Question Number 80442 by Power last updated on 03/Feb/20 Commented by john santu last updated on 03/Feb/20 $${f}\:'\left({a}\right)\:{hahaha} \\ $$ Commented by Power last updated…
Question Number 80417 by jagoll last updated on 03/Feb/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{1}+{x}\right)^{\frac{\mathrm{1}}{{x}}} −{e}}{{x}}\:=\:? \\ $$ Commented by MJS last updated on 03/Feb/20 $$\mathrm{I}\:\mathrm{tried}\:\mathrm{to}\:\mathrm{approximate}\:\mathrm{and}\:\mathrm{got}\:\mathrm{something} \\ $$$$\mathrm{very}\:\mathrm{close}\:\mathrm{to}\:−\frac{\mathrm{e}}{\mathrm{2}}\:\mathrm{but}\:\mathrm{I}\:\mathrm{cannot}\:\mathrm{prove}\:\mathrm{the} \\…
Question Number 145951 by ArielVyny last updated on 09/Jul/21 $$\underset{{n}\geqslant\mathrm{1}} {\sum}\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}}=?? \\ $$ Answered by Olaf_Thorendsen last updated on 09/Jul/21 $$\frac{\mathrm{1}}{\mathrm{1}+{x}}\:=\:\mathrm{1}−{x}+{x}^{\mathrm{2}} −{x}^{\mathrm{3}} +…\:=\:\underset{{n}=\mathrm{0}} {\overset{\infty}…
Question Number 80343 by jagoll last updated on 02/Feb/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{ln}\left(\mathrm{tan}\:{x}+\mathrm{1}\right)−\mathrm{sin}\:{x}}{{x}\mathrm{sin}\:{x}} \\ $$ Commented by abdomathmax last updated on 02/Feb/20 $${we}\:{have}\:{ln}^{'} \left(\mathrm{1}+{u}\right)=\frac{\mathrm{1}}{\mathrm{1}+{u}}=\mathrm{1}−{u}\:+{o}\left({u}^{\mathrm{2}} \right)\left({u}\sim\mathrm{0}\right) \\ $$$$\Rightarrow{ln}\left(\mathrm{1}+{u}\right)={u}−\frac{{u}^{\mathrm{2}}…
Question Number 80296 by john santu last updated on 02/Feb/20 $${what}\:{is}\:{the}\:{value}\:{of}\: \\ $$$$\underset{{x}\rightarrow−\infty\:} {\mathrm{lim}}\:{e}^{\frac{\mathrm{6}{x}^{\mathrm{2}} +{x}}{\mathrm{3}{x}+\mathrm{5}}} \:? \\ $$$$\mathrm{0}\:{or}\:\infty\:? \\ $$ Commented by Tony Lin last…
Question Number 80276 by M±th+et£s last updated on 01/Feb/20 $$\underset{{x}\rightarrow\mathrm{0}} {{lim}}\:\frac{{e}^{{x}} −{e}^{−{x}} −\mathrm{2}{x}}{{x}−{sin}\left({x}\right)}={L}\:\:>\mathrm{0}\:,\:{L}\in{R} \\ $$$${find}\:{L} \\ $$$$ \\ $$$${with}\:{out}\:{using}\:{hopital}\:{and}\:{Taylor}\:{methods} \\ $$ Commented by mathmax by…
Question Number 145806 by bramlexs22 last updated on 08/Jul/21 $$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{x}\left(\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{x}+\mathrm{1}}{\mathrm{x}+\mathrm{4}}\right)−\frac{\pi}{\mathrm{4}}\right)\:=?\: \\ $$ Answered by gsk2684 last updated on 08/Jul/21 $$\underset{{x}\rightarrow\infty} {\mathrm{lim}}{x}\left(\mathrm{tan}^{−\mathrm{1}} \left(\frac{\frac{{x}+\mathrm{1}}{{x}+\mathrm{4}}−\mathrm{1}}{\mathrm{1}+\frac{{x}+\mathrm{1}}{{x}+\mathrm{4}}}\right)\right) \\…
Question Number 145791 by mathmax by abdo last updated on 08/Jul/21 $$\mathrm{find}\:\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{0}} \:\:\:\int_{\mathrm{0}} ^{\mathrm{x}} \:\frac{\mathrm{e}^{\mathrm{t}} +\mathrm{e}^{−\mathrm{t}} −\mathrm{2}}{\mathrm{1}−\mathrm{cosx}}\mathrm{dx} \\ $$ Commented by mathmax by abdo last…
Question Number 145756 by puissant last updated on 07/Jul/21 Answered by phanphuoc last updated on 07/Jul/21 $${u}_{{n}} ={ln}\left({n}+\mathrm{1}\right) \\ $$ Commented by puissant last updated…
Question Number 80220 by TawaTawa last updated on 01/Feb/20 Commented by kaivan.ahmadi last updated on 01/Feb/20 $$\forall\epsilon>\mathrm{0}\exists\delta>\mathrm{0}\:{s}.{t}\:\mid{x}−\mathrm{3}\mid<\delta\Rightarrow\mid{x}^{\mathrm{2}} −\mathrm{9}\mid<\epsilon \\ $$$$\mid{x}^{\mathrm{2}} −\mathrm{9}\mid=\mid{x}−\mathrm{3}\mid\mid{x}+\mathrm{3}\mid<\epsilon \\ $$$${if}\:\delta=\mathrm{1}\Rightarrow\mid{x}−\mathrm{3}\mid<\mathrm{1}\Rightarrow−\mathrm{1}<{x}−\mathrm{3}<\mathrm{1}\Rightarrow\mathrm{2}<{x}<\mathrm{4}\Rightarrow \\ $$$$\mathrm{5}<{x}+\mathrm{3}<\mathrm{7}\Rightarrow\mid{x}+\mathrm{3}\mid<\mathrm{7}…