Topic 4.7 Using L'Hospital's Rule for Determining Limits of Indeterminate Forms: evaluate limits of indeterminate form using L'Hospital's rule.

What this topic is asking

The College Board (Topic 4.7) introduces L'Hospital's rule, a method for evaluating limits that produce an indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$. When direct substitution gives one of these forms, you may differentiate the numerator and denominator separately and take the limit of the new quotient. It connects the differentiation of Units 2 and 3 back to the limits of Unit 1.

The rule

Check the form first

A reliable procedure

Substitute to detect the form, differentiate numerator and denominator separately, then re-evaluate, repeating as needed.

Differentiate separately, not the quotient

The most damaging misunderstanding is treating L'Hospital's rule as the quotient rule. It is not: you differentiate the numerator and the denominator independently and form a brand-new fraction $\frac{f'}{g'}$, never $\left(\frac{f}{g}\right)'$. For $\frac{1 - \cos x}{x^2}$ you compute $\sin x$ over $2x$, not the quotient-rule derivative of the whole fraction. Keeping this straight is the single most important habit for the topic. It also explains why the rule can be applied repeatedly: each application produces a fresh simple quotient, and if that quotient is still $\frac{0}{0}$ or $\frac{\infty}{\infty}$, you differentiate top and bottom again, and so on, until the form resolves.

When L'Hospital is not the best tool, and other indeterminate forms

L'Hospital is powerful but not always the cleanest route. Some limits are faster by algebra - factoring and cancelling for $\frac{0}{0}$ rational limits, or dividing by the highest power for $\frac{\infty}{\infty}$ limits at infinity - and the AP exam rewards choosing efficiently (the "selecting procedures for limits" skill from Unit 1). For $\lim_{x\to\infty}\frac{3x^2 + 1}{2x^2 - x}$, L'Hospital works but dividing through by $x^2$ to get $\frac{3 + 1/x^2}{2 - 1/x} \to \frac{3}{2}$ is just as quick. Other indeterminate forms such as $0 \cdot \infty$, $\infty - \infty$, $1^\infty$, $0^0$ and $\infty^0$ are not directly L'Hospital-ready: you first rewrite them as a quotient ($0\cdot\infty$ becomes $\frac{0}{1/\infty}$, and the exponential forms are handled by taking $\ln$) so they become $\frac{0}{0}$ or $\frac{\infty}{\infty}$. AP Calculus AB focuses on the two basic forms, but recognizing that the others must be converted first prevents misapplication. Always confirm the form, choose between algebra and L'Hospital, and re-check the form after each differentiation.