Indeterminate Form

Core Sentence / 核心句

An indeterminate form indicates that direct substitution does not reveal the limit because competing quantities approach their values at unknown relative rates.

中文理解：未定式不是答案，而是直接代入的信息不足，需要继续比较各部分趋近的速度。

Common Forms / 常见形式

$$\frac{0}{0},\frac{\infty }{\infty },0\cdot \infty ,\infty -\infty ,1^{\infty },0^0,{\infty }^0$$

Watch Out / 易错点

• $0/0$ 不等于 0、1 或 infinity。
• 不是每个 undefined expression 都是可用 L’Hôpital 的 indeterminate form。
• 先确认 limit form，再选择 algebra、known limits 或 L’Hôpital’s Rule。

Search and Source Mentions / 搜索与出现位置

Mentioned In / 出现位置

• 微积分BC第1次正课 · Limit Evaluation Decision Map
  • $0/0$ and $\infty/\infty$ are signals to continue analysis, not final answers
• Chapter 7 · 0/0 Form
  • numerator and denominator both approach zero, but their relative rate is still unknown

Connected Notes / 连接

• 微积分BC第1次正课-极限与连续
• 07 - Limits, L’Hopital’s Rule, and Epsilon-Delta Definitions
• Limits
• L’Hôpital’s Rule
• Derivatives