Daniel Lobo
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Homework Statement
The problem wants me to find the limit below using series expansion.
##\lim_{x \to 0}(\frac{1}{x^2}\cdot \frac{\cos x}{(\sin x)^2})##
Homework Equations
The Attempt at a Solution
(1) For startes I'll group the two fractions inside the limit together
##\lim_{x \to0}\frac{sin^2(x)-x^2cos(x)}{x^2sin^2(x)}##[/B]
This limit is of the form 0/0 so I can use l'hopital's rule here, I'll expand both the numerator and the denominator into their respective power series and take enough derivatives so that the functions don't equal to zero at x=0.
(2) The numerator expanded:
##sin^2(x)-x^2cos(x)=\sum_{n=0}^\infty\frac{x^{4n+2}}{((2n+1)!)^2}-\sum_{n=0}^\infty\frac{(-1)^{n}x^{2n+2}}{(2n)!}=(x^2+\frac{x^6}{36}+...)-(x^2-\frac{x^4}{2}+...)=(\frac{x^4}{2}+\frac{x^6}{36}+...)##
(3) The x^(4)/2 is the only part of interest to me since it's the only part that will not be zero after taking 4 derivatives and setting x=0 and doing so will wield:
##\frac{\mathrm{d^4} }{\mathrm{d} x^4}(sin^2(x)-x^2cos(x))|_{x=0}=12##
The denominator expanded:
##x^2sin^2(x)=\sum_{n=0}^\infty\frac{x^{4n+4}}{(2n+1)^2!}=(x^4+\frac{x^8}{4}+...)##
(4) Doing the same as in (3):
##\frac{\mathrm{d^4} }{\mathrm{d} x^4}(x^2sin^2(x))|_{x=0}=24##
With these results I reach my answer of 12/24=1/2 but my book says the answer is 1/6 and I can't figure out what I'm doing wrong for the life of me, it's been 3 hours on this exercise already and would really apreciate some advice. Also apologies for the iffy english.