Find the limit $ \lim_{n \to \infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)^n$ For $a,b,c>0$, Find $$ \lim_{n \to \infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)^n$$
how can I find the limit of sequence above? 
Provide me a hint or full solution.
thanks ^^

$$\begin{align}\lim_{n \to \infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)^{n} &=\lim_{r \to 0^+}\left(\frac{a^{r}+b^{r}+c^{r}}{3}\right)^{1/r}\\ &=\lim_{r \to 0^+}\frac{1}{r}\cdot \exp\left( [\log \left(a^{r}+b^{r}+c^{r}\right)]\right)
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&=\exp\left( [\lim_{r \to 0}\frac{d}{dr}\log(a^{r}+b^{r}+c^{r})]\right)
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&=\exp\left( \frac{[\log(a)+\log(b)+\log(c)]}{3}\right) 
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&= (abc)^{1/3}
\end{align}$$
