Question Number 167085 by mathlove last updated on 06/Mar/22
$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\sqrt{{x}^{\mathrm{2}} +\mathrm{3}{x}}−\sqrt{{x}^{\mathrm{2}} +{x}}\right)^{{x}} =? \\ $$
Commented by cortano1 last updated on 06/Mar/22
$$\:\mathrm{L}=\mathrm{e}^{\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{3x}}−\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}}−\mathrm{1}\right).\mathrm{x}} \\ $$$$\mathrm{L}=\mathrm{e}^{\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\left(\mathrm{x}+\frac{\mathrm{3}}{\mathrm{2}}\right)−\left(\mathrm{x}+\frac{\mathrm{1}}{\mathrm{2}}\right)−\mathrm{1}\right).\mathrm{x}} \\ $$$$\mathrm{L}=\mathrm{e}^{\mathrm{0}} =\:\mathrm{1} \\ $$