L'Hopital's rule: the 0/0 form
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Question
What is an indeterminate form?
Answer
A limit shape like 0/0 or ∞/∞ where plain substitution fails — the limit may still exist and needs more work.
Question
State L'Hopital's rule.
Answer
If lim f/g is 0/0 or ∞/∞, then lim f/g = lim f′/g′ (differentiate top and bottom separately).
Question
Is L'Hopital the quotient rule?
Answer
No! Differentiate the top by itself and the bottom by itself, then divide. Never use the quotient rule.
Question
What must you check before using L'Hopital?
Answer
That substitution gives an indeterminate form 0/0 or ∞/∞.
Question
d/dx (tan x) = ?
Answer
sec²x.
Question
d/dx (arctan x) = ?
Answer
1/(1 + x²).
Question
Find lim (sin x)/x as x→0.
Answer
0/0, so → lim (cos x)/1 = cos 0 = 1.
Question
Find lim (arctan 2x)/(tan 3x) as x→0.
Answer
0/0 → lim [2/(1+4x²)] / [3 sec²3x] = 2/3.
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Full study notes for L'Hopital's rule: the 0/0 form
Topic 5.13 hub
Limits & l'Hopital (HL only)
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