Question Number 161533 by mathlove last updated on 19/Dec/21
Commented by cortano last updated on 19/Dec/21
$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\sqrt{\mathrm{1}+{x}^{\mathrm{2}} +{x}}\right)^{\mathrm{2020}} −\left(\sqrt{\mathrm{1}+{x}^{\mathrm{2}} −{x}}\right)^{\mathrm{2020}} }{{x}} \\ $$$$\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{1}+\mathrm{1010}\left({x}^{\mathrm{2}} +{x}\right)\right)−\left(\mathrm{1}+\mathrm{1010}\left({x}^{\mathrm{2}} −{x}\right)\right)}{{x}} \\ $$$$\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2020}{x}}{{x}}\:=\:\mathrm{2020} \\ $$
Answered by mathmax by abdo last updated on 19/Dec/21
$$\mathrm{f}\left(\mathrm{x}\right)=\frac{\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} +\mathrm{x}\right)^{\mathrm{1010}} −\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} −\mathrm{x}\right)^{\mathrm{1010}} }{\mathrm{x}}\:\Rightarrow \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\sim\frac{\mathrm{1}+\mathrm{1010}\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}\right)−\left(\mathrm{1}+\mathrm{1010}\left(\mathrm{x}^{\mathrm{2}} −\mathrm{x}\right)\right)}{\mathrm{x}} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\sim\frac{\mathrm{1010x}^{\mathrm{2}} +\mathrm{1010x}−\mathrm{1010x}^{\mathrm{2}} +\mathrm{1010x}}{\mathrm{x}}\:\Rightarrow \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\sim\frac{\mathrm{2020x}}{\mathrm{x}}\:\Rightarrow\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{0}} \mathrm{f}\left(\mathrm{x}\right)=\mathrm{2020} \\ $$