If a,b,c and k are non-zero real numbers and $\underset{\mathit{x}\mathit{\to }\mathit{\infty }}{\mathit{l}\mathit{i}\mathit{m}}x\left(a^{1/x}+b^{1/x}+c^{1/x}-3k^{1/x}\right)\mathit{=}\mathit{0}$ then $k=$

$\underset{\mathit{x}\mathit{\to }\mathit{\infty }}{\mathit{l}\mathit{i}\mathit{m}}x\left(a^{1/x}+b^{1/x}+c^{1/x}-3k^{1/x}\right)\mathit{=}\mathit{0}$

$\underset{x\to \infty }{Lim}\left(\frac{a^{{\displaystyle \frac{1}{x}}}+b^{{\displaystyle \frac{1}{x}}}+c^{{\displaystyle \frac{1}{x}}}-3k^{{\displaystyle \frac{1}{x}}}}{{\displaystyle \frac{1}{x}}}\right)=0$

Put $\frac{1}{x}=y$

$\underset{y\to 0}{Lim}\left(\frac{a^y-1+b^{{\displaystyle y}}-1+c^{{\displaystyle y}}-1-3k^{{\displaystyle y}}+3}{{\displaystyle y}}\right)=0$

$\underset{y\to 0}{Lim}\left(\frac{a^y-1}{{\displaystyle y}}+\frac{b^y-1}{y}+\frac{c^y-1}{y}-3\left(\frac{k^y-1}{y}\right)\right)=0$

$\log a+\log b+\log c-3\log k=0$

k=$(abc{)}^{1/3}$