Question Number 82800 by jagoll last updated on 24/Feb/20
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:\mathrm{cos}\:\mathrm{2}{x}}{{x}^{\mathrm{2}} } \\ $$
Commented by mathmax by abdo last updated on 24/Feb/20
$${let}\:{f}\left({x}\right)=\frac{\mathrm{1}−\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }{cos}\left(\mathrm{2}{x}\right)}{{x}^{\mathrm{3}} }\:\:{we}\:{have}\:{cos}\left(\mathrm{2}{x}\right)\sim\mathrm{1}−\mathrm{2}{x}^{\mathrm{2}} \\ $$$$\left(\mathrm{1}+{u}\right)^{\alpha} \:\sim\mathrm{1}+\alpha{u}\:+\frac{\alpha\left(\alpha−\mathrm{1}\right)}{\mathrm{2}}{u}^{\mathrm{2}} \:\:\:\:\left({u}\sim{o}\right)\:\Rightarrow \\ $$$$\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:\sim\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} \:+\frac{\frac{\mathrm{1}}{\mathrm{2}}\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)}{\mathrm{2}}{x}^{\mathrm{4}} \:=\mathrm{1}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{8}}{x}^{\mathrm{4}} \\ $$$${cosu}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} \:{u}^{\mathrm{2}{n}} }{\left(\mathrm{2}{n}\right)!}\:\Rightarrow{cosu}\:\sim\mathrm{1}−\frac{{u}^{\mathrm{2}} }{\mathrm{2}}\:+\frac{{u}^{\mathrm{4}} }{\mathrm{4}!}\:\Rightarrow \\ $$$${cos}\left(\mathrm{2}{x}\right)\sim\mathrm{1}−\mathrm{2}{x}^{\mathrm{2}} \:+\frac{\left(\mathrm{2}{x}\right)^{\mathrm{4}} }{\mathrm{4}!}\:=\mathrm{1}−\mathrm{2}{x}^{\mathrm{2}} \:+\frac{\mathrm{16}{x}^{\mathrm{4}} }{\mathrm{4}.\mathrm{6}}\:=\mathrm{1}−\mathrm{2}{x}^{\mathrm{2}} \:+\frac{\mathrm{2}}{\mathrm{3}}{x}^{\mathrm{4}} \:\Rightarrow \\ $$$$\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }{cos}\left(\mathrm{2}{x}\right)\sim\left(\mathrm{1}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}}−\frac{{x}^{\mathrm{4}} }{\mathrm{8}}\right)\left(\mathrm{1}−\mathrm{2}{x}^{\mathrm{2}} +\frac{\mathrm{2}}{\mathrm{3}}{x}^{\mathrm{4}} \right) \\ $$$$=\mathrm{1}−\mathrm{2}{x}^{\mathrm{2}} \:+\frac{\mathrm{2}}{\mathrm{3}}{x}^{\mathrm{4}} +\frac{{x}^{\mathrm{2}} }{\mathrm{2}}−{x}^{\mathrm{4}} +\frac{\mathrm{1}}{\mathrm{3}}{x}^{\mathrm{6}} −\frac{{x}^{\mathrm{4}} }{\mathrm{8}}\:+\frac{\mathrm{1}}{\mathrm{4}}{x}^{\mathrm{6}} −\frac{{x}^{\mathrm{8}} }{\mathrm{12}} \\ $$$$=\mathrm{1}+\left(−\mathrm{2}+\frac{\mathrm{1}}{\mathrm{2}}\right){x}^{\mathrm{2}} \:+\left(\frac{\mathrm{2}}{\mathrm{3}}−\mathrm{1}−\frac{\mathrm{1}}{\mathrm{8}}\right){x}^{\mathrm{4}} +\left(\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{4}}\right){x}^{\mathrm{6}} \:−\frac{{x}^{\mathrm{8}} }{\mathrm{12}} \\ $$$$=\mathrm{1}−\frac{\mathrm{3}}{\mathrm{2}}{x}^{\mathrm{2}} \:+\frac{\mathrm{5}}{\mathrm{24}}{x}^{\mathrm{4}} \:+\frac{\mathrm{7}}{\mathrm{12}}{x}^{\mathrm{6}} −\frac{{x}^{\mathrm{8}} }{\mathrm{12}}\:\Rightarrow \\ $$$${f}\left({x}\right)\sim\frac{\frac{\mathrm{3}}{\mathrm{2}}{x}^{\mathrm{2}} +\frac{\mathrm{5}}{\mathrm{24}}{x}^{\mathrm{4}} \:+\frac{\mathrm{7}}{\mathrm{12}}{x}^{\mathrm{6}} −\frac{{x}^{\mathrm{8}} }{\mathrm{12}}}{{x}^{\mathrm{3}} }\:=\frac{\mathrm{3}}{\mathrm{2}{x}}+\frac{\mathrm{5}}{\mathrm{24}}{x}+\frac{\mathrm{7}}{\mathrm{12}}{x}^{\mathrm{3}} \:−\frac{\mathrm{1}}{\mathrm{12}}{x}^{\mathrm{5}} \:\Rightarrow \\ $$$${lim}_{{x}\rightarrow\mathrm{0}} \:\:{f}\left({x}\right)\:=\infty \\ $$
Commented by jagoll last updated on 24/Feb/20
$${sorry}\:{my}\:{question}\:{is}\:{typo}.\: \\ $$$${i}\:{will}\:{correct} \\ $$
