Question and Answers Forum

All Questions      Topic List

Limits Questions

Previous in All Question      Next in All Question      

Previous in Limits      Next in Limits      

Question Number 216055 by efronzo1 last updated on 26/Jan/25

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

$$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:^{\mathrm{2}} \mathrm{2x}}{\:\sqrt[{\mathrm{3}}]{\mathrm{cos}\:\mathrm{x}}−\sqrt[{\mathrm{4}}]{\mathrm{cos}\:\mathrm{x}}}\:=? \\ $$

Answered by golsendro last updated on 27/Jan/25

= lim_(x→0) ((4cos^2 x(1−cos^2 x))/( ((cos x))^(1/3) −((cos x))^(1/4) )) = 8. lim_(x→0) ((1−cos x)/( ((cos x))^(1/3) −((cos x))^(1/4) )) Let cos x= t^(12) = 8 lim_(t→1) ((1−t^(12) )/(t^4 −t^3 )) = 8 lim_(t→1) (((1−t)(t^(11) +t^(10) +...+1))/(t^3 (t−1))) = −8.(12)= −96

$$\:\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{4cos}\:^{\mathrm{2}} \mathrm{x}\left(\mathrm{1}−\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}\right)}{\:\sqrt[{\mathrm{3}}]{\mathrm{cos}\:\mathrm{x}}−\sqrt[{\mathrm{4}}]{\mathrm{cos}\:\mathrm{x}}} \\ $$$$\:\:=\:\mathrm{8}.\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:\mathrm{x}}{\:\sqrt[{\mathrm{3}}]{\mathrm{cos}\:\mathrm{x}}−\sqrt[{\mathrm{4}}]{\mathrm{cos}\:\mathrm{x}}} \\ $$$$\:\mathrm{Let}\:\mathrm{cos}\:\mathrm{x}=\:\mathrm{t}^{\mathrm{12}} \\ $$$$\:\:=\:\mathrm{8}\:\underset{\mathrm{t}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{t}^{\mathrm{12}} }{\mathrm{t}^{\mathrm{4}} −\mathrm{t}^{\mathrm{3}} } \\ $$$$\:\:=\:\mathrm{8}\:\underset{\mathrm{t}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\left(\mathrm{1}−\mathrm{t}\right)\left(\mathrm{t}^{\mathrm{11}} +\mathrm{t}^{\mathrm{10}} +...+\mathrm{1}\right)}{\mathrm{t}^{\mathrm{3}} \left(\mathrm{t}−\mathrm{1}\right)} \\ $$$$\:\:=\:−\mathrm{8}.\left(\mathrm{12}\right)=\:−\mathrm{96} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com