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Let $a_1,a_2,\ldots,a_n$ be positive real numbers.Prove that

$$\lim_{x\to 0}\bigg(\dfrac{a_1^x+a_2^x+\cdots+a_n^x}{n}\bigg)^{\frac{1}{x}}=\sqrt[n]{a_1a_2\cdots a_n}$$
Got no clue where to begin from though looks like somewhat AM-GM property! Please help!

Hawk
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$$1\xleftarrow[x\to 0]{}\frac{n\max\{a_1^x,...,a_n^x\}}n\ge\frac{a_1^x+\ldots +a_n^x}n\ge\sqrt[n] {a_1^x\cdot\ldots\cdot a_n^x}=\left(\sqrt[n]{a_1\cdot\ldots\cdot a_n}\right)^x\xrightarrow[x\to 0]{}1$$

I get something different...

DonAntonio
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