Question Number 165868 by cortano1 last updated on 09/Feb/22
$$\:\:\:\:\:\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:\mathrm{x}}{\mathrm{sin}\:\mathrm{x}−\sqrt[{\mathrm{3}}]{\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}}}\:=? \\ $$
Answered by qaz last updated on 10/Feb/22
$$\underset{\mathrm{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\frac{\mathrm{cos}\:\mathrm{x}}{\mathrm{sin}\:\mathrm{x}−\sqrt[{\mathrm{3}}]{\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}}} \\ $$$$=\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{−\mathrm{sin}\:\mathrm{x}}{\mathrm{cos}\:\mathrm{x}−\sqrt[{\mathrm{3}}]{\mathrm{cos}\:\mathrm{x}−\mathrm{sin}\:\mathrm{x}}} \\ $$$$=\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{−\mathrm{x}}{\frac{\mathrm{1}}{\mathrm{3}}\mathrm{ln}\left(\mathrm{1}−\mathrm{sin}\:\mathrm{x}\right)} \\ $$$$=−\mathrm{3}\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{x}}{\mathrm{sin}\:\mathrm{x}} \\ $$$$=−\mathrm{3} \\ $$
Answered by Rohit143Jo last updated on 10/Feb/22
![Ans: lim_(x→(π/2)) ((Cos x)/(Sin x − ((Sin x + Cos x))^(1/3) )) = lim_(x→(π/2)) (((−Sin x))/(Cos x − ((Cos x − Sin x)/(3 (Sin x +Cos x)^(2/3) )))) [L′Hospital Rule] = lim_(x→(π/2)) ((3.(−Sin x).(Sin x + Cos x)^(2/3) )/(3Cos x.(Sin x + Cos x)^(2/3) − Cos x + Sin x)) = ((3.(−Sin (π/2)).(Sin (π/2) + Cos (π/2))^(2/3) )/(3.Cos (π/2).(Sin (π/2) + Cos (π/2))^(2/3) − Cos (π/2) + Sin (π/2))) = (−3).](https://www.tinkutara.com/question/Q165930.png)
$${Ans}:\:\:{lim}_{{x}\rightarrow\frac{\pi}{\mathrm{2}}} \:\:\frac{{Cos}\:{x}}{{Sin}\:{x}\:−\:\sqrt[{\mathrm{3}}]{{Sin}\:{x}\:+\:{Cos}\:{x}}} \\ $$$$\:\:\:\:\:\:\:=\:{lim}_{{x}\rightarrow\frac{\pi}{\mathrm{2}}} \:\:\frac{\left(−{Sin}\:{x}\right)}{{Cos}\:{x}\:−\:\frac{{Cos}\:{x}\:−\:{Sin}\:{x}}{\mathrm{3}\:\left({Sin}\:{x}\:+{Cos}\:{x}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} }}\:\left[{L}'{Hospital}\:{Rule}\right] \\ $$$$\:\:\:\:\:\:\:=\:{lim}_{{x}\rightarrow\frac{\pi}{\mathrm{2}}} \:\:\frac{\mathrm{3}.\left(−{Sin}\:{x}\right).\left({Sin}\:{x}\:+\:{Cos}\:{x}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} }{\mathrm{3}{Cos}\:{x}.\left({Sin}\:{x}\:+\:{Cos}\:{x}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} \:−\:{Cos}\:{x}\:+\:{Sin}\:{x}}\: \\ $$$$\:\:\:\:\:\:\:=\:\:\frac{\mathrm{3}.\left(−{Sin}\:\frac{\pi}{\mathrm{2}}\right).\left({Sin}\:\frac{\pi}{\mathrm{2}}\:+\:{Cos}\:\frac{\pi}{\mathrm{2}}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} }{\mathrm{3}.{Cos}\:\frac{\pi}{\mathrm{2}}.\left({Sin}\:\frac{\pi}{\mathrm{2}}\:+\:{Cos}\:\frac{\pi}{\mathrm{2}}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} \:−\:{Cos}\:\frac{\pi}{\mathrm{2}}\:+\:{Sin}\:\frac{\pi}{\mathrm{2}}} \\ $$$$\:\:\:\:\:\:\:=\:\left(−\mathrm{3}\right). \\ $$
Answered by greogoury55 last updated on 15/Feb/22
