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Question Number 90060 by jagoll last updated on 21/Apr/20

lim_(x→0)  ((ln (1+sin x))/(((2+x))^(1/(3  ))  − ((2+3x))^(1/3) )) = ?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{ln}\:\left(\mathrm{1}+\mathrm{sin}\:\mathrm{x}\right)}{\sqrt[{\mathrm{3}\:\:}]{\mathrm{2}+\mathrm{x}}\:−\:\sqrt[{\mathrm{3}}]{\mathrm{2}+\mathrm{3x}}}\:=\:? \\ $$

Commented by john santu last updated on 21/Apr/20

lim_(x→0)  ((x−(1/2)x^2 +(1/6)x^3 +o(x^3 ))/(((2 ))^(1/(3  ))  {((1+(x/2)))^(1/(3  )) −((1+((3x)/2)))^(1/(3  )) }))  = ((4)^(1/(3  )) /2) lim_(x→0)  ((x−(1/2)x^2 +(1/6)x^3 +o(x^3 ))/((1+(x/6))−(1+(x/2))))  = ((4)^(1/(3  )) /2) lim_(x→0)  ((−3x(1−(1/2)x+(1/6)x^2 +o(x^2 )))/x)  = −((3 (4)^(1/(3  )) )/2) .

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}−\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{6}}{x}^{\mathrm{3}} +{o}\left({x}^{\mathrm{3}} \right)}{\sqrt[{\mathrm{3}\:\:}]{\mathrm{2}\:}\:\left\{\sqrt[{\mathrm{3}\:\:}]{\mathrm{1}+\frac{{x}}{\mathrm{2}}}−\sqrt[{\mathrm{3}\:\:}]{\mathrm{1}+\frac{\mathrm{3}{x}}{\mathrm{2}}}\right\}} \\ $$$$=\:\frac{\sqrt[{\mathrm{3}\:\:}]{\mathrm{4}}}{\mathrm{2}}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}−\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{6}}{x}^{\mathrm{3}} +{o}\left({x}^{\mathrm{3}} \right)}{\left(\mathrm{1}+\frac{{x}}{\mathrm{6}}\right)−\left(\mathrm{1}+\frac{{x}}{\mathrm{2}}\right)} \\ $$$$=\:\frac{\sqrt[{\mathrm{3}\:\:}]{\mathrm{4}}}{\mathrm{2}}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{−\mathrm{3}{x}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}{x}+\frac{\mathrm{1}}{\mathrm{6}}{x}^{\mathrm{2}} +{o}\left({x}^{\mathrm{2}} \right)\right)}{{x}} \\ $$$$=\:−\frac{\mathrm{3}\:\sqrt[{\mathrm{3}\:\:}]{\mathrm{4}}}{\mathrm{2}}\:.\: \\ $$

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