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IBMathsDPAnalysis & Approaches (AA)HLFlashcards5. CalculusLimits using l'Hôpital's Rule & Maclaurin Series

Limits using l'Hôpital's Rule & Maclaurin Series (DP IB Analysis & Approaches (AA))

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  • FrontFurther Limits

    Define the term indeterminate form.

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  • Define the term indeterminate form.

    An indeterminate form is a quotient of the form 0 over 0 or fraction numerator plus-or-minus infinity over denominator plus-or-minus infinity end fraction that results from attempting to evaluate a limit.

    Such a form does not provide a definitive answer as to what the limit might be.

  • True or False?

    l'Hôpital's Rule can be applied to any limit.

    False.

    l'Hôpital's Rule can not be applied to any limit.

    It can only be applied to the limit of a quotient fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction for which usual limit evaluation techniques return one of the indeterminate forms 0 over 0 or fraction numerator plus-or-minus infinity over denominator plus-or-minus infinity end fraction.

  • What should your first step be when applying l'Hôpital's Rule?

    When applying l'Hôpital's Rule, your first step should be to check that the limit of the quotient results in one of the indeterminate forms 0 over 0 or fraction numerator plus-or-minus infinity over denominator plus-or-minus infinity end fraction.

  • Assuming limit as x rightwards arrow a of fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction returns an indeterminate form, what other expression does l'Hôpital's Rule say that limit is equal to?

    Assuming limit as x rightwards arrow a of fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction returns an indeterminate form, then l'Hôpital's Rule states that limit as x rightwards arrow a of fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction equals limit as x rightwards arrow a of fraction numerator f to the power of apostrophe open parentheses x close parentheses over denominator g to the power of apostrophe open parentheses x close parentheses end fraction (if that second limit exists).

  • True or False?

    l'Hôpital's Rule can be applied multiple times to the same limit.

    True.

    l'Hôpital's Rule can be applied multiple times to the same limit, as long as the result continues to be an indeterminate form.

  • True or False?

    l'Hôpital's Rule can be used to evaluate limits as x approaches infinity.

    True.

    l'Hôpital's Rule can be used to evaluate limits as x approaches infinity, provided the limit results in an indeterminate form.

  • True or False?

    Maclaurin series can also be used to evaluate limits for expressions of the form fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction.

    True.

    Maclaurin series can also be used to evaluate limits for expressions of the form fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction, including cases where the limit initially returns an indeterminate form 0 over 0 or fraction numerator plus-or-minus infinity over denominator plus-or-minus infinity end fraction.

    The functions f open parentheses x close parentheses and g open parentheses x close parentheses are replaced by their Maclaurin series, and algebra is used to simplify the quotient before attempting to evaluate the limit.

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