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Question Number 90564 by mathmax by abdo last updated on 24/Apr/20

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

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

Commented by jagoll last updated on 25/Apr/20

lim_(x→0) (1/((1+cos 2x)^(2/3) +((2+2cos 2x))^(1/(3 )) +2^(2/3) )) × lim_(x→0) (((1+cos 2x)−2)/(x^2 sin 3x)) = (1/(3((4)^(1/(3 )) ))) × lim_(x→0) ((cos 2x−1)/x^2 ) ×lim_(x→0) (1/(sin 3x)) =∞

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{cos}\:\mathrm{2}{x}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} +\sqrt[{\mathrm{3}\:\:}]{\mathrm{2}+\mathrm{2cos}\:\mathrm{2}{x}}+\mathrm{2}^{\frac{\mathrm{2}}{\mathrm{3}}} }\:× \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{1}+\mathrm{cos}\:\mathrm{2}{x}\right)−\mathrm{2}}{{x}^{\mathrm{2}} \mathrm{sin}\:\mathrm{3}{x}}\:= \\ $$$$\frac{\mathrm{1}}{\mathrm{3}\left(\sqrt[{\mathrm{3}\:\:}]{\mathrm{4}}\right)}\:×\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{cos}\:\mathrm{2}{x}−\mathrm{1}}{{x}^{\mathrm{2}} }\:×\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}}{\mathrm{sin}\:\mathrm{3}{x}}\:=\infty \\ $$$$ \\ $$$$ \\ $$

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