Question Number 27912 by abdo imad last updated on 16/Jan/18 $${find}\:{the}\:{sum}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\propto} \frac{\mathrm{1}}{{n}}\left(\:\:\frac{\sqrt{\mathrm{2}}}{\mathrm{1}+{i}}\right)^{{n}} . \\ $$ Commented by abdo imad last updated on 23/Jan/18 $${for}\:{z}\in\:{iR}\:\:{and}\:\:\mid{z}\mid\leqslant\:\mathrm{1}\:{we}\:{have}\:\:\sum_{{n}=\mathrm{1}}…
Question Number 158894 by mathlove last updated on 10/Nov/21 Commented by cortano last updated on 10/Nov/21 $$\:\underset{{x}\rightarrow{y}} {\mathrm{lim}}\:\frac{\left(\sqrt[{\mathrm{3}}]{{x}}\right)^{\mathrm{2}} −\left(\sqrt[{\mathrm{3}}]{{y}}\right)^{\mathrm{2}} }{{x}−{y}} \\ $$$$=\:\underset{{x}\rightarrow{y}} {\mathrm{lim}}\:\frac{\left(\sqrt[{\mathrm{3}}]{{x}}+\sqrt[{\mathrm{3}}]{{y}}\right)\left(\sqrt[{\mathrm{3}}]{{x}}−\sqrt[{\mathrm{3}}]{{y}}\right)}{\left(\sqrt[{\mathrm{3}}]{{x}}−\sqrt[{\mathrm{3}}]{{y}}\right)\left(\sqrt[{\mathrm{3}}]{{x}^{\mathrm{2}} }+\sqrt[{\mathrm{3}}]{{xy}}+\sqrt[{\mathrm{3}}]{{y}}\right)} \\…
Question Number 27792 by abdo imad last updated on 14/Jan/18 $${find}\:{lim}_{{x}−>\mathrm{1}} \int_{{x}} ^{{x}^{\mathrm{2}} } \:\:\frac{{dt}}{{ln}\left({t}\right)}\:\:. \\ $$ Commented by abdo imad last updated on 15/Jan/18…
Question Number 27789 by abdo imad last updated on 14/Jan/18 $${find}\:\:{lim}_{{x}−>\mathrm{0}} \left(\mathrm{1}+{sinx}\right)^{\frac{\mathrm{1}}{{x}}} \:\:. \\ $$ Commented by abdo imad last updated on 15/Jan/18 $${we}\:{have}\:\left(\mathrm{1}+{sinx}\right)^{\frac{\mathrm{1}}{{x}}} \:=\:{e}^{\frac{\mathrm{1}}{{x}}{ln}\left(\mathrm{1}+{sinx}\right)}…
Question Number 27786 by abdo imad last updated on 14/Jan/18 $${find}\:{lim}_{{x}−>\mathrm{0}} \:\left(\frac{{tanx}}{{x}}\right)^{\frac{\mathrm{1}}{{x}^{\mathrm{2}} }} . \\ $$ Commented by abdo imad last updated on 17/Jan/18 $${we}\:{have}\:\:\left(\frac{{tanx}}{{x}}\right)^{\frac{\mathrm{1}}{{x}^{\mathrm{2}}…
Question Number 27787 by abdo imad last updated on 14/Jan/18 $${find}\:\:{lim}_{{x}−>\mathrm{0}} \frac{{e}^{{x}} \:\:−{x}−\mathrm{1}}{{x}^{\mathrm{2}} }\:. \\ $$ Answered by prakash jain last updated on 14/Jan/18 $${e}^{{x}}…
Question Number 158847 by mathlove last updated on 09/Nov/21 Answered by mr W last updated on 09/Nov/21 $$\underset{{x}\rightarrow{a}} {\mathrm{lim}}\frac{{x}^{{x}} −{a}^{{a}} }{{x}−{a}}={a}^{{a}} \left(\mathrm{ln}\:{a}+\mathrm{1}\right) \\ $$$$\underset{{y}\rightarrow{x}} {\mathrm{lim}}\frac{{y}^{{y}}…
Question Number 93287 by john santu last updated on 12/May/20 Commented by mathmax by abdo last updated on 12/May/20 $${let}\:{f}\left({x}\right)\:=\frac{{arctan}\left({x}^{\mathrm{2}} \left(\mathrm{1}−{cosx}\right)\right)}{\left(\mathrm{1}−\sqrt{{cosx}}\right){ln}\left(\frac{{sinx}}{{x}}\right)}{we}\:{have}\:\mathrm{1}−{cosx}\:\sim\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\:\Rightarrow{x}^{\mathrm{2}} \left(\mathrm{1}−{cosx}\right)\sim\frac{{x}^{\mathrm{4}} }{\mathrm{2}}\:\Rightarrow \\…
Question Number 93258 by john santu last updated on 12/May/20 Commented by john santu last updated on 12/May/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{3sin}\:\mathrm{x}−\mathrm{4sin}\:^{\mathrm{3}} \mathrm{x}+\mathrm{4sin}\:^{\mathrm{3}} \mathrm{x}−\mathrm{3ln}\left(\mathrm{1}+\mathrm{x}\right)}{\left(\mathrm{e}^{\mathrm{x}} −\mathrm{1}\right)\mathrm{sin}\:\mathrm{x}} \\ $$$$\underset{{x}\rightarrow\mathrm{0}}…
Question Number 27727 by NECx last updated on 13/Jan/18 $${What}\:{are}\:{the}\:{conditions}\:{whereby} \\ $$$${the}\:{limit}\:{of}\:{a}\:{function}\:{does}\:{not} \\ $$$${exist}\:{at}\:{a}\:{poont}? \\ $$ Answered by prakash jain last updated on 13/Jan/18 $$\mathrm{When}\:\mathrm{LHL}\:\mathrm{is}\:\mathrm{not}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{RHL}.…