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Cogito

AP Calculus AB · Chapter 4 · Lesson 3

Practice Worksheet

L'Hospital's Rule

Derivatives resolving an indeterminate form.

12 problems · about 21 minutes · LIM-4.A

NameDate

Figure — use these to answer the problems

An indeterminate form hides a real limit. The rule finds it by comparing how fast the two parts change.
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y = 1/x + 0
  • Point(2, 0.50)
a1
Point on the curve2
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. Limit of (x² − 25) ÷ (x − 5) as x approaches 5, using the rule?

    Answer

  2. What must be checked before applying the rule?

    • That the limit gives 0 over 0 or infinity over infinity.
    • That both functions are continuous everywhere.

  3. Limit of (x² − 9) ÷ (x − 3) as x approaches 3, using the rule?

    Answer

  4. Limit of sin x ÷ x as x approaches 0, using the rule?

    Answer

  5. A limit gives 0 ÷ 0. Does the rule apply? 1 yes, 0 no.

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. A limit gives 7 ÷ 3. Does the rule apply? 1 yes, 0 no.

    Answer

  2. Limit of (x³ − 8) ÷ (x − 2) as x approaches 2, using the rule?

    Answer

  3. Should you use the quotient rule when applying it? 1 yes, 0 no.

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Put the process for applying the rule in order.

    Write 1 to 4 in the boxes to put these in order.

    • Substitute and check the form.
    • Confirm it is 0 over 0 or infinity over infinity.
    • Differentiate the top and bottom separately.
    • Evaluate the new limit, repeating if still indeterminate.

  2. Limit of (x² − 16) ÷ (x − 4) as x approaches 4, using the rule?

    Answer

  3. The Form

    A limit gives 5 ÷ 0. Does the rule apply? 1 yes, 0 no.

    Answer

  4. The Application

    Limit of (x² − 4) ÷ (x − 2) as x approaches 2, using the rule?

    Answer