<s>[INST] Find $\lim\limits_{x\to 0}\left (\frac{1^x+2^x+3^x+\dots+n^x}{n} \right)^{\frac1x}$ Consider the following expression.
$$\lim\limits_{x\to 0} \left (\frac{1^x+2^x+3^x+\dots+n^x}{n} \right )^{\frac 1 x}$$
How to solve this?
Let $y= \left (\frac {1^x+2^x+\cdots +n^x} {n} \right)^{1/x}$
I tried taking  $\ln$ on both sides. We get that $$\ln(y)=\frac{1}{x}\ln \left (\frac {1^x+2^x+\cdots +n^x} {n} \right ).$$
Taking $\lim$ on both sides we get $$\ln(y)=\lim_{x\to 0}\frac{1}{x}\ln \left (\frac {1^x+2^x+\cdots +n^x} {n} \right ).$$
Now applying the LH rule, we get $$\ln(y)=\lim_{x\to 0}\frac{n}{1^x+2^x+\cdots +n^x}({1^x\ln(1)+\cdots +n^x\ln(n)})$$
Is this a right way to go?
 [/INST] I write this answer only because the wrong one is accepted.
There is an error in your final expression. It should be:
$$\lim_{x\to 0}\frac{n}{1^x+2^x+\cdots +n^x}\frac{1^x\ln(1)+\cdots +n^x\ln(n)}{\color{red}n}=\frac{\ln n!}{n}.$$
Correspondingly:
$$
\lim\limits_{x\to 0} \left (\frac{1^x+2^x+3^x+\dots+n^x}{n} \right )^{\frac 1 x}=\sqrt[n]{n!}.
$$
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