Repeated L'Hopital & rewriting forms
Practice Flashcards
Flip to reveal answersWhen do you apply L'Hopital more than once?
Track your progress — Sign up free to save your progress and get smart review reminders based on spaced repetition.
All 8 Flashcards — Repeated L'Hopital & rewriting forms
Sign up free to track progress and get spaced-repetition review schedules.
Question
When do you apply L'Hopital more than once?
Answer
When, after differentiating, substitution STILL gives 0/0 (or ∞/∞) — keep going until you get a number.
Question
When must you STOP applying L'Hopital?
Answer
The moment substitution gives a finite value — applying it further would give a wrong answer.
Question
Find lim (1 − cos x)/x² as x→0.
Answer
0/0 → (sin x)/(2x) → 0/0 → (cos x)/2 = 1/2.
Question
L'Hopital needs which form?
Answer
A quotient that is 0/0 or ∞/∞ — never a bare product or difference.
Question
How do you handle a 0·∞ limit?
Answer
Rewrite the product as a fraction: f·g = f/(1/g), making 0/0 or ∞/∞, then apply L'Hopital.
Question
Find lim (x→0⁺) x ln x.
Answer
0·(−∞) → (ln x)/(1/x) → (1/x)/(−1/x²) = −x → 0.
Question
If a limit (top)/x² is finite as x→0, what must the top do?
Answer
It must → 0 as x→0; otherwise the 0 in the bottom forces ∞. Set top → 0 to find unknowns.
Question
Find lim sin²(kx)/x² as x→0.
Answer
It equals k² (e.g. via L'Hopital twice or sin kx ≈ kx). So if it's 16, k = ±4.
Read the notes
Full study notes for Repeated L'Hopital & rewriting forms
Topic 5.13 hub
Limits & l'Hopital (HL only)
More from Topic 5.13
All flashcards in this topic
Math AA exam skills
Paper structures & tips
Track your progress with spaced repetition
Sign up free — Aimnova tells you exactly which cards to review and when, so you remember everything before your IB exam.
Start Free