Key takeaway: L’Hôpital’s rule says that when a quotient’s limit is indeterminate, of the form 0/0 or infinity/infinity, you may take derivatives of the top and the bottom separately and find the limit again.
Grade 12 – AP Calculus ABAP Calculus AB Unit 4Calculus
Definition of L'Hôpital's Rule
L’Hôpital’s rule rescues limits that look undecided. If substituting the limit value produces \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), the quotient could approach nearly any number, and the form alone settles nothing. Comparing how fast the top and bottom change settles it.
Derivatives measure those speeds, so swapping each part for its derivative often reveals the true value. If the new quotient is still undecided, the same step can be repeated. Other forms need a different approach.
Formula and notation
Let \(p\) and \(q\) be differentiable near \(k\), with \(\frac{p}{q}\) indeterminate at \(k\). Then
\[ \lim_{x\to k}\frac{p(x)}{q(x)}=\lim_{x\to k}\frac{p^{\prime}(x)}{q^{\prime}(x)}, \]
whenever the second limit exists or is infinite.
Key facts
- Use derivatives of the numerator and denominator separately; this is not the quotient rule.
- Test the form first: only 0/0 and infinity/infinity qualify for l’Hôpital’s rule.
- Keep repeating the step as long as the form stays indeterminate.
- Products, differences and powers must first be rewritten as fractions.
L'Hôpital's Rule example
Evaluate \(\lim_{x\to 0}\frac{1-\cos 3x}{x^2}\).
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Substituting \(x=0\) gives \(\frac{1-1}{0}\), an indeterminate \(\frac{0}{0}\).
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Replace top and bottom by their derivatives: \(\frac{3\sin 3x}{2x}\), still \(\frac{0}{0}\).
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Once more: \(\frac{9\cos 3x}{2}\), which approaches \(\frac{9}{2}\) as \(x\to 0\).
The limit equals \(\frac{9}{2}\).
Common mistake
Watch out
A student sees \(\lim_{x\to 2}\frac{x^2+5}{x+1}\) and differentiates top and bottom to get \(\frac{2x}{1}\to 4\).
Why it is wrong
Substituting \(x=2\) gives \(\frac{9}{3}=3\), a definite number, so the form is not indeterminate and l’Hôpital’s rule must not be used. Check the form first. The limit is 3, not 4.
Related terms and lessons
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