Question Number 172361 by Mikenice last updated on 25/Jun/22
Answered by mr W last updated on 26/Jun/22
$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{x}−\frac{{x}^{\mathrm{3}} }{\mathrm{3}}+{o}\left({x}^{\mathrm{5}} \right)−\left({x}+\frac{{x}^{\mathrm{3}} }{\mathrm{6}}+{o}\left({x}^{\mathrm{5}} \right)\right)}{{x}^{\mathrm{3}} } \\ $$$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{−\left(\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{6}}\right){x}^{\mathrm{3}} +{o}\left({x}^{\mathrm{5}} \right)}{{x}^{\mathrm{3}} } \\ $$$$=−\left(\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{6}}\right) \\ $$$$=−\frac{\mathrm{1}}{\mathrm{2}} \\ $$
Commented by bagjagugum123 last updated on 26/Jun/22
$${What}\:{is}\:{o}\:{Sir}? \\ $$
Commented by mr W last updated on 26/Jun/22
$${o}\left({x}^{\mathrm{5}} \right)\:{means}\:{following}\:{terms}\:{are}\:{of} \\ $$$$\mathrm{5}^{{th}} \:{order}\:{or}\:{higher}. \\ $$
Commented by mr W last updated on 26/Jun/22
Commented by bagjagugum123 last updated on 26/Jun/22
$${Thank}\:{you}\:{Sir} \\ $$
Answered by mahdipoor last updated on 25/Jun/22
$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\frac{\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{1}}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}}{\mathrm{3}{x}^{\mathrm{2}} }\:=\frac{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }−\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{\mathrm{3}{x}^{\mathrm{2}} \left({x}^{\mathrm{2}} +\mathrm{1}\right)\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}= \\ $$$$=\frac{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)−\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{\mathrm{3}{x}^{\mathrm{2}} \left({x}^{\mathrm{2}} +\mathrm{1}\right)\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\left(\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }+\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\right)}= \\ $$$$=\frac{−{x}^{\mathrm{2}} \left({x}^{\mathrm{2}} +\mathrm{3}\right)}{\mathrm{3}{x}^{\mathrm{2}} \left({x}^{\mathrm{2}} +\mathrm{1}\right)\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\left(\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }+\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\right)}= \\ $$$$=−\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$ \\ $$