学生认为该极限为 $0/0$ 型并使用洛必达法则对分子和分母分别求导。结合洛必达法则的假设条件，解释该操作为何无效，并用替代方法求出极限值。 [5]
Q6HARD PROOF exponential indeterminate forms via logarithms利用对数处理指数型不定式 [10 marks]

Use the logarithm technique: set $y$ equal to the expression, take $\ln$, evaluate $\lim\ln y$ using L'Hopital, and then recover $\lim y=e^{\lim\ln y}$. Show this full chain in each part.使用对数技巧：令 $y$ 等于该表达式，取对数，用洛必达法则求 $\lim\ln y$，再还原 $\lim y=e^{\lim\ln y}$。在每小题中完整展示这一推导链。

(a) $\displaystyle\lim_{x\to 0^{+}} x^{x}$  (form $0^{0}$)（$0^{0}$ 型） [3]
(b) $\displaystyle\lim_{x\to\infty} x^{1/x}$  (form $\infty^{0}$)（$\infty^{0}$ 型） [3]
(c) $\displaystyle\lim_{x\to\infty}\left(1+\frac{3}{x}\right)^{x}$  (form $1^{\infty}$)（$1^{\infty}$ 型） [4]
Q7HARD PROOF Newton's method iteration and convergence condition牛顿迭代法与收敛条件 [8 marks]

Newton's method approximates a root of $f(x)=0$ via the iteration $x_{n+1}=x_{n}-\dfrac{f(x_{n})}{f'(x_{n})}$.牛顿迭代法通过迭代 $x_{n+1}=x_{n}-\dfrac{f(x_{n})}{f'(x_{n})}$ 近似求解方程 $f(x)=0$ 的根。

(a) Starting from $x_{0}=1$, perform two iterations of Newton's method applied to $f(x)=x^{2}-3$ to approximate $\sqrt{3}$. Give each iterate to four decimal places.从 $x_{0}=1$ 出发，对 $f(x)=x^{2}-3$ 进行两次牛顿迭代以近似 $\sqrt{3}$。每次迭代结果保留四位小数。 [4]
(b) Explain geometrically what Newton's method computes at each step: what is $x_{n+1}$ in terms of the graph of $f$?从几何角度解释牛顿迭代法每步的含义：从 $f$ 的图形来看，$x_{n+1}$ 是什么？ [2]
(c) Identify one condition on $f$ near the starting point $x_{0}$ that can cause Newton's method to fail or diverge, and give a brief example.指出初始点 $x_{0}$ 附近 $f$ 的一个条件，使得牛顿迭代法失效或发散，并给出一个简短例子。 [2]
PART III  ·  APPLICATIONS AND SYNTHESIS第三部分  ·  应用与综合Extended problems · 28 marks综合拓展题 · 28 分

Repeated L'Hopital, Growth Rates, and Synthesis多次洛必达、增长率与综合运用

Set up each problem cleanly. Carry exact values through intermediate steps and simplify only at the end. When L'Hopital is applied more than once, re-verify the indeterminate form at each step.清晰列出每题的求解框架。中间步骤保持精确值，仅在最后化简。当洛必达法则被多次使用时，每步均须重新验证不定式形式。

Q8HARD APPLIED repeated L'Hopital and a looping caution多次洛必达与循环陷阱警示 [8 marks]

Evaluate each limit by applying L'Hopital's rule the appropriate number of times. In (c), show why the direct iteration loops and use an alternative approach.对每个极限使用适当次数的洛必达法则求值。在 (c) 中，说明为何直接迭代会陷入循环，并使用替代方法。