Answered by TANMAY PANACEA last updated on 24/Feb/20
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}−{cos}^{\mathrm{2}} \mathrm{2}{x}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{{x}^{\mathrm{3}} }×\frac{\mathrm{1}}{\mathrm{1}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:×{cos}\mathrm{2}{x}} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{sin}^{\mathrm{2}} \mathrm{2}{x}−{x}^{\mathrm{2}} {cos}^{\mathrm{2}} \mathrm{2}{x}}{{x}^{\mathrm{3}} }×\frac{\mathrm{1}}{\mathrm{1}+\mathrm{1}} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left({sin}\mathrm{2}{x}+{xcos}\mathrm{2}{x}\right)}{\mathrm{1}}×\frac{{sin}\mathrm{2}{x}−{xcos}\mathrm{2}{x}}{{x}^{\mathrm{3}} } \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}×\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left(\mathrm{2}{x}−\frac{\mathrm{8}{x}^{\mathrm{3}} }{\mathrm{3}!}+\frac{\mathrm{32}{x}^{\mathrm{5}} }{\mathrm{5}!}..\right)−{x}\left(\mathrm{1}−\frac{\mathrm{4}{x}^{\mathrm{2}} }{\mathrm{2}!}+\frac{\mathrm{16}{x}^{\mathrm{4}} }{\mathrm{4}!}..\right)}{{x}^{\mathrm{3}} } \\ $$$${in}\:{numinator}\:{x}\:{can}\:{not}\:{be}\:{eliminated} \\ $$$$ \\ $$
Commented by jagoll last updated on 24/Feb/20
$${sir}\:{the}\:{originall}\:{question}\: \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\sqrt{\mathrm{1}+{x}^{\mathrm{2}} \:}\mathrm{cos}\:\mathrm{2}{x}}{{x}^{\mathrm{2}} } \\ $$
Commented by TANMAY PANACEA last updated on 24/Feb/20
$${ok}.. \\ $$
Answered by john santu last updated on 24/Feb/20
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\left(\mathrm{1}−\frac{\mathrm{4}{x}^{\mathrm{2}} }{\mathrm{2}!}+\frac{\mathrm{16}{x}^{\mathrm{4}} }{\mathrm{4}!}−{o}\left(\left(\mathrm{2}{x}\right)^{\mathrm{4}} \right)\right)\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} +{o}\left({x}^{\mathrm{2}} \right)\right)}{{x}^{\mathrm{2}} } \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} +{o}\left({x}^{\mathrm{2}} \right)−\mathrm{2}{x}^{\mathrm{2}} −{x}^{\mathrm{4}} +…\right)}{{x}^{\mathrm{2}} } \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left(\frac{\mathrm{3}}{\mathrm{2}}{x}^{\mathrm{2}} −{o}\left({x}^{\mathrm{2}} \right)+{x}^{\mathrm{4}} +…\right)}{{x}^{\mathrm{2}} }=\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$
Answered by john santu last updated on 24/Feb/20
Commented by jagoll last updated on 24/Feb/20
$${yess}..{thank}\:{you}\:{mister} \\ $$
Answered by Henri Boucatchou last updated on 24/Feb/20
$$\frac{\mathrm{1}−\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }{cos}\mathrm{2}{x}}{{x}^{\mathrm{2}} }=\frac{\mathrm{1}−\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }+\mathrm{2}\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }{sin}^{\mathrm{2}} {x}}{{x}^{\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:=\frac{−{x}^{\mathrm{2}} }{{x}^{\mathrm{2}} \left(\mathrm{1}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)}+\mathrm{2}\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:\left(\frac{{sinx}}{{x}}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:=−\frac{\mathrm{1}}{\mathrm{1}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}+\mathrm{2}\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\left(\frac{{sinx}}{{x}}\right)^{\mathrm{2}} \:\rightarrow−\frac{\mathrm{1}}{\mathrm{2}}+\mathrm{2}=\mathrm{1},\mathrm{5}\:\:\left(\boldsymbol{{x}}\rightarrow\mathrm{0}\right) \\ $$