If a limit gives 0 ÷ 0 or infinity ÷ infinity, the limit of f ÷ g equals the limit of f′ ÷ g′, when that second limit exists.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Check the form first
The rule applies only to those two indeterminate forms. Using it on a form like 3 ÷ 0 gives a confidently wrong answer.
It is not the quotient rule
Differentiate the top and bottom separately. Applying the quotient rule here is a different and incorrect calculation.
It may be applied again
If the new limit is still indeterminate, apply the rule again. Each pass must be justified by rechecking the form.
Other forms
Forms like 0 times infinity can be rewritten as a quotient first, which brings them within reach of the rule.
Not always the best tool
Factoring or algebra is often faster. The rule is powerful rather than automatic.
Check the form first
L'Hôpital's rule applies only to 0/0 and ∞/∞. Applying it to a form that is not indeterminate produces a confidently wrong answer, so the exam expects the form to be verified in writing.
It can be applied repeatedly
If differentiating once still leaves an indeterminate form, differentiate again — checking the form each time. Each application must be justified separately, not assumed to carry over.
It is not the quotient rule
You differentiate numerator and denominator separately, not as a quotient. Applying the quotient rule instead is a frequent and complete error, and the two give quite different expressions.
Other indeterminate forms
Products of the form 0·∞ and differences ∞ − ∞ can be rewritten as quotients first. Rearranging into 0/0 or ∞/∞ is the standard preliminary step before the rule becomes applicable.
Step 2: Try It Yourself
Tap and try it out.
- Point(2, 0.50)
Step 3: Watch an Example
One step at a time.
Watch Ines Apply the Rule
Ines evaluates the limit of (x² − 4) ÷ (x − 2) as x approaches 2.
- Step 1
Substituting gives 0 ÷ 0, so the rule applies.
Step 4: Your Turn
Practice makes it stick.
The Form
Problem 1 of 2
A limit gives 5 ÷ 0. Does the rule apply? 1 yes, 0 no.
The Application
Problem 2 of 2
Limit of (x² − 4) ÷ (x − 2) as x approaches 2, using the rule?
Resolve the Form
1 of 8
Limit of (x² − 9) ÷ (x − 3) as x approaches 3, using the rule?
2 of 8
Limit of sin x ÷ x as x approaches 0, using the rule?
3 of 8
A limit gives 0 ÷ 0. Does the rule apply? 1 yes, 0 no.
4 of 8
A limit gives 7 ÷ 3. Does the rule apply? 1 yes, 0 no.
5 of 8
Limit of (x³ − 8) ÷ (x − 2) as x approaches 2, using the rule?
6 of 8
Should you use the quotient rule when applying it? 1 yes, 0 no.
7 of 8
Put the process for applying the rule in order.
- 1Confirm it is 0 over 0 or infinity over infinity.
- 2Differentiate the top and bottom separately.
- 3Evaluate the new limit, repeating if still indeterminate.
- 4Substitute and check the form.
8 of 8
Limit of (x² − 16) ÷ (x − 4) as x approaches 4, using the rule?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Limit of (x² − 25) ÷ (x − 5) as x approaches 5, using the rule?
Question 2 of 2
What must be checked before applying the rule?
What You Learned
- The rule applies only to 0 ÷ 0 and infinity ÷ infinity.
- Differentiate the top and bottom separately, not with the quotient rule.
- Recheck the form before each further application.