L'Hopital's Rule is used to evaluate limits of indeterminate forms such as 0/0 and ∞/∞ by taking the derivative of the numerator and denominator. It can be applied repeatedly as long as the resulting form is still indeterminate. Examples are provided for common indeterminate forms like 0/0, ∞/∞, 0*∞, 1/∞, and 00 which may require converting to 0/0 or ∞/∞ first before applying the rule. L'Hopital's Rule can only be used when the basic criteria of an indeterminate form is met.

1. AP Calculus BCL’Hopital’sRule

2. L’Hopital’s RuleFDWK 8.2, Larson 7.7Basic formula: lim𝑥->𝑎𝑓(𝑥)𝑔(𝑥)=lim𝑥->𝑎𝑓′(𝑥)𝑔′(𝑥)Used to evaluate limits of indeterminate forms 00,∞∞. It can also be used with the forms 0∙∞,1∞,00, and ∞−∞.It can be used repeatedly, but only when the new form is also indeterminate.Note that it is different than the quotient rule.

3. ExamplesIndeterminate Form 0/0Indeterminate Form ∞/∞Applying L’Hopital’s Rule more than onceIndeterminate Form 0•∞ convert to 0/0 or ∞/∞Indeterminate Form 1∞ use logarithmic differentiationIndeterminate Form 00 use logarithmic differentiationIndeterminate Form ∞-∞ find a common denominator

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9. Wrapping It UpYou can only use L’Hopital’s Rule when the basic criteria are metEven though there are many indeterminate forms, we only apply L’Hopital’s Rule to the basic cases of 0/0 or ∞/∞The other types of indeterminate forms must first be converted into 0/0 or ∞/ ∞, before L’Hopital’s Rule can be applied