Question Number 146635 by mathdanisur last updated on 14/Jul/21
$$\underset{{x}\rightarrow\infty} {{lim}}\left(\frac{\mathrm{2}{x}\:+\:\mathrm{1}}{{x}\:+\:\mathrm{1}}\right)^{\frac{\mathrm{1}}{\boldsymbol{{x}}}} =\:? \\ $$
Answered by Olaf_Thorendsen last updated on 14/Jul/21
$$\frac{\mathrm{1}}{{x}}\mathrm{ln}\left(\frac{\mathrm{2}{x}+\mathrm{1}}{{x}+\mathrm{1}}\right)\:=\:\frac{\mathrm{1}}{{x}}\left[\mathrm{ln}\left(\mathrm{2}+{x}\right)−\mathrm{ln}\left({x}+\mathrm{1}\right)\right] \\ $$$$=\:\frac{\mathrm{1}}{{x}}\left[\mathrm{ln2}+\mathrm{ln}{x}+\mathrm{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}{x}}\right)−\mathrm{ln}{x}−\mathrm{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\right)\right] \\ $$$$\sim\:\frac{\mathrm{1}}{{x}}\left[\mathrm{ln2}+\frac{\mathrm{1}}{\mathrm{2}{x}}−\frac{\mathrm{1}}{{x}}\right]\:\underset{\infty} {\rightarrow}\:\mathrm{0} \\ $$$$\Rightarrow\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\mathrm{2}{x}+\mathrm{1}}{{x}+\mathrm{1}}\right)^{\frac{\mathrm{1}}{{x}}} \:=\:{e}^{\mathrm{0}} \:=\:\mathrm{1} \\ $$
Commented by mathdanisur last updated on 14/Jul/21
$${cool}\:{thank}\:{you}\:{Ser} \\ $$