L'Hopital's Rule
Grade 12 · Math · Study Guide
Evaluate indeterminate limit forms by differentiating numerator and denominator separately.
L'Hopital's Rule applies when a limit produces an indeterminate form 0/0 or ∞/∞. In those cases: lim(x→a) f(x)/g(x) = lim(x→a) f′(x)/g′(x), provided the new limit exists. You differentiate the numerator and denominator independently — not using the quotient rule.
Other indeterminate forms (0·∞, ∞-∞, 0⁰, 1^∞, ∞⁰) must first be rewritten as 0/0 or ∞/∞ before applying the rule. For products, write f·g = f/(1/g). For exponential forms, take the natural log first.
Always verify the indeterminate form before applying the rule. If substitution gives a determinate value or a form like 5/0 (not 0/0 or ∞/∞), L'Hopital's Rule does not apply.
| Form | How to Handle |
|---|---|
| 0/0 | Apply L'Hopital directly |
| ∞/∞ | Apply L'Hopital directly |
| 0·∞ | Rewrite as 0/(1/∞) or ∞/(1/0) |
| ∞−∞ | Combine into a single fraction |
| 0⁰, 1^∞, ∞⁰ | Take ln, apply L'Hopital, then exponentiate |
Key terms
- Indeterminate form
- A limit expression like 0/0 or ∞/∞ whose value cannot be determined by direct substitution.
- L'Hopital's Rule
- If lim f/g → 0/0 or ∞/∞, then lim f/g = lim f′/g′.
- 0·∞ form
- Rewrite as 0/(1/∞) or ∞/(1/0) to get 0/0 or ∞/∞ before applying the rule.
- 1^∞ form
- Take ln of the expression to convert to a 0·∞ form, then exponentiate the result.
Practice questions
Evaluate lim(x→0) sin(x)/x using L'Hopital's Rule.
- 0
- 1
- ∞
- Undefined
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0/0 form. Differentiate: lim cos(x)/1 = cos(0) = 1.
Evaluate lim(x→∞) x/eˣ.
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∞/∞ form. L'Hopital: lim 1/eˣ = 0 as x→∞.
Evaluate lim(x→0) (eˣ - 1)/x.
- 0
- e
- 1
- ∞
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0/0 form. Differentiate: lim eˣ/1 = e⁰ = 1.
L'Hopital's Rule can be applied to a limit of the form 5/0.
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L'Hopital's Rule only applies to 0/0 or ∞/∞ indeterminate forms, not 5/0 (which is simply undefined/infinite).
Evaluate lim(x→0) (1 - cos x)/x².
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0/0 form. L'Hopital: lim sin x/(2x). Still 0/0. Again: lim cos x/2 = 1/2.
Fun facts
- L'Hopital's Rule was published in 1696 by the Marquis de L'Hopital, but the technique was actually discovered by Johann Bernoulli, who sold it to him.
- You can apply L'Hopital's Rule repeatedly if the new limit is still indeterminate — just keep differentiating until you get a determinate form.
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