Still 0/0? Differentiate again: After one round of L'Hopital, substitute again.
If you get a number, stop — that's the limit. If you're back at 0/0 (or ∞/∞), the rule still applies: differentiate top and bottom a second time, and repeat as needed.
Each pass peels off one layer of the vanishing.
Stop the instant substitution gives a finite value — applying the rule when it's no longer 0/0 gives a wrong answer.
IB-style question — apply twice
Find the limit of (1 − cos x)/x² as x approaches 0.
Step by step
- Substitution: (1 − cos 0)/0 = 0/0, so use L'Hopital.
- First pass — top derivative sin x, bottom derivative 2x.
- Substitute again: sin 0 / 0 = 0/0 — STILL indeterminate, so apply again.
- Second pass — top derivative cos x, bottom derivative 2.
- Now substitution gives a number — stop.
Final answer
The limit is 1/2.
IB-style question — x sin x over (1 − cos x)
Find the limit of (x sin x)/(1 − cos x) as x approaches 0.
Step by step
- Substitution: (0·0)/(1 − 1) = 0/0, so L'Hopital applies.
- Top derivative (product rule): sin x + x cos x. Bottom derivative: sin x.
- Substitute again: (0 + 0)/0 = 0/0 — apply once more.
- Differentiate again. Top: cos x + (cos x − x sin x) = 2cos x − x sin x. Bottom: cos x.
- Substitute x = 0.
Final answer
The limit is 2.
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L'Hopital only eats fractions: The rule needs a quotient that is 0/0 or ∞/∞. A product like 0·∞ isn't a fraction yet.
The fix: turn the product into a fraction. Write one factor over the reciprocal of the other:
f · g = f / (1/g).
Choose whichever way lands you in 0/0 (or ∞/∞). Then L'Hopital takes over.
IB-style question — a 0·∞ form
Find the limit of x ln x as x approaches 0 from the positive side.
Step by step
- As x → 0⁺: x → 0 and ln x → −∞, so this is the form 0·(−∞) — not yet a fraction.
- Rewrite as a fraction by putting ln x over 1/x.
- Now it's −∞/∞, so L'Hopital applies. Top derivative 1/x; bottom derivative −1/x².
- Simplify the complex fraction: (1/x)·(−x²) = −x.
Final answer
The limit is 0 — so x ln x → 0 as x → 0⁺.
Why simplify before re-substituting: After L'Hopital you often get a messy stacked fraction. Tidy it first (cancel, combine) before substituting — it's easier and avoids dividing-by-zero panics that the algebra would have cancelled anyway.