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lim-x-pi-2-cos-x-1-sin-x-2-3-




Question Number 116029 by bemath last updated on 30/Sep/20
 lim_(x→(π/2))  ((cos x)/((1−sin x)^(2/3) )) =?
$$\:\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:{x}}{\left(\mathrm{1}−\mathrm{sin}\:{x}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} }\:=?\: \\ $$
Answered by bobhans last updated on 30/Sep/20
set x = (π/2) + z  lim_(z→0)  ((−sin z)/((1−cos z)^(2/3) )) = lim_(z→0)  ((−z)/(((z^2 /2))^(2/3) )) doesn′t exist
$${set}\:{x}\:=\:\frac{\pi}{\mathrm{2}}\:+\:{z} \\ $$$$\underset{{z}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{−\mathrm{sin}\:{z}}{\left(\mathrm{1}−\mathrm{cos}\:{z}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} }\:=\:\underset{{z}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{−{z}}{\left(\frac{{z}^{\mathrm{2}} }{\mathrm{2}}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} }\:{doesn}'{t}\:{exist} \\ $$

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