This is a beta course, so its structure, chapters, and examples may continue to change.

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.

$\infty$ （Infinity）：数学符号，读作“无穷大”，表示值可以无限增大。

Core statement

定理1

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

then

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

几何解释

推论

证明

符号说明

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

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.

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).

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

Reference Answer(1 个标签)

L'Hospital's rule

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

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

Reference Answer(1 个标签)

L'Hospital's rule

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

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

Reference Answer(1 个标签)

L'Hospital's rule

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

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