How to evaluate an indeterminate limit
A decision process for 0/0 and ∞/∞ that starts with algebra before reaching for a theorem.
How to recognize the method, run it, and know when it is the wrong choice.
Reviewed July 11, 2026Substitute once. If the form is indeterminate, simplify the expression or expose a known limit; do not perform arithmetic with the symbols 0/0 or ∞/∞.
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.
Evaluate (x² − 9)/(x − 3) as x approaches 3.
Direct substitution identifies the obstruction.
Cancel for nearby inputs, which is exactly where a limit lives.
Now substitution is legal.
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.
Three takeaways
- The form diagnoses a problem; it does not determine the answer.
- Preserve equality for nearby nonzero inputs.
- Simplify, then substitute again.