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Question

limx0sec5xsec3xsec3xsecx

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Solution

limx0sec5xsec3xsec3xsecx=limx01cos5x1cos3x1cos3x1cosx

=limx0(cos3xcos5xcos3xcos5xcosxcos3xcosxcos3x)=limx0(cos3xcos5xcosxcos3x×cosxcos3xcos3xcos5x)

[Applying Componendo and dividendo formula]

=limx0(2sin4xsin(x)2sin(2x)sin(x)×cosxcos5x)=limx0(sin4xsin2x×cosxcos5x)

=limx0sin4x×limx0cosxlimx0sin2x×limx0cos5x=(lim4x0sin4x4x×4xlim2x0sin2x2x×2x)×(limx0cosxlimx0cos5x)

=(1×4x)×11×2x×1 [limx0sinxx=1 andlimx0cosx=cos0=1]

=4x2x=2


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