Skip to content
Calculus I / Applications of Differentiation

Indeterminate Forms and L'Hôpital's Rule

The compound-interest limit lim⁡n→∞(1+r/n)n=er\lim_{n \to \infty}(1 + r/n)^n = e^r is the mathematical bridge between discrete and continuous compounding. Its form is 1∞1^\infty — a base approaching 11 raised to an exponent approaching ∞\infty — which is a competition that can resolve to any positive number. Such expressions are called indeterminate forms: ratios, products, or powers where naïve substitution produces a meaningless symbol (0/00/0, ∞/∞\infty/\infty, 0⋅∞0 \cdot \infty, 1∞1^\infty, 000^0, ∞0\infty^0, ∞−∞\infty - \infty) that reveals nothing about the limit's value.

L'Hôpital's Rule resolves the two most common forms, 0/00/0 and ∞/∞\infty/\infty, by replacing the limit of a ratio with the limit of the ratio of derivatives. The other forms reduce to these two by algebraic manipulation, and the compound-interest limit falls out as a direct corollary. This rule and the continuous-compounding formula A=PertA = Pe^{rt} are thereafter the foundation of present-value calculations throughout finance.

Learning objectives. By the end of this note you should be able to:
  • identify the seven indeterminate forms and explain why direct substitution fails for each;
  • state L'Hôpital's Rule precisely and apply it to 0/00/0 and ∞/∞\infty/\infty forms, including iterated applications;
  • convert 0⋅∞0 \cdot \infty and ∞−∞\infty - \infty forms into a quotient before applying the rule;
  • handle indeterminate powers (000^0, ∞0\infty^0, 1∞1^\infty) via the logarithm-or-exponential device;
  • derive lim⁡n→∞(1+r/n)n=er\lim_{n \to \infty}(1 + r/n)^n = e^r rigorously and interpret it as continuous compounding.

In this lesson

  1. Why limits of ratios can be indeterminate
  2. L'Hôpital's Rule
  3. The 0/00/0 form
  4. The ∞/∞\infty/\infty form
  5. The 0⋅∞0 \cdot \infty form
  6. The ∞−∞\infty - \infty form
  7. Indeterminate powers
  8. The compound-interest limit (a key financial derivation)
  9. Growth rate hierarchy

Before you start

The full lesson, with worked examples and exercise solutions, is for members.

Read the full lesson

New to SelfMadeQuant.com? See membership options or explore the curriculum.