洛必达法则求解不定式 — AP 微积分 BC

1. 针对0/0和∞/∞型的核心洛必达法则 ★★☆☆☆ ⏱ 4 min

洛必达法则是一种基于微分的技巧，用于计算不定式的极限，这类极限直接代入后无法得到确定的极限值。该法则仅能直接应用于$\frac{0}{0}$和$\frac{\infty}{\infty}$两种不定式；所有其他不定式必须先改写为这两种结构之一，才能使用洛必达法则。

The rule can be applied repeatedly: if after one application you still get an indeterminate $\frac{0}{0}$ or $\frac{\infty}{\infty}$, you can apply it again as long as all conditions hold.

Exam tip: Always confirm the indeterminate form explicitly on FRQ answers; AP graders require this step to award full points for using L'Hopital's rule.

2. Indeterminate Products: $0 \cdot \infty$ ★★★☆☆ ⏱ 3 min

An indeterminate product occurs when one term approaches 0 and the other approaches $\pm\infty$, written $0 \cdot \infty$. The product's limit is indeterminate because 0 pulls the product toward 0 while $\infty$ pulls it toward infinity, so the result can be any finite value, 0, or infinity. To apply L'Hopital's rule, rewrite the product as a fraction by moving one term to the denominator, resulting in either $\frac{0}{0}$ or $\frac{\infty}{\infty}$.

Both conversions are mathematically valid, but one is almost always simpler to differentiate, so choose the conversion that minimizes extra work like the quotient rule.

Exam tip: When converting a product, always leave the simpler term to differentiate in the numerator to avoid introducing extra chain rule or quotient rule errors.

3. Indeterminate Differences: $\infty - \infty$ ★★★☆☆ ⏱ 4 min

Indeterminate differences occur when we have the difference of two terms, both approaching $+\infty$ or both approaching $-\infty$, written $\infty - \infty$. This is indeterminate because the two infinite terms compete, and the result can be 0, any finite number, or $\pm\infty$. To solve this type of limit, convert the difference into a single fraction, almost always by combining terms over a common denominator, factoring, or multiplying by a conjugate to eliminate radicals. The resulting fraction will almost always be a $\frac{0}{0}$ or $\frac{\infty}{\infty}$ indeterminate form suitable for L'Hopital's rule.

Exam tip: Never differentiate each term of the difference separately; you must always combine into a single fraction first before applying L'Hopital's rule.

4. Indeterminate Powers: $0^0, 1^\infty, \infty^0$ ★★★★☆ ⏱ 3 min

These three indeterminate forms are exponential expressions of the form $\lim_{x \to a} f(x)^{g(x)}$, where the limits of the base and exponent create one of the three indeterminate combinations. To solve these, use the natural logarithm to convert the exponential into a product, which you can then convert to $\frac{0}{0}$ or $\frac{\infty}{\infty}$ to apply L'Hopital's rule. The standard process is: (1) Let $L = \lim f(x)^{g(x)}$, (2) Take natural log of both sides: $\ln L = \lim g(x) \ln f(x)$, (3) Solve the resulting indeterminate product limit, (4) Exponentiate to get $L = e^{\ln L}$.

Exam tip: Don't forget to undo the natural logarithm at the end; forgetting the final exponentiation is one of the most common mistakes AP graders see on this question type.

5. AP-Style Concept Check ★★★☆☆ ⏱ 2 min