What is the limit of $ f(x) = \sqrt[n]{(x+a\_1)(x+a\_2)...(x+a\_n)} - x$ when $x$ goes to plus infinity? The number $n$ is fixed and $a\_i$ are some constants. 

All I know is the answer:
$(a\_1 + a\_2 + ... + a\_n) / n$

I figured out that I can limit the root from above with the arithmetic mean. And this gives me some nice expression as an upper limit. But OK... this only proves the limit is $\leq (a\_1 + a\_2 + ... + a\_n) / n$.

But how do I limit it from below with the same value. I tried using the harmonic mean but it seems it does not lead me to anything nice. So... any ideas?