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What is an indeterminate form?
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All Flashcards in Topic 5.13
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5.13.18 cards
What is an indeterminate form?
A limit shape like 0/0 or ∞/∞ where plain substitution fails — the limit may still exist and needs more work.
State L'Hopital's rule.
If lim f/g is 0/0 or ∞/∞, then lim f/g = lim f′/g′ (differentiate top and bottom separately).
Is L'Hopital the quotient rule?
No! Differentiate the top by itself and the bottom by itself, then divide. Never use the quotient rule.
What must you check before using L'Hopital?
That substitution gives an indeterminate form 0/0 or ∞/∞.
d/dx (tan x) = ?
sec²x.
d/dx (arctan x) = ?
1/(1 + x²).
Find lim (sin x)/x as x→0.
0/0, so → lim (cos x)/1 = cos 0 = 1.
Find lim (arctan 2x)/(tan 3x) as x→0.
0/0 → lim [2/(1+4x²)] / [3 sec²3x] = 2/3.
5.13.28 cards
When do you apply L'Hopital more than once?
When, after differentiating, substitution STILL gives 0/0 (or ∞/∞) — keep going until you get a number.
When must you STOP applying L'Hopital?
The moment substitution gives a finite value — applying it further would give a wrong answer.
Find lim (1 − cos x)/x² as x→0.
0/0 → (sin x)/(2x) → 0/0 → (cos x)/2 = 1/2.
L'Hopital needs which form?
A quotient that is 0/0 or ∞/∞ — never a bare product or difference.
How do you handle a 0·∞ limit?
Rewrite the product as a fraction: f·g = f/(1/g), making 0/0 or ∞/∞, then apply L'Hopital.
Find lim (x→0⁺) x ln x.
0·(−∞) → (ln x)/(1/x) → (1/x)/(−1/x²) = −x → 0.
If a limit (top)/x² is finite as x→0, what must the top do?
It must → 0 as x→0; otherwise the 0 in the bottom forces ∞. Set top → 0 to find unknowns.
Find lim sin²(kx)/x² as x→0.
It equals k² (e.g. via L'Hopital twice or sin kx ≈ kx). So if it's 16, k = ±4.
Topic 5.13 study notes
Full notes & explanations for Limits & l'Hopital (HL only)
Math AA exam skills
Paper structures, command terms & tips
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