L'Hopital's Rule
Learn L'Hopital's Rule for evaluating indeterminate limits of the form 0/0 or infinity/infinity with worked examples.
The Formula
limx→c f(x)/g(x) = limx→c f'(x)/g'(x)
L'Hopital's Rule is a powerful technique in calculus for evaluating limits that produce indeterminate forms. When directly substituting a value into a limit yields 0/0 or ∞/∞, the limit cannot be determined by simple substitution. L'Hopital's Rule states that under these conditions, you can take the derivative of the numerator and the derivative of the denominator separately, then evaluate the limit of that new fraction.
The rule was published in 1696 by Guillaume de l'Hopital in France, though it was actually discovered by Johann Bernoulli. It applies when three conditions are met: the original limit produces an indeterminate form (0/0 or ∞/∞), both f and g are differentiable near the point c, and g'(x) is not zero near c (except possibly at c itself).
An important detail is that L'Hopital's Rule can be applied repeatedly. If after one application the result is still an indeterminate form, you can differentiate the numerator and denominator again. However, you must verify the indeterminate form exists each time before reapplying the rule.
The rule also works for one-sided limits and for limits as x approaches positive or negative infinity. It does not apply to other indeterminate forms like 0 × ∞ or 1∞ directly, but these can often be algebraically rearranged into a 0/0 or ∞/∞ form first.
Common mistakes include applying the rule when the limit is not actually indeterminate, or using the quotient rule instead of differentiating numerator and denominator independently. Always check the form before applying.
Variables
| Symbol | Meaning |
|---|---|
| f(x) | The numerator function |
| g(x) | The denominator function |
| f'(x) | The derivative of the numerator |
| g'(x) | The derivative of the denominator |
| c | The value x approaches (can be a number or ±∞) |
Example 1
Evaluate limx→0 sin(x) / x
Direct substitution gives sin(0)/0 = 0/0 (indeterminate)
Apply L'Hopital's Rule: differentiate top and bottom
limx→0 cos(x) / 1 = cos(0) / 1 = 1
limx→0 sin(x) / x = 1
Example 2
Evaluate limx→∞ x² / ex
Direct evaluation gives ∞/∞ (indeterminate)
Apply L'Hopital's Rule: limx→∞ 2x / ex → still ∞/∞
Apply again: limx→∞ 2 / ex = 0
limx→∞ x² / ex = 0 (exponentials grow faster than polynomials)
When to Use It
L'Hopital's Rule is essential in many areas of mathematics and applied science.
- Evaluating limits in calculus courses when direct substitution fails
- Analyzing the behavior of functions near singularities or at infinity
- Comparing growth rates of functions (polynomial vs exponential vs logarithmic)
- Physics and engineering problems involving rates that approach zero or infinity
- Proving other important limits like limx→0 (ex − 1)/x = 1