Question Number 186983 by cortano12 last updated on 12/Feb/23
Answered by CElcedricjunior last updated on 12/Feb/23
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\boldsymbol{{x}}\sqrt[{\mathrm{3}}]{\boldsymbol{{x}}−\mathrm{1}}+\sqrt[{\mathrm{4}}]{\left(\boldsymbol{{x}}+\mathrm{1}\right)}−\mathrm{1}}{\boldsymbol{{x}}^{\mathrm{2}} \sqrt[{\mathrm{3}}]{\boldsymbol{{x}}+\mathrm{1}}+\sqrt[{\mathrm{4}}]{\boldsymbol{{x}}+\mathrm{1}}−\mathrm{1}}=\boldsymbol{{k}} \\ $$$$\boldsymbol{{k}}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\boldsymbol{{x}}\left(−\mathrm{1}−\frac{\mathrm{1}}{\mathrm{3}}\boldsymbol{{x}}+\frac{\mathrm{1}}{\mathrm{9}}\boldsymbol{{x}}^{\mathrm{2}} \right)+\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{4}}\boldsymbol{{x}}−\frac{\mathrm{3}}{\mathrm{32}}\boldsymbol{{x}}^{\mathrm{2}} \right)−\mathrm{1}}{\boldsymbol{{x}}^{\mathrm{2}} \left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}}\boldsymbol{{x}}−\frac{\:\:\mathrm{1}}{\mathrm{9}}\boldsymbol{{x}}^{\mathrm{2}} \right)+\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{4}}\boldsymbol{{x}}−\frac{\mathrm{3}}{\mathrm{32}}\boldsymbol{{x}}^{\mathrm{2}} \right)−\mathrm{1}}\bigstar{celebre}\:{cedric}\:{junior} \\ $$$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{−\boldsymbol{{x}}−\frac{\mathrm{1}}{\mathrm{3}}\boldsymbol{{x}}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{9}}\boldsymbol{{x}}^{\mathrm{3}} +\frac{\mathrm{1}}{\mathrm{4}}\boldsymbol{{x}}−\frac{\mathrm{3}}{\mathrm{32}}\boldsymbol{{x}}^{\mathrm{2}} }{\boldsymbol{{x}}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{3}}\boldsymbol{{x}}^{\mathrm{3}} −\frac{\mathrm{1}}{\mathrm{9}}\boldsymbol{{x}}^{\mathrm{4}} +\frac{\mathrm{1}}{\mathrm{4}}\boldsymbol{{x}}−\frac{\mathrm{3}}{\mathrm{32}}} \\ $$$$=\frac{\frac{\mathrm{1}}{\mathrm{4}}−\mathrm{1}}{\frac{\mathrm{1}}{\mathrm{4}}}=−\mathrm{3} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\boldsymbol{{x}}\sqrt[{\mathrm{3}}]{\boldsymbol{{x}}−\mathrm{1}}+\sqrt[{\mathrm{4}}]{\boldsymbol{{x}}+\mathrm{1}}−\mathrm{1}}{\boldsymbol{{x}}^{\mathrm{2}} \sqrt[{\mathrm{3}}]{\boldsymbol{{x}}+\mathrm{1}}+\sqrt[{\mathrm{4}}]{\boldsymbol{{x}}+\mathrm{1}}−\mathrm{1}}=−\mathrm{3}\:\:\:\bigstar{Moivre} \\ $$
Answered by ARUNG_Brandon_MBU last updated on 12/Feb/23
$$\mathscr{L}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{x}\sqrt[{\mathrm{3}}]{{x}−\mathrm{1}}+\sqrt[{\mathrm{4}}]{{x}+\mathrm{1}}−\mathrm{1}}{{x}^{\mathrm{2}} \sqrt[{\mathrm{3}}]{{x}+\mathrm{1}}+\sqrt[{\mathrm{4}}]{{x}+\mathrm{1}}−\mathrm{1}} \\ $$$$\:\:\:\:\:=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{−{x}\left(\mathrm{1}−\frac{{x}}{\mathrm{3}}\right)+\left(\mathrm{1}+\frac{{x}}{\mathrm{4}}\right)−\mathrm{1}}{{x}^{\mathrm{2}} \left(\mathrm{1}+\frac{{x}}{\mathrm{3}}\right)+\left(\mathrm{1}+\frac{{x}}{\mathrm{4}}\right)−\mathrm{1}} \\ $$$$\:\:\:\:\:=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\frac{{x}^{\mathrm{2}} }{\mathrm{3}}−\frac{\mathrm{3}{x}}{\mathrm{4}}}{\frac{{x}^{\mathrm{3}} }{\mathrm{3}}+{x}^{\mathrm{2}} +\frac{{x}}{\mathrm{4}}}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\frac{{x}}{\mathrm{3}}−\frac{\mathrm{3}}{\mathrm{4}}}{\frac{{x}^{\mathrm{2}} }{\mathrm{3}}+{x}+\frac{\mathrm{1}}{\mathrm{4}}}=−\mathrm{3} \\ $$
Answered by cortano12 last updated on 14/Feb/23