Lesson 64 — L'Hôpital's Rule
L'Hôpital's Rule for 0/0 and ∞/∞ indeterminate forms. Other indeterminate forms like 0·∞, ∞−∞, 1^∞, 0^0, ∞^0, and notable limits such as sin(x)/x and e^x/x^n.
Used in: 2nd Year High School · Japanese Equiv. Math III · German Equiv. Analysis Klasse 12 · Singapore Equiv. H2 Maths
Rigorous notation, full derivation, hypotheses
Statement, proof, and extensions
Formal Statement
"L'Hôpital's rule simplifies the evaluation of limits of quotients when both numerator and denominator approach 0 or ∞. The key is recognizing the indeterminate form, applying the rule, and checking that the resulting limit actually exists." — OpenStax Calculus Vol. 1, §4.8
Proof Idea (0/0 case)
By Cauchy's Mean Value Theorem: if , for near there exists between and such that:
As , we have , and the limit becomes: (when the ratio of derivatives converges).
Extension to Other Indeterminate Forms
"Note carefully: L'Hôpital's Rule says that the limit of a quotient of functions equals the limit of the quotient of their derivatives, provided the original limit is in the form 0/0 or ∞/∞. This is not the same as the derivative of a quotient." — Active Calculus §2.8
Solved Examples
Exercise list
30 exercises · 7 with worked solution (25%)
- Ex. 64.1
Calculate .
- Ex. 64.2
Calculate .
- Ex. 64.3
Calculate .
- Ex. 64.4
Calculate .
- Ex. 64.5
Calculate .
- Ex. 64.6Answer key
Calculate .
- Ex. 64.7Answer key
Calculate .
- Ex. 64.8
Calculate .
- Ex. 64.9
Calculate .
- Ex. 64.10
Calculate .
- Ex. 64.11Answer key
Prove via L'Hôpital that .
- Ex. 64.12
Calculate .
- Ex. 64.13
Someone wants to calculate using L'Hôpital. What is wrong?
- Ex. 64.14
What is the indeterminate form of ?
- Ex. 64.15
Calculate .
- Ex. 64.16Answer key
Calculate .
- Ex. 64.17Answer key
Calculate .
- Ex. 64.18
Calculate .
- Ex. 64.19
Calculate for positive integer .
- Ex. 64.20
Calculate .
- Ex. 64.21Answer key
Calculate .
- Ex. 64.22
Calculate .
- Ex. 64.23
Calculate .
- Ex. 64.24
Calculate .
- Ex. 64.25
Try calculating using L'Hôpital. What happens? What is the correct answer?
- Ex. 64.26
Calculate .
- Ex. 64.27
Calculate .
- Ex. 64.28
Sketch the proof of L'Hôpital's Rule for the case, using Cauchy's Mean Value Theorem.
- Ex. 64.29
Use L'Hôpital's Rule to prove the fundamental limit .
- Ex. 64.30Answer key
Show that for any (exponential dominates any power). Use iterated L'Hôpital and argue about the number of applications needed.
Sources
- Boelkins, Matt. Active Calculus 2.0. Grand Valley State University, 2022. CC-BY-NC-SA. activecalculus.org/single/sec-2-8-LHR.html
- OpenStax. Calculus Volume 1. Strang, Herman et al., 2023. CC-BY-NC-SA. openstax.org/books/calculus-volume-1/pages/4-8-lhopitals-rule
- Hartman, G. et al. APEX Calculus. Virginia Military Institute, 2023. CC-BY-NC. apexcalculus.com