Wednesday, 24 August 2016

real analysis - Evaluation of limlimitsxrightarrow0fractan(x)xx3



One of the previous posts made me think of the following question: Is it possible to evaluate this limit without L'Hopital and Taylor?



limx0tan(x)xx3


Answer




The statement tan(x)xx3c as x0 is equivalent to
tan(x)=x+cx3+o(x3) as x0, so this is a statement about a
Taylor polynomial of tan(x), and I'm not sure what would count as doing
that "without Taylor". However, one thing you could do is start from sin(x)=x+o(x)

integrate to get cos(x)=1x2/2+o(x2)
then sec(x)=11x2/2+o(x2)=1+x2/2+o(x2)
sec2(x)=(1+x2/2+o(x2))2=1+x2+o(x2)
and integrate again to get
tan(x)=x+x3/3+o(x3)


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