Question Number 195021 by tri26112004 last updated on 22/Jul/23
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}−{sin}\:{x}}{{x}^{{n}} }=¿\:\left({n}\in{N}^{\ast} \right) \\ $$
Answered by MM42 last updated on 22/Jul/23
$${if}\:\:{x}\rightarrow\mathrm{0}\:\Rightarrow\:{x}−{sinx}\:\sim\:\frac{\mathrm{1}}{\mathrm{6}}{x}^{\mathrm{3}} \\ $$$$\Rightarrow{if}\:\:\:{n}<\mathrm{3}\:\Rightarrow\:{lim}=\mathrm{0} \\ $$$${if}\:\:{n}=\mathrm{3}\:\:\Rightarrow{lim}=\frac{\mathrm{1}}{\mathrm{6}} \\ $$$${if}\:\:{n}>\mathrm{3}\:\Rightarrow\:{lim}=\infty \\ $$