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IBMathsDPAnalysis & Approaches (AA)HLRevision Notes5. CalculusLimits using l'Hôpital's Rule & Maclaurin SeriesLimits using Maclaurin Series

Limits using Maclaurin Series (DP IB Analysis & Approaches (AA)): Revision Note

Roger B

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Roger B

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DP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): HL

Limits Using a Maclaurin Series

How do I evaluate a limit using Maclaurin series?

  • Limits of the form limit as x rightwards arrow a of fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction or limit as x rightwards arrow infinity of fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction may sometimes be evaluated by using Maclaurin series

  • Usually this will be in a situation where attempting to evaluate the limit in the usual way returns an indeterminate form 0 over 0 or fraction numerator plus-or-minus infinity over denominator plus-or-minus infinity end fraction.

  • In such a case:

    • STEP 1: Find the Maclaurin series for f left parenthesis x right parenthesis and g left parenthesis x right parenthesis

    • STEP 2: Rewrite fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction using the Maclaurin series in the numerator and denominator

    • STEP 3: Use algebra to simplify your new expression for fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction as far as possible

    • STEP 4: Evaluate the limit using your simplified form of the expression

Examiner Tips and Tricks

  • Some limits of an indeterminate form can also be evaluated using l’Hôpital’s Rule

  • If an exam question does not specify a method to use, then you are free to use whichever method you prefer

Worked Example

Use Maclaurin series to evaluate the limit

limit as x rightwards arrow 0 of fraction numerator x cubed over denominator negative 2 x plus sin space 2 x end fraction

5-12-1-ib-aa-hl-maclaurin-series-limits-we-solution

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    Roger B

    Author: Roger B

    Expertise: Maths Content Creator

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.