Saturday, June 24, 2017

real analysis - How to evaluate limxto0frac(sin(2x)2sin(x))4(3+cos(2x)4cos(x))3?




lim



without L'Hôpital.



I've tried using equivalences with {(\sin(2x)-2\sin(x))^4} and arrived at -x^{12} but I don't know how to handle {(3+\cos(2x)-4\cos(x))^3}. Using \cos(2x)=\cos^2(x)-\sin^2(x) hasn't helped, so any hint?


Answer



Hint: Note that
3+\cos(2x)-4\cos(x) = 3 + 2\cos^2(x) - 1 - 4\cos(x) = 2(\cos(x)-1)^2,
and that

\sin(2x) - 2\sin(x) = 2\sin(x)\cos(x)-2\sin(x) = 2\sin(x)(\cos(x)-1).


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