How do I evaluate $\lim_{x\to 0}\left( \frac{1^x+2^x+3^x+\cdots+ n^x}{n}\right)^{a/x} $? $$\lim_{x\to 0}\left( \frac{1^x+2^x+3^x+\cdots+ n^x} n \right)^{a/x} $$ 

Hint: Apply $\ln$ to get
$$a\cdot\frac{\ln (1^x + \cdots +n^x) - \ln n}{x}.$$
Let $f(x) = \ln (1^x + \cdots +n^x).$ Then the above equals
$$a\cdot\frac{f(x) - f(0)}{x-0},$$
which hopefully looks familiar.
