L'Hôpital's Rule

L’Hôpital’s rule converts certain indeterminate limits into derivative ratios, especially the $\tfrac{0}{0}$ and $\tfrac{\infty}{\infty}$ types.

Core statement

定理 1

Let $f,g$ be defined and differentiable on a punctured neighborhood of $a$ with $g'(x)\neq 0$. If

1. $\lim_{x\to a} f(x) = \lim_{x\to a} g(x) = 0$ or $\pm\infty$, and
2. $\lim_{x\to a} \dfrac{f'(x)}{g'(x)}$ exists (finite or $\pm\infty$),

then

$$\lim_{x\to a} \frac{f(x)}{g(x)} = \lim_{x\to a} \frac{f'(x)}{g'(x)}.$$

Indeterminate forms and quick examples

1. $\tfrac{0}{0}$: $\lim_{x\to 0} \dfrac{\sin x}{x} = \lim \dfrac{\cos x}{1} = 1$.
2. $\tfrac{\infty}{\infty}$: $\lim_{x\to \infty} \dfrac{x^2}{e^x} \to \lim \dfrac{2x}{e^x} \to \lim \dfrac{2}{e^x} = 0$.
3. $0\cdot\infty$: rewrite as quotient, e.g. $\lim_{x\to 0^+} x\ln x = \lim \dfrac{\ln x}{1/x} = 0$.
4. $\infty - \infty$: combine terms, then apply the rule.
5. $0^0$, $\infty^0$, $1^\infty$: take logs to turn into $\tfrac{0}{0}$ or $\tfrac{\infty}{\infty}$, e.g. $x^x = e^{x\ln x}$.

Tips

• Multiple rounds: check conditions before each application.
• Pair with Taylor: series often gives the answer faster (e.g., $\sin x - x$).
• Substitution: change variables to expose an indeterminate form.

Common mistakes

• Skipping hypothesis checks (continuity/differentiability or $g'(x)\ne 0$).
• Applying it when the limit is already determinate.
• Infinite looping: if repeated applications stall, switch methods (Taylor, squeeze, substitution).

Limitations

• Not for limits that are already determinate.
• Not usable if derivatives fail to exist or the derivative ratio limit fails.
• Some problems are better handled by expansions or comparison tests.

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练习题

练习 1

Compute $\lim_{x \to 0} \dfrac{\sin x}{x}$ with L’Hôpital.

参考答案

$\tfrac{0}{0}$ type; differentiate numerator/denominator: $\lim_{x\to 0} \dfrac{\cos x}{1} = 1$.

练习 2

Compute $\lim_{x \to \infty} \dfrac{x^2}{e^x}$.

参考答案

$\tfrac{\infty}{\infty}$ type. First application: $\lim \dfrac{2x}{e^x}$, second: $\lim \dfrac{2}{e^x}=0$.

练习 3

Compute $\lim_{x \to 0^+} x \ln x$.

参考答案

Rewrite as $\dfrac{\ln x}{1/x}$ ($\tfrac{\infty}{\infty}$). Derivatives: $\dfrac{1/x}{-1/x^2} = -x \to 0$.

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总结

本文出现的符号

符号	类型	读音/说明	在本文中的含义
$\infty$	数学符号	Infinity（无穷大）	表示量可以无限增大

中英对照

中文术语	英文术语	音标	说明
洛必达法则	L’Hôpital’s rule	/loʊpiːˈtɑːlz ruːl/	通过导数处理不定式极限
不定式	indeterminate form	/ɪnˈdɛtərmənət fɔːm/	形式无法直接给出极限的表达
零除零	zero over zero	/ˈzɪərəʊ ˈəʊvə ˈzɪərəʊ/	$\frac{0}{0}$ 型
无穷除无穷	infinity over infinity	/ɪnˈfɪnəti ˈəʊvə ɪnˈfɪnəti/	$\frac{\infty}{\infty}$ 型
乘积不定式	zero times infinity	/ˈzɪərəʊ taɪmz ɪnˈfɪnəti/	$0 \cdot \infty$ 型
幂指不定式	exponential indeterminate	/ɪkˌspɒnɛnʃəl ɪnˈdɛtərmənət/	$0^0,\ \infty^0,\ 1^\infty$ 型
泰勒展开	Taylor expansion	/ˈteɪlər ɪkˈspænʃən/	用多项式逼近函数的方法

PreviousMean Value Theorems for Derivatives Next导数的应用学习导数在研究函数性质中的应用，包括单调性、极值、凹凸性、拐点、最大最小值等。