Menu Close

lim-x-0-1-cos-x-cos-2x-cos-3x-1-3-cos-4x-1-4-x-2-




Question Number 115999 by bemath last updated on 30/Sep/20
lim_(x→0)  ((1−cos x (√(cos 2x)) ((cos 3x))^(1/(3 ))  ((cos 4x))^(1/(4 )) )/x^2 )
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:{x}\:\sqrt{\mathrm{cos}\:\mathrm{2}{x}}\:\sqrt[{\mathrm{3}\:}]{\mathrm{cos}\:\mathrm{3}{x}}\:\sqrt[{\mathrm{4}\:}]{\mathrm{cos}\:\mathrm{4}{x}}}{{x}^{\mathrm{2}} } \\ $$
Answered by bobhans last updated on 30/Sep/20
short cut ′mr john santu ′  lim_(x→0)  ((1−cos x (√(cos 2x)) ((cos 3x))^(1/(3 ))  ((cos 4x))^(1/(4 )) )/x^2 )  = (1/2)(1^2 +(2^2 /2)+(3^2 /3)+(4^2 /4))  =(1/2)(1+2+3+4)=5
$${short}\:{cut}\:'{mr}\:{john}\:{santu}\:' \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:{x}\:\sqrt{\mathrm{cos}\:\mathrm{2}{x}}\:\sqrt[{\mathrm{3}\:}]{\mathrm{cos}\:\mathrm{3}{x}}\:\sqrt[{\mathrm{4}\:}]{\mathrm{cos}\:\mathrm{4}{x}}}{{x}^{\mathrm{2}} } \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}^{\mathrm{2}} +\frac{\mathrm{2}^{\mathrm{2}} }{\mathrm{2}}+\frac{\mathrm{3}^{\mathrm{2}} }{\mathrm{3}}+\frac{\mathrm{4}^{\mathrm{2}} }{\mathrm{4}}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}\right)=\mathrm{5} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *