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Question Number 184795 by Spillover last updated on 11/Jan/23

Evaluate lim_(x→0) ((1−cos x(√(cos 2x)) )/x^2 )

$${Evaluate}\: \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}−\mathrm{cos}\:{x}\sqrt{\mathrm{cos}\:\mathrm{2}{x}}\:}{{x}^{\mathrm{2}} } \\ $$

Commented by MJS_new last updated on 11/Jan/23

(3/2)

$$\frac{\mathrm{3}}{\mathrm{2}} \\ $$

Answered by cortano1 last updated on 12/Jan/23

lim_(x→0) ((1−cos x+cos x−cos x(√(cos 2x)))/x^2 ) = lim_(x→0) ((sin^2 x)/((1+cos x)x^2 )) +lim_(x→0) ((cos x(1−(√(cos 2x))))/x^2 ) = (1/2) +lim_(x→0) ((cos x(1−cos 2x))/((1+(√(cos 2x)))x^2 )) =(1/2) +(1/2).lim_(x→0) ((2sin^2 x)/x^2 ) =(1/2)(1+2)=(3/2)

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:{x}+\mathrm{cos}\:{x}−\mathrm{cos}\:{x}\sqrt{\mathrm{cos}\:\mathrm{2}{x}}}{{x}^{\mathrm{2}} } \\ $$$$=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:^{\mathrm{2}} {x}}{\left(\mathrm{1}+\mathrm{cos}\:{x}\right){x}^{\mathrm{2}} }\:+\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:{x}\left(\mathrm{1}−\sqrt{\mathrm{cos}\:\mathrm{2}{x}}\right)}{{x}^{\mathrm{2}} } \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{2}}\:+\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{cos}\:{x}\left(\mathrm{1}−\mathrm{cos}\:\mathrm{2}{x}\right)}{\left(\mathrm{1}+\sqrt{\mathrm{cos}\:\mathrm{2}{x}}\right){x}^{\mathrm{2}} } \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\:+\frac{\mathrm{1}}{\mathrm{2}}.\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2sin}\:^{\mathrm{2}} {x}}{{x}^{\mathrm{2}} } \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+\mathrm{2}\right)=\frac{\mathrm{3}}{\mathrm{2}} \\ $$

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