L'Hôpital's Rule
Evaluate indeterminate limits using L'Hôpital's Rule.
Learn when and how to apply the derivative-based limit technique with examples.
The Formula
lim(x→c) f(x)/g(x) = lim(x→c) f′(x)/g′(x)
L'Hôpital's Rule provides a method for evaluating limits that produce indeterminate forms like 0/0 or ∞/∞. Instead of struggling with algebraic manipulation, you simply differentiate the numerator and denominator separately and try the limit again.
The rule was published in 1696 by Guillaume de L'Hôpital, though it was actually discovered by Johann Bernoulli. L'Hôpital had paid Bernoulli a salary for his mathematical discoveries.
The rule can be applied repeatedly if the result is still indeterminate. However, it only works when the original limit is genuinely 0/0 or ∞/∞ — applying it to other forms gives wrong answers.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| f(x) | Numerator function | — |
| g(x) | Denominator function | — |
| f′(x) | Derivative of the numerator | — |
| g′(x) | Derivative of the denominator | — |
| c | The point the limit approaches (can be ±∞) | — |
Conditions for Use
- The limit must produce 0/0 or ∞/∞ (check this first!)
- f and g must be differentiable near c
- g′(x) ≠ 0 near c (except possibly at c itself)
- The limit of f′(x)/g′(x) must exist (or be ±∞)
Example 1
Evaluate lim(x→0) sin(x) / x
Check the form: sin(0)/0 = 0/0 ✓ — L'Hôpital's Rule applies
Differentiate: f′(x) = cos(x), g′(x) = 1
Evaluate: lim(x→0) cos(x) / 1 = cos(0) / 1
The limit equals 1
Example 2
Evaluate lim(x→∞) x² / e^x
Check the form: ∞/∞ ✓ — apply L'Hôpital's Rule
First application: lim(x→∞) 2x / e^x — still ∞/∞
Second application: lim(x→∞) 2 / e^x
Now 2/∞ is no longer indeterminate
The limit equals 0 — exponential growth always overtakes polynomial growth
Example 3
Evaluate lim(x→0) (e^x − 1) / x
Check: (e⁰ − 1)/0 = 0/0 ✓
Differentiate: f′(x) = e^x, g′(x) = 1
Evaluate: lim(x→0) e^x / 1 = e⁰ = 1
The limit equals 1
When to Use It
- Evaluating limits that produce 0/0 or ∞/∞ indeterminate forms
- Comparing growth rates of functions (polynomial vs exponential vs logarithmic)
- Simplifying complex limit problems in calculus courses
- Analyzing asymptotic behavior of functions
- Deriving important limits like lim(x→0) sin(x)/x = 1