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lim-x-0-cos-x-1-x-2-2-x-4-24-x-6-




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

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