Question Number 133583 by bemath last updated on 23/Feb/21
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:\mathrm{x}−\mathrm{1}+\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}−\frac{\mathrm{x}^{\mathrm{4}} }{\mathrm{24}}}{\mathrm{x}^{\mathrm{6}} } \\ $$
Answered by EDWIN88 last updated on 23/Feb/21
$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}+\frac{\mathrm{x}^{\mathrm{4}} }{\mathrm{24}}−\frac{\mathrm{x}^{\mathrm{6}} }{\mathrm{720}}+\mathrm{R}\left(\mathrm{x}^{\mathrm{6}} \right)−\mathrm{1}+\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}−\frac{\mathrm{x}^{\mathrm{4}} }{\mathrm{24}}}{\mathrm{x}^{\mathrm{6}} } \\ $$$$=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{−\frac{\mathrm{x}^{\mathrm{6}} }{\mathrm{720}}+\mathrm{R}\left(\mathrm{x}^{\mathrm{6}} \right)}{\mathrm{x}^{\mathrm{6}} }\:=\:−\frac{\mathrm{1}}{\mathrm{720}} \\ $$