Cogito
AP Calculus AB · Chapter 4 · Lesson 3
L'Hospital's Rule
Derivatives resolving an indeterminate form.
12 problems · about 21 minutes · LIM-4.A
What this lesson teaches
The student applies L'Hospital's Rule to indeterminate limits and checks its conditions.
- 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.
Warm Up
Straightforward practice. Get the method working first.
5 problemsLimit of (x² − 25) ÷ (x − 5) as x approaches 5, using the rule?
Answer 10
Why 10.
What must be checked before applying the rule?
Answer That the limit gives 0 over 0 or infinity over infinity.
Why The indeterminate form.
Limit of (x² − 9) ÷ (x − 3) as x approaches 3, using the rule?
Answer 6
Why 2x at x = 3.
Limit of sin x ÷ x as x approaches 0, using the rule?
Answer 1
Why cos x ÷ 1 at 0.
A limit gives 0 ÷ 0. Does the rule apply? 1 yes, 0 no.
Answer 1
Why One of the two forms.
Build It Up
The same ideas with more to keep track of.
3 problemsA limit gives 7 ÷ 3. Does the rule apply? 1 yes, 0 no.
Answer 0
Why That is already an answer.
Limit of (x³ − 8) ÷ (x − 2) as x approaches 2, using the rule?
Answer 12
Why 3x² at x = 2.
Should you use the quotient rule when applying it? 1 yes, 0 no.
Answer 0
Why Differentiate top and bottom separately.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the process for applying the rule in order.
Answer 1. Substitute and check the form. 2. Confirm it is 0 over 0 or infinity over infinity. 3. Differentiate the top and bottom separately. 4. Evaluate the new limit, repeating if still indeterminate.
Why The form must be verified before differentiating.
Limit of (x² − 16) ÷ (x − 4) as x approaches 4, using the rule?
Answer 8
Why 2x at x = 4.
The Form: A limit gives 5 ÷ 0. Does the rule apply? 1 yes, 0 no.
Answer 0
Why No.
The Application: Limit of (x² − 4) ÷ (x − 2) as x approaches 2, using the rule?
Answer 4
Why 4.