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AP Calculus AB: 12.1.2 An Introduction to L'Hôpital's Rule

Mathematics14 CardsCreated 3 months ago

This content introduces L’Hôpital’s Rule as a powerful technique for evaluating limits involving indeterminate forms such as 0/0 or ∞/∞. It explains when and how to apply the rule using derivatives and emphasizes verifying the presence of an indeterminate form before applying it.

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An Introduction to L’Hôpital’s Rule

  • A limit of a function is called an indeterminate form when it produces a mathematically meaningless expression.

  • L’Hôpital’s rule enables you to evaluate indeterminate forms quickly by using derivatives.

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Key Terms

Term
Definition

An Introduction to L’Hôpital’s Rule

  • A limit of a function is called an indeterminate form when it produces a mathematically meaningless expression.

  • L’Hôpital’s ...

note

  • The limits of simple functions can be evaluated easily with simple algebra and direct numerical substitution.

  • For complicate...

Evaluate limx→2 x^6−6/410x−5

0

Evaluate limx→∞ x3+3x+1/4x2+2

∞

Evaluate limx→0 x100+7x2/x50−4x2

−7/4

Evaluate.

limx→9 x−9/x2−11x+18

1/7

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AP Calculus AB: 12.1.2 An Introduction to L'Hôpital's Rule
TermDefinition

An Introduction to L’Hôpital’s Rule

  • A limit of a function is called an indeterminate form when it produces a mathematically meaningless expression.

  • L’Hôpital’s rule enables you to evaluate indeterminate forms quickly by using derivatives.

note

  • The limits of simple functions can be evaluated easily with simple algebra and direct numerical substitution.

  • For complicated functions, it might not be easy to factor or simplify their respective limits, and in some cases, algebraic manipulation can be time-consuming and lead to errors.

  • Suppose a function has a numerator and a denominator and its limit produces an indeterminate form. Then you can apply L’Hôpital’s rule.

  • If the top and the bottom are differentiable, the limit of the function is equal to the limit of the derivative of the numerator divided by the derivative of the denominator.

  • Instead of using algebra, L’Hôpital’s rule enables you to evaluate indeterminate forms quickly by using derivatives.

  • If the quotient of the derivatives still yields an indeterminate form, L’Hôpital’s rule can be applied again.

  • To avoid mistakes, make sure the limit produces an
    indeterminate form before using L’Hôpital’s rule.

  • CAUTION: Notice that in the example above, the quotient rule is not applied when taking the derivatives. L’Hôpital’s rule doesn’t instruct you to take the derivative of the function. This makes L’Hôpital’s rule easier to use, since you do not have to remember the quotient rule.

Evaluate limx→2 x^6−6/410x−5

0

Evaluate limx→∞ x3+3x+1/4x2+2

∞

Evaluate limx→0 x100+7x2/x50−4x2

−7/4

Evaluate.

limx→9 x−9/x2−11x+18

1/7

Evaluate limx→5x2−7x+10/x3−25x

0.06

Evaluate limx→0 x6−x5+2x+4x5+3x−1

−4

Evaluate limx→1 x3−2x+1/x4+3x−4

1/7

Evaluate limx→2 x3−2x2/10x−20.

2/5

Evaluate.

limx→1 x74−1/x148−1

1/2

Evaluate.

limx→0 x2−2x2x−1

0

Evaluate limx→2 x3−2x2+3x−6x2−4x+4

The limit does not exist

Evaluate limx→3x3−9x2+27x−27x−3.

0