L'Hôpital's Rule
L'Hôpital's Rule is a technique for evaluating limits of indeterminate forms (0/0 or ∞/∞) by differentiating the numerator and denominator separately and then taking the limit of the resulting ratio. The rule states that if lim f(x)/g(x) yields an indeterminate form and g'(x) ≠ 0, then the limit equals lim f'(x)/g'(x). It is named after the French mathematician Guillaume de l'Hôpital, who published it in 1696, though it was originally developed by Johann Bernoulli.
Key Formula
lim[x→a] f(x)/g(x) = lim[x→a] f'(x)/g'(x)
LaTeX: \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}
| Symbol | Meaning | Unit |
|---|---|---|
| f(x) | Numerator function | dimensionless |
| g(x) | Denominator function | dimensionless |
| f'(x) | Derivative of numerator | dimensionless |
| g'(x) | Derivative of denominator | dimensionless |
| a | Point at which limit is evaluated | dimensionless |
Worked Example
Problem
Evaluate lim(x→0) sin(x)/x.
Solution
Step 1: Direct substitution gives sin(0)/0 = 0/0 — indeterminate form. Step 2: Apply L'Hôpital's Rule: differentiate numerator and denominator separately. Numerator: d/dx[sin(x)] = cos(x). Denominator: d/dx[x] = 1. Step 3: Evaluate the new limit: lim(x→0) cos(x)/1 = cos(0)/1 = 1/1 = 1.
Answer
lim(x→0) sin(x)/x = 1
Indeterminate Forms and L'Hôpital's Rule Applicability
| Indeterminate Form | Direct Form | Transformation Needed | Apply L'Hôpital? | Example |
|---|---|---|---|---|
| 0/0 | f(a)=0, g(a)=0 | None | Yes | sin(x)/x at x=0 |
| ∞/∞ | Both diverge | None | Yes | ln(x)/x at x=∞ |
| 0 · ∞ | Product form | Rewrite as 0/0 or ∞/∞ | After rewrite | x·ln(x) at x=0⁺ |
| 1^∞ | Exponential form | Take logarithm | After rewrite | (1+1/x)^x at x=∞ |
| ∞ − ∞ | Difference form | Common denominator | After rewrite | 1/x − 1/sin(x) |
Interactive Tools
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Named after French mathematician Guillaume François Antoine, Marquis de l'Hôpital (1661–1704), who published the rule in his 1696 textbook "Analyse des Infiniment Petits." The underlying theorem was actually derived by Johann Bernoulli.