Question Number 59187 by maxmathsup by imad last updated on 05/May/19 $${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\frac{{ln}\left({arctan}\left(\mathrm{1}+{x}\right)\right)−{ln}\left(\frac{\pi}{\mathrm{4}}\right)}{{x}^{\mathrm{2}} } \\ $$ Commented by kaivan.ahmadi last updated on 05/May/19 $${hop} \\…
Question Number 59182 by maxmathsup by imad last updated on 05/May/19 $${let}\:{f}\left({x}\right)\:={arctan}\left(\mathrm{1}+{ix}\right) \\ $$$${determine}\:{Re}\left({f}\left({x}\right)\right)\:{and}\:{Im}\left({f}\left({x}\right)\right){dx} \\ $$ Commented by maxmathsup by imad last updated on 10/Jun/19…
Question Number 124580 by Boucatchou last updated on 04/Dec/20 $$ \\ $$$$\:\:\:\:\:\:\:\:{Let}\:\:{f}\left({x}\right)={x}^{\mathrm{3}} +\mathrm{2}{x}−\mathrm{6},\:\:\:\:{f}^{−\mathrm{1}} \left({x}\right)=? \\ $$ Answered by MJS_new last updated on 04/Dec/20 $${x}={y}^{\mathrm{3}} +\mathrm{2}{y}−\mathrm{6}…
Question Number 124444 by benjo_mathlover last updated on 03/Dec/20 $$\:{If}\:{f}\left(\frac{\mathrm{2}}{{x}}\right)\:=\:\frac{\mathrm{3}}{\mathrm{2}+\mathrm{4}{x}}\:{and}\:{f}^{−\mathrm{1}} \left({p}\right)=\mathrm{2}\: \\ $$$${then}\:{p}\:=? \\ $$ Answered by bemath last updated on 03/Dec/20 $$\:\Leftrightarrow\:{f}^{−\mathrm{1}} \left({p}\right)=\mathrm{2}\:{then}\:{f}\left(\mathrm{2}\right)={p} \\…
Question Number 58753 by maxmathsup by imad last updated on 29/Apr/19 $${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\frac{\mathrm{1}−{cos}\left({x}\right){cos}\left({x}^{\mathrm{2}} \right)….{cos}\left({x}^{{n}} \right)}{{x}^{{n}} }\:\:\:{with}\:{n}\:{natural}\:{integr}\:\geqslant\mathrm{2} \\ $$ Commented by tanmay last updated on 30/Apr/19…
Question Number 124102 by mnjuly1970 last updated on 30/Nov/20 $$\:\:\:\:\:\:\:\:\:\:\:…{advanced}\:\:\:{calculus}… \\ $$$$\:\:{p}\:,\:{q}\:{are}\:{positive}\:{integers}\:{and} \\ $$$${p}\geqslant{q}\:\::\:{let}\::\phi\left({p},{q}\right)=\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{{p}} \left\{\frac{\mathrm{1}}{{x}}\right\}^{{q}} {dx} \\ $$$$\:\:{prove}\:{that}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\phi\left({n},{n}\right)\overset{?} {=}\mathrm{1}−\frac{\mathrm{1}}{{n}+\mathrm{1}}\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\zeta\left({k}+\mathrm{1}\right)…
Question Number 58530 by maxmathsup by imad last updated on 24/Apr/19 $${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\frac{\mathrm{1}−{cos}\left(\mathrm{2}{x}\right)\:{cos}\left(\mathrm{3}{x}^{\mathrm{3}} \right)}{{x}^{\mathrm{2}} } \\ $$ Answered by tanmay last updated on 24/Apr/19 $${cos}\mathrm{2}{x}=\mathrm{1}−\frac{\left(\mathrm{2}{x}\right)^{\mathrm{2}}…
Question Number 124066 by Bird last updated on 30/Nov/20 $${let}\:{f}\left({x}\right)={arctan}\left(\frac{\mathrm{2}}{{x}}\right) \\ $$$${calculate}\:{f}^{\left({n}\right)} \left({x}\right)\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{1}\right) \\ $$$${find}\:{f}^{\left(\mathrm{7}\right)} \left(\frac{\mathrm{1}}{\mathrm{7}}\right) \\ $$ Answered by Olaf last updated on…
Question Number 124065 by Bird last updated on 30/Nov/20 $${determine}\:{tbe}\:{sewuence}\:{u}_{{n}} \\ $$$${wich}\:{verify}\:{u}_{{n}} +{u}_{{n}+\mathrm{1}} =\frac{\left(−\mathrm{1}\right)^{{n}} }{\:\sqrt{{n}}} \\ $$$${n}>\mathrm{0} \\ $$ Answered by mindispower last updated on…
Question Number 58354 by maxmathsup by imad last updated on 21/Apr/19 $${let}\:{U}_{{n}} =\frac{\mathrm{1}^{\mathrm{2}} \:+\mathrm{2}^{\mathrm{2}} \:+\mathrm{3}^{\mathrm{2}} \:+….+{n}^{\mathrm{2}} }{\mathrm{1}^{\mathrm{4}} \:+\mathrm{2}^{\mathrm{4}} \:+\mathrm{3}^{\mathrm{4}} \:+….+{n}^{\mathrm{4}} } \\ $$$$\left.\mathrm{1}\right){find}\:{lim}_{{n}\rightarrow+\infty} {U}_{{n}} \\…