How to evaluate an indeterminate limit
A decision process for 0/0 and ∞/∞ that starts with algebra before reaching for a theorem.
LaTeX article Updated July 11, 2026
Indeterminate does not mean nonexistent
The same form can hide many outcomes. A 0/0 limit may be zero, finite and nonzero, infinite, or nonexistent. The form tells you the current expression has not revealed enough information.
Write the form beside your work, then change the expression while preserving equality for nearby inputs.
A reliable order of operations
For rational expressions, factor and cancel. For radicals, multiply by a conjugate. For trigonometric expressions, look for sin u / u or 1 minus cos u patterns. For growth at infinity, divide by the dominant power.
L’Hôpital’s rule is useful only after its hypotheses are met. It should confirm structure, not replace basic algebra automatically.
Know when to stop
After simplification, substitute again. If the new expression is continuous at the target, the limit is finished. Repeated manipulation after the obstruction is gone creates mistakes rather than rigor.
Worked example
Common mistakes
- Writing 0/0 = 0 or 1.
- Canceling terms across addition instead of common factors.
- Using L’Hôpital’s rule before checking the form and hypotheses.
Keep these ideas
- The form diagnoses a problem; it does not determine the answer.
- Preserve equality for nearby nonzero inputs.
- Simplify, then substitute again.