Monday, 28 August 2017

calculus - Compute this limit limxto0fracsin(x2+frac1x)sinfrac1xx using L'Hôpital's rule



I have asked this problem before, but I can't understand the explanation, I couldn't understand how the sin multiply for cos, and too multiply for A + and - B: sin(A)sin(B)=2sin(AB2)cos(A+B2) and I don't understand in this step how/why the AB and A+B was replaced by x22 and x22+1x :

lim


Answer



Hint:its easy to prove \sin(x+y)+\sin(x-y)=2\sin(x)\cos(y)
then put y=\frac{A+B}{2},x=\frac{A-B}{2}
\sin(x)\sim x \ cosx\sim1-\frac{x^2}{2} because \lim_{x\to0}\frac{sinx}{x}=\lim_{x\to0}\frac{cosx}{1-\frac{x^2}{2}}=1


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