(a) $\displaystyle\lim_{x\to 0}\frac{e^{x}-1-x-\tfrac{1}{2}x^{2}}{x^{3}}$ [3]
(b) $\displaystyle\lim_{x\to 0}\frac{1-\cos x}{x^{2}}$ via L'Hopital (apply twice and check each step).用洛必达法则求解（使用两次，每步均验证）。 [3]
(c) Explain why applying L'Hopital's rule directly to $\displaystyle\lim_{x\to\infty}\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}$ loops without resolving the limit, and evaluate the limit correctly.解释为何对 $\displaystyle\lim_{x\to\infty}\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}$ 直接使用洛必达法则会陷入循环而无法求出极限，并正确求出该极限。 [2]
Q9HARD APPLIED growth rates, 0 times infty, and infty minus infty增长率、$0\cdot\infty$ 型与 $\infty-\infty$ 型 [10 marks]

This question explores the hierarchy of growth rates and two further indeterminate forms.本题探究增长率的层级关系及另外两种不定式类型。

(a) Using L'Hopital's rule, prove that $e^{x}$ grows faster than any fixed power: show that $\displaystyle\lim_{x\to\infty}\frac{x^{n}}{e^{x}}=0$ for every positive integer $n$, by applying L'Hopital $n$ times and identifying the pattern.利用洛必达法则证明 $e^{x}$ 比任意固定幂次增长更快：通过 $n$ 次使用洛必达法则并归纳规律，证明对每个正整数 $n$ 均有 $\displaystyle\lim_{x\to\infty}\frac{x^{n}}{e^{x}}=0$。 [4]
(b) Evaluate the $\infty-\infty$ form: $\displaystyle\lim_{x\to 0^{+}}\left(\frac{1}{\sin x}-\frac{1}{x}\right)$. Combine over a common denominator, then apply L'Hopital if needed.求 $\infty-\infty$ 型极限：$\displaystyle\lim_{x\to 0^{+}}\left(\frac{1}{\sin x}-\frac{1}{x}\right)$。先通分，再视需要使用洛必达法则。 [3]
(c) Evaluate $\displaystyle\lim_{x\to\infty}\left(\ln(x+1)-\ln x\right)$ by first writing the difference as a single logarithm, without using L'Hopital's rule. State the final answer and confirm it is consistent with the growth-rate hierarchy.求 $\displaystyle\lim_{x\to\infty}\left(\ln(x+1)-\ln x\right)$，先将差化为单个对数，不使用洛必达法则。写出最终答案并确认其与增长率层级的一致性。 [3]
Q10HARD APPLIED linearization error bound and the natural exponential limit线性化误差界与自然指数极限 [10 marks]

This question connects linearization to the definition of $e$ and to error control.本题将线性化与 $e$ 的定义及误差控制联系起来。

(a) Let $f(x)=(1+x)^{1/x}$ for $x>0$. Evaluate $\displaystyle\lim_{x\to 0^{+}}f(x)$ using the logarithm technique, identifying the indeterminate form and applying L'Hopital's rule.设 $f(x)=(1+x)^{1/x}$，$x>0$。使用对数技巧求 $\displaystyle\lim_{x\to 0^{+}}f(x)$，需识别不定式类型并使用洛必达法则。 [4]
(b) Using the linearization of $g(t)=\ln(1+t)$ at $t=0$, show that $\ln(1+t)\approx t$ for small $t$. Hence give an intuitive explanation of why the result in (a) equals $e$.利用 $g(t)=\ln(1+t)$ 在 $t=0$ 处的线性化，证明当 $t$ 较小时 $\ln(1+t)\approx t$。由此直观解释 (a) 的结果等于 $e$ 的原因。 [3]
(c) The linearization $L(x)=1+x$ approximates $e^{x}$ near $x=0$. Compute the absolute error $|e^{0.2}-L(0.2)|$ to four decimal places ($e^{0.2}\approx 1.2214$), and compare it with the quadratic approximation $Q(x)=1+x+\tfrac{1}{2}x^{2}$. Which is more accurate and by what factor?线性近似 $L(x)=1+x$ 在 $x=0$ 附近近似 $e^{x}$。计算绝对误差 $|e^{0.2}-L(0.2)|$（精确到四位小数，$e^{0.2}\approx 1.2214$），并与二次近似 $Q(x)=1+x+\tfrac{1}{2}x^{2}$ 比较。哪个更精确？精确多少倍？ [3]