Question Number 82988 by jagoll last updated on 26/Feb/20
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} \:}\mathrm{cos}\:\mathrm{x}}{\mathrm{x}^{\mathrm{4}} } \\ $$
Commented by jagoll last updated on 26/Feb/20
$$\mathrm{my}\:\mathrm{answer}\:\mathrm{is}\:\frac{\mathrm{1}}{\mathrm{3}}.\:\mathrm{that}\:\mathrm{is}\:\mathrm{correct}? \\ $$
Commented by mr W last updated on 26/Feb/20
$$=\frac{\mathrm{1}−\left(\mathrm{1}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}}−\frac{{x}^{\mathrm{4}} }{\mathrm{8}}+{o}\left({x}^{\mathrm{4}} \right)\right)\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}+\frac{{x}^{\mathrm{4}} }{\mathrm{24}}+{o}\left({x}^{\mathrm{4}} \right)\right)}{{x}^{\mathrm{4}} } \\ $$$$=\frac{\mathrm{1}−\left(\mathrm{1}−\frac{{x}^{\mathrm{4}} }{\mathrm{3}}+{o}\left({x}^{\mathrm{4}} \right)\right)}{{x}^{\mathrm{4}} } \\ $$$$=\frac{\frac{{x}^{\mathrm{4}} }{\mathrm{3}}+{o}\left({x}^{\mathrm{4}} \right)}{{x}^{\mathrm{4}} } \\ $$$$=\frac{\mathrm{1}}{\mathrm{3}} \\ $$
Commented by jagoll last updated on 27/Feb/20
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir} \\ $$
Answered by jagoll last updated on 26/Feb/20
