L'Hôpital's Rule

Definition

A rule that uses derivatives to evaluate limits of indeterminate forms $0/0$ or $\infty/\infty$.

How to read: “zero over zero or infinity over infinity” Meaning: Indeterminate forms where the limit cannot be determined by direct substitution, requiring algebraic manipulation or L’Hôpital’s Rule.

Why It Matters

Provides a powerful algebraic shortcut for solving complex limits that cannot be simplified by factoring or rationalization.

Core Concepts

• If $\lim_{x \to c} f(x)/g(x)$ results in $0/0$ or $\infty/\infty$, then: $\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$ How to read: “the limit as x approaches c of f of x over g of x equals the limit as x approaches c of f prime of x over g prime of x” Meaning: The limit of a ratio of functions is equal to the limit of the ratio of their derivatives, provided the indeterminate conditions are met.
• Must verify the indeterminate form before applying.
• Can be applied multiple times sequentially if the form persists.

Connected Concepts

Derivatives of Exponential Functions, Limit Laws