Cách khác: Sử dụng khai triển Taylor-Maclaurin
\(\cos x=1-\dfrac{x^2}{2!}+o\left(x^2\right)\)
\(\cos2x=1-\dfrac{4x^2}{2!}+o\left(x^2\right)\)
\(\cos4x=1-\dfrac{16x^2}{2!}+o\left(x^2\right)\)
\(\Rightarrow1-\cos x\cos2x\cos4x=1-\left(1-\dfrac{x^2}{2}+o\left(x^2\right)\right)\left(1-2x^2+o\left(x^2\right)\right)\left(1-8x^2+o\left(x^2\right)\right)\)
\(=1-\left(1-\dfrac{5x^2}{2}+o\left(x^2\right)\right)\left(1-8x^2+o\left(x^2\right)\right)=1-1+8x^2+\dfrac{5x^2}{2}+o\left(x^2\right)=\dfrac{21x^2}{2}+o\left(x^2\right)\)
\(\Rightarrow\lim\limits_{x\rightarrow0}\dfrac{1-\cos x\cos2x\cos4x}{1-\cos2x}=\lim\limits_{x\rightarrow0}\dfrac{\dfrac{21}{2}x^2+o\left(x^2\right)}{2x^2+o\left(x^2\right)}=\dfrac{21}{4}\)
\(=\lim\limits_{x\rightarrow0}\dfrac{1-cosx+\left(1-cos2x\right)cosx+\left(1-cos4x\right)cosx.cos2x}{1-cos2x}\)
\(=\lim\limits_{x\rightarrow0}\dfrac{1-cosx}{1-cos2x}+\lim\limits_{x\rightarrow0}\dfrac{\left(1-cos2x\right)}{1-cos2x}.cosx+\lim\limits_{x\rightarrow0}\dfrac{\left(1-cos4x\right)}{1-cos2x}.cosx.cos2x\)
\(=\lim\limits_{x\rightarrow0}\dfrac{1-cosx}{2\left(1-cosx\right)\left(1+cosx\right)}+\lim\limits_{x\rightarrow0}\left(cosx\right)+\lim\limits_{x\rightarrow0}\dfrac{2\left(1-cos2x\right)\left(1+cos2x\right)}{1-cos2x}.cosx.cos2x\)
\(=\lim\limits_{x\rightarrow0}\left(\dfrac{1}{2\left(1+cosx\right)}\right)+\lim\limits_{x\rightarrow0}\left(cosx\right)+\lim\limits_{x\rightarrow0}2cosx.cos2x.\left(1+cos2x\right)\)
\(=\dfrac{1}{4}+1+4=\dfrac{21}{4}\)
\(1-cosx.cos2x.cos3x...cosnx\)
\(=1-cosx+cosx\left(1-cos2x\right)+cosx.cos2x\left(1-cos3x\right)+...+cosx.cos2x...cos\left(n-1\right)x\left(1-cosnx\right)\)
Hay nói chung là:
\(1-a.b.c.d...m.n\)
\(=1-a+a\left(1-b\right)+ab\left(1-c\right)+abc\left(1-d\right)+...+ab...m\left(1-n\right)\)
