$\lim _{x \rightarrow \infty} x\left(a^{1 / x}+b^{1 / x}+c^{1 / x}-3 k^{1 / x}\right)=0$
$\begin{aligned} & \text { If } y=\frac{1}{x} \Rightarrow y \rightarrow 0 \\ & \Rightarrow \lim _{y \rightarrow 0}\left(\frac{a^y+b^y+c^y-3 k^y}{y}\right)=0 \\ & \Rightarrow \lim _{y \rightarrow 0}\left[\left(\frac{a^y-1}{y}\right)+\left(\frac{b^y-1}{y}\right)+\left(\frac{c^y-1}{y}\right)+\left(\frac{3-3 k^y}{y}\right)\right]=0 \\ & \Rightarrow \log a+\log b+\log c+\lim _{y \rightarrow 0} \frac{-3\left(k^y-1\right)}{y}=0 \\ & \Rightarrow \quad \log (a b c)=3 \cdot \lim _{y \rightarrow 0} \frac{k^y-1}{y} \\ & \Rightarrow \frac{1}{3} \log (a b c)=\lim _{y \rightarrow 0} \frac{k^y-1}{y} \\ & \Rightarrow \log (a b c)^{1 / 3}=\log k \\ & \therefore \quad k=(a b c)^{1 / 3} . \\ & \end{aligned}$