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Date May 2019 Marks available 9 Reference code 19M.3.AHL.TZ0.Hca_4
Level Additional Higher Level Paper Paper 3 Time zone Time zone 0
Command term Find Question number Hca_4 Adapted from N/A

Question

Using L’Hôpital’s rule, find limx0(tan3x3tanxsin3x3sinx).

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

limx0(tan3x3tanxsin3x3sinx)

limx0(3sec23x3sec2x3cos3x3cosx)(=limx0(sec23xsec2xcos3xcosx))      M1A1A1

Note: Award M1 for attempt at differentiation using l'Hopital's rule, A1 for numerator, A1 for denominator.

 

METHOD 1

using l’Hopital’s rule again

=limx0(18sec23xtan3x6sec2xtanx9sin3x+3sinx)(=limx0(6sec23xtan3x2sec2xtanx3sin3x+sinx))      A1A1

EITHER

=limx0(108sec23xtan23x+54sec43x12sec2xtan2x6sec4x - 27cos3x+3cosx)      A1A1

Note: Not all terms in numerator need to be written in final fraction. Award A1 for 54sec43x+6sec4x. However, if the terms are written, they
must be correct to award A1.

attempt to substitute x=0         M1

=4824

OR

ddx(18sec23xtan3x6sec2xtanx)|x=0=48      (M1)A1

ddx(9sin3x+3sinx)|x=0=24      A1

THEN

(limx0(tan3x3tanxsin3x3sinx))=2      A1

 

METHOD 2

=limx0(3cos23x3cos2x3cos3x3cosx)      M1

=limx0(cos2xcos23xcos23xcos2x(cos3xcosx))      A1

=limx0(cosx+cos3xcos23xcos2x)      M1A1

attempt to substitute x=0         M1

=21

=2      A1

 

[9 marks]

Examiners report

[N/A]

Syllabus sections

Topic 5 —Calculus » AHL 5.13—Limits and L’Hopitals
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Topic 5 —Calculus

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