Question Number 160875 by cortano last updated on 08/Dec/21
$$\:\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\sqrt[{\mathrm{n}}]{\mathrm{5}^{\mathrm{n}} +\mathrm{7}^{\mathrm{n}} }\:=? \\ $$
Answered by MJS_new last updated on 08/Dec/21
$$\left(\mathrm{5}^{{n}} +\mathrm{7}^{{n}} \right)^{\mathrm{1}/{n}} =\left(\left(\frac{\mathrm{5}^{{n}} }{\mathrm{7}^{{n}} }+\mathrm{1}\right)\mathrm{7}^{{n}} \right)^{\mathrm{1}/{n}} =\mathrm{7}\left(\mathrm{1}+\left(\frac{\mathrm{5}}{\mathrm{7}}\right)^{{n}} \right)^{\mathrm{1}/{n}} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{\mathrm{5}}{\mathrm{7}}\right)^{\mathrm{n}} =\mathrm{0} \\ $$$$\Rightarrow \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{5}^{{n}} +\mathrm{7}^{{n}} \right)^{\mathrm{1}/{n}} \:=\mathrm{7} \\ $$