Question Number 46422 by maxmathsup by imad last updated on 25/Oct/18
$${let}\:{f}_{{n}} \left({x}\right)=\left(\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }−{x}\right)^{{n}\:} \:\:\:{with}\:{n}\:{integr}\:{natural} \\ $$$$\left.\mathrm{1}\right)\:{find}\:\:\int\:\:{f}_{{n}} \left({x}\right){dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{f}_{{n}} ^{−\mathrm{1}} \left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{find}\:{lim}_{{x}\rightarrow+\infty} \:{f}_{{n}} \left({x}\right) \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\:{f}_{{n}} ^{'} \left({x}\right)\:\:{and}\:\:{f}_{{n}} ^{'} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{5}\right)\:{give}\:{the}\:{equation}\:{of}\:{tangent}\:{at}\:{point}\:{E}\left(\mathrm{0},{f}\left(\mathrm{0}\right)\right) \\ $$