Back to AI Flashcard MakerMathematics /AP Calculus AB: 12.2.1 L'Hôpital's Rule and Indeterminate Products

AP Calculus AB: 12.2.1 L'Hôpital's Rule and Indeterminate Products

Mathematics10 CardsCreated 3 months ago

This content explains how to handle indeterminate products, which need to be transformed into quotients before applying L’Hôpital’s Rule. It covers strategies to rewrite limits involving products like 0·∞ into a quotient form, enabling the use of derivatives to evaluate tricky limits.

L’Hôpital’s Rule and Indeterminate Products

  • Some indeterminate forms have to be transformed before you can apply L’Hôpital’s rule.

  • When applying L’Hôpital’s rule to an indeterminate product, express one of the factors as a fraction.

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Key Terms

Term
Definition

L’Hôpital’s Rule and Indeterminate Products

  • Some indeterminate forms have to be transformed before you can apply L’Hôpital’s rule.

  • When applying L’Hôpital’s rule to an ...

note

  • An example of a camouflaged indeterminate form is the indeterminate product 0 · . It is indeterminate because you cannot tell who wins. Zer...

Evaluate limx→∞xsin2/x

2

Evaluate limx→1 (x−1)^3(1−x)^−2.

0

Evaluate limx→∞ e^−x√x.

0

Evaluate limx→∞ x^−1e^x.

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TermDefinition

L’Hôpital’s Rule and Indeterminate Products

  • Some indeterminate forms have to be transformed before you can apply L’Hôpital’s rule.

  • When applying L’Hôpital’s rule to an indeterminate product, express one of the factors as a fraction.

note

  • An example of a camouflaged indeterminate form is the indeterminate product 0 · . It is indeterminate because you cannot tell who wins. Zero times anything is zero, but anything times infinity is infinity, so what is the limit?

  • If you write as , then your limit produces the standard
    indeterminate quotient L’Hôpital’s rule. , which allows you to use

  • This limit also produces the indeterminate product 0

  • Here, it makes sense to write cot θ as the reciprocal of tan θ. Then you have the other standard indeterminate quotient, 0/0.

Evaluate limx→∞xsin2/x

2

Evaluate limx→1 (x−1)^3(1−x)^−2.

0

Evaluate limx→∞ e^−x√x.

0

Evaluate limx→∞ x^−1e^x.

Evaluate limx→0 2xcotx.

2

Evaluate limx→0 1/xcotx

1

Evaluate limx→0 xlnx.

0

Evaluate limx→0 x^−2(x−3)^3.

−∞