CalcRef

Limits & Continuity · Special Limits

L'Hopital's Rule

If 00 or ±∞±∞:lim⁡x→af(x)g(x)=lim⁡x→af′(x)g′(x)\text{If } \frac{0}{0} \text{ or } \frac{\pm\infty}{\pm\infty}: \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}

L'Hopital's Rule: When a limit gives an indeterminate form 0/0 or ∞/∞, the limit equals the ratio of the derivatives (if that limit exists).

Conditions. The limit must produce 0/0 or ±∞/±∞. g'(x) ≠ 0 near a. The derivative limit must exist (or be ±∞).
  • 0 variables
  • 3 worked examples
  • 18 in Limits & Continuity

Worked examples

Find lim(x→0) sin(x)/x.
  1. Direct substitution gives 0/0 -indeterminate
  2. Apply L'Hopital: lim(x→0) cos(x)/1
  3. = cos(0)/1 = 1

Answer: 1

2 more worked examplesPremium

Find lim(x→∞) ln(x)/x.

Step-by-step solution locked.

Find lim(x→0) (eˣ - 1)/x.

Step-by-step solution locked.

Unlock the full working. CalcRef Premium opens the step-by-step solutions above, and 141 worked solutions across the whole library - the intermediate lines, not just the answer. The formula statement, its conditions, its variables and the first worked example stay free on every page.
See PremiumBrowse the free reference

See all 18 Limits & Continuity formulas on one printable sheet - the same statements and conditions as these pages, typeset and laid out for paper.

Practice

Quiz yourself on Limits & Continuity

L'Hopital's Rule is one of 18 Limits & Continuity formulas in CalcRef, out of 136 in the library. Today's free round is 10 mixed multiple-choice questions drawn from the same dataset as this page, refreshed every day and replayable as often as you like - no account needed.

Take today's free quizAll 18 Limits & Continuity formulas