How can I calculate \(\displaystyle\lim_{{{x}\to{0}}}\frac{{\log{{\left({\cos{{\left({x}\right)}}}\right)}}}}{{\log{{\left({\cos{{\left({3}{x}\right)}}}\right)}}}}\) without l'Hopital?

Oliver Carson

Oliver Carson

Answered question

2022-03-26

How can I calculate limx0log(cos(x))log(cos(3x)) without l'Hopital?

Answer & Explanation

cineworld93uowb

cineworld93uowb

Beginner2022-03-27Added 16 answers

We need some limiting property of log(x). The only limiting property we will use is that
limu0log(1+u)u=1
With 1+u=cos2(x) , we get
limx0log(cos(x))log(cos(3x))=limx0log(cos(x))log(4cos3(x)-3cos(x))
=limx0log(cos(x))log(4cos2(x)3)+log(cos(x))
=limx012log(cos2(x))log(4cos2(x)3)+12log(cos2(x))
=limu0log(1+u)2log(1+4u)+log(1+u)
=limu0log(1+u)u8limu0log(1+4u)4u+limu0log(1+u)u
=18+1
=19

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