---
title: "State L'Hospital's rule"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:state-lhospital-rule"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/state-lhospital-rule
site: BCT Engineering
---

# State L'Hospital's rule

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2074 Chaitra, 1 marks; 2073 Chaitra (asked 2 times)
- Video: [L'hopital's rule](https://www.youtube.com/watch?v=Gh48aOvWcxw) — The Organic Chemistry Tutor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/state-lhospital-rule
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

State L'Hospital's rule.

## Short answer

L'Hospital's rule: if lim f(x)/g(x) is an indeterminate form 0/0 or ∞/∞, then lim f(x)/g(x) = lim f'(x)/g'(x) provided the latter limit exists. The rule can be applied repeatedly, and other indeterminate forms must be converted to 0/0 or ∞/∞ first.

## Answer

**L'Hospital's rule.** Let f and g be differentiable near a (except possibly at a), with g′(x) ≠ 0 near a. If lim_{x→a} f(x)/g(x) is of the indeterminate form 0/0 (or ∞/∞), then

lim_{x→a} f(x)/g(x) = lim_{x→a} f′(x)/g′(x),

provided the limit on the right exists (finite or infinite). The rule can be applied repeatedly if the form 0/0 or ∞/∞ persists.

Other forms (0·∞, ∞−∞, 1^∞, 0⁰, ∞⁰) are first converted to 0/0 or ∞/∞, using logarithms for the exponential forms.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. L'Hospital's rule applies directly to which forms?
   A. 0/0 or infinity/infinity
   B. infinity - infinity
   C. 1^infinity
   D. 0 * infinity
   Answer: A. Only quotient forms 0/0 and infinity/infinity can be used directly. (Key point: Applicable forms)

2. Under the rule, lim f/g equals:
   A. lim f' * lim g'
   B. f(a)/g(a)
   C. lim (f/g)'
   D. lim f'/g' if it exists
   Answer: D. We differentiate numerator and denominator separately. (Key point: Statement)

3. How is a 1^infinity form handled?
   A. Differentiate base and exponent separately
   B. Replace by 1
   C. Apply the rule directly
   D. Take logarithms to convert to 0/0 or infinity/infinity
   Answer: D. Exponential forms are logged first. (Key point: Other forms)