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Question Number 116039 by bobhans last updated on 30/Sep/20
lim_(x→(π/2))  (((cos^4 (x)))^(1/(3 )) /((1−sin (x))^(2/3) )) ?
$$\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\:\frac{\sqrt[{\mathrm{3}\:}]{\mathrm{cos}\:^{\mathrm{4}} \left({x}\right)}}{\left(\mathrm{1}−\mathrm{sin}\:\left({x}\right)\right)^{\frac{\mathrm{2}}{\mathrm{3}}} }\:? \\ $$
Answered by Dwaipayan Shikari last updated on 30/Sep/20
lim_(x→(π/2)) (((cosx)^(4/3) )/((1−sinx)^(2/3) ))=(((cos^2 x)^(2/3) )/((1−sinx)^(2/3) ))=(1+sinx)^(2/3) =2^(2/3)
$$\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\frac{\left(\mathrm{cosx}\right)^{\frac{\mathrm{4}}{\mathrm{3}}} }{\left(\mathrm{1}−\mathrm{sinx}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} }=\frac{\left(\mathrm{cos}^{\mathrm{2}} \mathrm{x}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} }{\left(\mathrm{1}−\mathrm{sinx}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} }=\left(\mathrm{1}+\mathrm{sinx}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} =\mathrm{2}^{\frac{\mathrm{2}}{\mathrm{3}}} \\ $$

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